Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Abtahi, Seyed Arman
Doctor of Philosophy - PhD
University of British Columbia
Both biological swimming microorganisms and artificial active particles capable of propulsion have recently been active areas of research in fluid dynamics. The influence of nearby boundaries or other active particles or swimmers is critical in many important cases for understanding various phenomena. Bacteria, for example, accumulate near surfaces and interact closely to form biofilms. Accordingly, this dissertation addresses a few gaps in the literature of microorganism hydrodynamics. First, we focus on the suspension of active particles near solid boundaries. We start with the study of the hydrodynamics of two unequal swimmers at close proximity by considering the squirming model. In comparison to passive particles, microorganisms have surface activity that provides the impetus for swimmers to move. In order to satisfy both the boundary conditions on the swimmers' skin and the Stokes equations, a regular perturbation in the velocity and pressure field of the inner creeping flow between the swimmers is employed. A relatively large size ratio leads to the hydrodynamics of a squirmer near a flat surface. Afterwards, we calculate the far-field hydrodynamics of a suspension of squirmers near a solid boundary. In the course of calculations, an extension of Stokesian Dynamics (SD) method, developed for passive particles, is utilized for active particles. This technique, called active SD, offers a good starting point for further research questions in the field. The transition between near-field and far-field solutions is also discussed. Thereafter, we propose a rigid and a flexible toy model to examine the effect of elasticity on the dynamics of a swimmer. Eventually, the developed method can help to establish a relation between hydrodynamics and biological behaviours, e.g., biofilm formation. Ultimately, we focus on axisymmetric particles in non-Newtonian fluids as a first step toward extending the SD method for poly disperse nonspherical active particles in complex fluids. We investigate the dynamics of a prolate spheroid in a shear flow of a shear-thinning Carreau fluid. The motion of a prolate particle is developed analytically for asymptotically weak shear thinning and then integrated numerically.
Hydrodynamic interactions:microswimmers near boundariesbySeyed Arman AbtahiB.Sc., Isfahan University of Technology, 2014M.Sc., Western University, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2022© Seyed Arman Abtahi 2022The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:submitted by in partial fulfillment of the requirements forthe degree ofinExamining Committee:Additional Supervisory Committee Members:iiHydrodynamic interactions: microswimmers near boundariesSeyed Arman AbtahiDoctor of PhilosophyMechanical EngineeringGwynn J. Elfring, Associate Professor, Mechanical Engineering, UBCSupervisorNeil J. Balmforth, Professor, Mathematics, UBCSupervisory Committee Member Brian Wetton, Professor, Mathematics, UBCUniversity ExaminerMiranda Holmes-Cerfon, Associate Professor, Mathematics, UBCUniversity ExaminerHenry Fu, Associate Professor, Mechanical Engineering, The University of UtahExternal ExaminerCarl Ollivier-Gooch, Professor, Mechanical Engineering, UBCSupervisory Committee MemberSupervisory Committee MemberKendal Bushe, Associate Professor, Mechanical Engineering, UBCAbstractBoth biological swimming microorganisms and artificial active particles capableof propulsion have recently been active areas of research in fluid dynamics. Theinfluence of nearby boundaries or other active particles or swimmers is critical inmany important cases for understanding various phenomena. Bacteria, for exam-ple, accumulate near surfaces and interact closely to form biofilms. Accordingly,this dissertation addresses a few gaps in the literature of microorganism hydrody-namics. First, we focus on the suspension of active particles near solid boundaries.We start with the study of the hydrodynamics of two unequal swimmers at closeproximity by considering the squirming model. In comparison to passive particles,microorganisms have surface activity that provides the impetus for swimmers tomove. In order to satisfy both the boundary conditions on the swimmers’ skin andthe Stokes equations, a regular perturbation in the velocity and pressure field ofthe inner creeping flow between the swimmers is employed. A relatively large sizeratio leads to the hydrodynamics of a squirmer near a flat surface. Afterwards,we calculate the far-field hydrodynamics of a suspension of squirmers near a solidboundary. In the course of calculations, an extension of Stokesian Dynamics (SD)method, developed for passive particles, is utilized for active particles. This tech-nique, called active SD, offers a good starting point for further research questionsin the field. The transition between near-field and far-field solutions is also dis-cussed. Thereafter, we propose a rigid and a flexible toy model to examine the ef-fect of elasticity on the dynamics of a swimmer. Eventually, the developed methodcan help to establish a relation between hydrodynamics and biological behaviours,e.g., biofilm formation.Ultimately, we focus on axisymmetric particles in non-Newtonian fluids as afirst step toward extending the SD method for poly disperse nonspherical activeparticles in complex fluids. We investigate the dynamics of a prolate spheroid ina shear flow of a shear-thinning Carreau fluid. The motion of a prolate particle isdeveloped analytically for asymptotically weak shear thinning and then integratednumerically.iiiLay SummaryThis dissertation investigates the dynamics and hydrodynamic interactions of swim-ming microorganisms near solid boundaries. First, we focus on two active particlesinteracting at close distances, and then on the suspension of active particles nearsolid boundaries. Then, we describe how flexible swimmers deform as a result ofhydrodynamic interactions with a wall. Finally, as a step towards more compli-cated dynamics of nonspherical active particles in complex fluids, we investigatethe dynamics of a passive prolate spheroid in shear flow of a shear-thinning Carreaufluid.ivPrefaceThe contents of this dissertation are the results of the research of the author, S.Arman Abtahi, under the supervision of Professor Gwynn J. Elfring. The followingarticles has been published and/or are in progress:• S. A. Abtahi and G. J. Elfring. Jeffery orbits in shear-thinning fluids. Phys.Fluids, 31(10):103106, 2019The author of this dissertation was the principal contributor. G. J. Elfringsupervised the research and was involved in the analysis and editing of thepaper.• S. A. Abtahi and G. J. Elfring. Active particles near a wall. to be submitted,2022The author of this dissertation was the principal contributor. G. J. Elfringsupervised the research and was involved in the analysis and editing of thepaper.• S. A. Abtahi and G. J. Elfring. Confined flexible swimmers. to be submitted,2022The author of this dissertation was the principal contributor. G. J. Elfringsupervised the research and was involved in the analysis and editing of thepaper.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Near-field hydrodynamics of swimmers . . . . . . . . . . . . . . . . 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Axisymmetric problems . . . . . . . . . . . . . . . . . . . . . . 152.4 Asymmetric problems . . . . . . . . . . . . . . . . . . . . . . . 152.5 Interactive moments . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Squirming near a wall . . . . . . . . . . . . . . . . . . . . . . . 202.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Active particles near a wall . . . . . . . . . . . . . . . . . . . . . . . 253.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Active Stokesian Dynamics . . . . . . . . . . . . . . . . . . . . 263.3 Far-field hydrodynamics . . . . . . . . . . . . . . . . . . . . . . 293.3.1 Unbounded fluid . . . . . . . . . . . . . . . . . . . . . . 30viTable of Contents3.3.2 Bounded fluid . . . . . . . . . . . . . . . . . . . . . . . 363.4 Near-field hydrodynamics . . . . . . . . . . . . . . . . . . . . . 383.4.1 Pairwise particle–particle interaction . . . . . . . . . . . 383.4.2 Pairwise particle–wall interaction . . . . . . . . . . . . . 493.5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.6 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.7 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . 573.8 Contact models . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.8.1 Adhesive contact model . . . . . . . . . . . . . . . . . . 703.8.2 Rough contact model . . . . . . . . . . . . . . . . . . . 723.8.3 Smooth contact model . . . . . . . . . . . . . . . . . . . 733.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 A linked-swimmer near a wall . . . . . . . . . . . . . . . . . . . . . 794.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Dynamics of a linked swimmer . . . . . . . . . . . . . . . . . . 804.3 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . 834.3.1 First squirming mode Bα1 . . . . . . . . . . . . . . . . . . 844.3.2 Second squirming mode Bα2 . . . . . . . . . . . . . . . . 864.3.3 First azimuthal squirming mode Cα1 . . . . . . . . . . . . 894.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975 An elastic swimmer near a wall (Active Stokesian Dynamics method) 1005.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Dynamics of a elastic swimmer . . . . . . . . . . . . . . . . . . 1015.3 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.1 First squirming mode Bα1 . . . . . . . . . . . . . . . . . . 1045.3.2 Second squirming mode Bα2 . . . . . . . . . . . . . . . . 1095.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146 An elastic swimmer near a wall (far-field approach) . . . . . . . . . 1156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . 1166.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2.2 Background velocity . . . . . . . . . . . . . . . . . . . . 1176.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.4 Swimming in an unbounded fluid . . . . . . . . . . . . . . . . . 1326.5 An alternative formulation . . . . . . . . . . . . . . . . . . . . . 1326.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137viiTable of Contents7 Jeffery orbits in shear-thinning fluids . . . . . . . . . . . . . . . . . 1397.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . 1427.2.1 A spheroid in shear . . . . . . . . . . . . . . . . . . . . 1427.2.2 Shear-thinning fluids . . . . . . . . . . . . . . . . . . . . 1437.2.3 Motion of a particle in a complex fluid . . . . . . . . . . 1447.2.4 Asymptotic solution . . . . . . . . . . . . . . . . . . . . 1457.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.3.1 Jeffery Orbits . . . . . . . . . . . . . . . . . . . . . . . . 1467.3.2 Dynamics of a prolate spheroid in shear flow of a shear-thinning fluid . . . . . . . . . . . . . . . . . . . . . . . . 1487.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518 Machine learning for the interactions of two squirmers . . . . . . . 1539 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 155Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160AppendicesA Near-field hydrodynamics of swimmers . . . . . . . . . . . . . . . . 178A.1 Slip velocity break down . . . . . . . . . . . . . . . . . . . . . . 178A.2 Axisymmetric problems . . . . . . . . . . . . . . . . . . . . . . 180A.3 Asymmetric problems . . . . . . . . . . . . . . . . . . . . . . . 182A.4 Far-field swim force near a surface . . . . . . . . . . . . . . . . . 184B Active particles near a wall . . . . . . . . . . . . . . . . . . . . . . . 187B.1 Details of far-field hydrodynamics . . . . . . . . . . . . . . . . . 187B.2 Details of near-field hydrodynamics . . . . . . . . . . . . . . . . 199B.3 Tensorial notation . . . . . . . . . . . . . . . . . . . . . . . . . 204C Jeffery orbits in shear-thinning fluids . . . . . . . . . . . . . . . . . 209C.1 Spheroidal multipoles . . . . . . . . . . . . . . . . . . . . . . . 209C.2 A prolate spheroid in Stokes flow . . . . . . . . . . . . . . . . . 214C.2.1 Jeffery Orbits . . . . . . . . . . . . . . . . . . . . . . . . 215C.3 Shear-thinning correction . . . . . . . . . . . . . . . . . . . . . 215viiiList of Tables2.1 Breaking the slip velocity on the surface of sphere α into seven cases. 132.2 The right hand side term in eq.2.6 and the particular solutions of pfor different problems. . . . . . . . . . . . . . . . . . . . . . . . 163.1 Order unity corrections of scalar functions for two nearly touchingequal spheres, λ = 1. . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Order unity corrections of scalar functions for a sphere near a wallλ→ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51ixList of Figures1.1 The asymmetric flexural motion exhibited by a single Volvox cil-ium. Reprinted from [182] . . . . . . . . . . . . . . . . . . . . . 11.2 The drag anisotropy for a slender filament leads to a net motion.Reprinted from [129] . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Axisymmetric surface velocity profile vs and the analytical flowfields for (a) a pusher (b) a neutral squirmer, and (c) a puller in thelab frame. Reprinted from [127, 224] . . . . . . . . . . . . . . . . 32.1 Squirmers α and β placed in creeping flow of a Newtonian fluid. . 122.2 Sub-problems with boundary conditions summarized in table 2.1. . 142.3 Force and torque acting on the active swimmer of problem 4. . . . 182.4 a) Swim force and velocity of two identical converging squirm-ers. b) Swim torque and angular velocity of two identical passingsquirmers. The angular velocity reorients the swimmers away. Re-sults are for B1 = 1. Thick (thin) lines are corresponded to force(velocity) moments. Dashed-lines and dotted-dashed-lines showfar-field and lubrication solutions, respectively, while solid linesshow the transition between two solutions. . . . . . . . . . . . . . 212.5 a) Trajectories for a neutral squirmer B2/B1 = 0 (gray head), apuller B2/B1 = 20 (red head), and a pusher B2/B1 = −20 (bluehead), near an obstacle with β = 20. b) The inner region flow fieldfor different head orientations. The green arrows show transna-tional and angular velocities of the squirmers. Note that the size ofarrows do not illustrate the velocity amplitude. . . . . . . . . . . . 212.6 Velocity and force moments of a swimmer swimming a) towardsa wall and b) parallel to a wall. The angular velocity reorients theswimmer away from the wall. Results are for B1 = 1 and sameconditions as in fig.2.4. . . . . . . . . . . . . . . . . . . . . . . . 23xList of Figures2.7 Trajectories of pushers (blue head), pullers (red head), and neutral(gray head) squirmers near a wall for different initial orientations χ.Narrow lines throughout the trajectories show the direction of theswimmers. Results are for initial height of 5 and B2/B1 = −20, 20, 0 243.1 Active particles above a stationary wall. . . . . . . . . . . . . . . 363.2 a) The order O(1) correction for XααFU plotted against 1/M as M →∞ for λ = 1 demonstrated by dot marks for up to M = 300 anddashed-dotted line for up to M = 500. b) A closer look at M−1 →0. Solid, dashed, and dotted lines are MR, DS, and linear fits to theseries data for M = 300, respectively. . . . . . . . . . . . . . . . 453.3 Coefficients Cm, Domb–Sykes rDSm , and Mercer-Roberts rMRm plotsfor the series in the scalar function XααFU as a function of M−1. Thesolid lines in (b) and (c) are linear asymptotes, which intercepts thevertical axis at 1/r and has a slope of a. . . . . . . . . . . . . . . 473.4 TheO(1) contributions for different scalar functions plotted against1/M as M → ∞ for λ = 1 demonstrated by dot marks. Solid,dashed, and dotted lines are MR, DS, and linear fits to the seriesdata, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5 TheO(1) contributions for different scalar functions plotted against1/M as M → ∞ for λ = 0.1, 0.01, 0.0067, 0.005 showed by dots,stars, circles, and triangles, respectively. Dotted lines are linear fitsto the series data. . . . . . . . . . . . . . . . . . . . . . . . . . . 503.6 Values of scalar functions as a function of spacing ϵ, for two iden-tical spheres λ = 1 shown by dots, two spheres with λ = 0.1 shownby stars, and for a sphere near a wall λ = 0.005 shown by circles.Asymptotes for λ = 1, 0.005 are shown with solid lines that areaccurate for ϵ < 10−2 . . . . . . . . . . . . . . . . . . . . . . . . 523.7 a) Trajectories of the sedimentation of three passive spheres withinitial positions of x⊥ = 0, and x∥ = −5, 0, 7. Red dots are the fullF-T-S solution of Durlofsky et al. [71]. b) The rotation rate of apair of spheres above a surface. . . . . . . . . . . . . . . . . . . . 56xiList of Figures3.8 a) Trajectories of non-interacting shakers near a wall with initialposition of h = 5 and different initial orientations θi(p) for a pusher-shaker B2 = −1 shown by blue heads, and a puller-shaker B2 = 1shown by red heads. Points on the trajectories show increments oft = 100 as well as the swimmer’s orientation. b) (h, θ(p)) - phaseportraits of a shaker. c) (h,U⊥)-phase portraits of shakers withinitial orientations of θi(p) = {0, π}. Dashed lines correspond toasymptotes, while dotted-lines are merely far-field solutions. Tran-sition length is showed between two dashed lines corresponding tothe change of far-field and near-field solutions. . . . . . . . . . . . 593.9 Trajectories of non-interacting swimmers near a wall (located atx⊥ = 0) at initial position of h = 5 for different initial orientationsp. Results are for B1 = 0.1, B2 = ±1,±10, and thin dashed lines arefor the case of a neutral swimmer B2 = 0. Points on the trajectoriesshow increments of t = 30 as well as the swimmer’s orientation. . 603.10 Phase portrait of trajectories in (θ(p), h) plane for squirmers withdifferent β. Thick blue lines show pushers and thin red lines showpullers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.11 Contour plot of hc as a function of B1 and B2 for a pusher orientedparallel to the wall, p = −e⊥. . . . . . . . . . . . . . . . . . . . . 623.12 a) Trajectories of a buoyant non-motile particle when a pusher B2 =−1, a neutral squirmer B2 = 0, and a puller B2 = 1, passes by. b) θ-phase portrait (on the left axis) and the variation of r as a functionof θ (on the right axis) for the non-motile particle. Results are forB1 = 0.1, λ = 2, and the total non-dimensional time is t = 150.Points on the trajectories show increments of t/5 and the particle’sorientation. Transition length is showed between two dashed linescorresponding to the change of far-field and near-field solutions. . 633.13 Trajectories of different types of swimmers, when they are passingby each other. Results are for B1 = 0.1, B2 = −1, 0, 1 and the pointson the trajectories show increments of t = 30 and the swimmer’direction. Swimmers are initially at r∥ = 2, and r⊥ = 10. . . . . . 643.14 Trajectories of two pushers passing by each other for different ini-tial positions r∥. Results are for B1 = 0.1, B2 = −1, and the initialdistance is r⊥ = 10, and r∥ = {5, 4, 3, 2, 1, 0.2, 0} for figs. a to g,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.15 a) Variation of r and b) θ(p), θ(er) as a function of t, c) U and r-phase portrait, and d) θ(er)-phase portrait for two pushers shown infig.3.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66xiiList of Figures3.16 Trajectories of different types of swimmers when they are passingby each other in vicinity of a wall with normal e⊙. Results are forB1 = 0.1, B2 = −1, 0, 1 and the points on the trajectories showincrements of t = 30 and the swimmers’ direction. Swimmers areinitially at hα = hβ = 5, r∥ = 2, and r⊥ = 10. . . . . . . . . . . . . 673.17 Trajectories of two pushers passing by each other in vicinity ofa wall for different initial positions. Results are for B1 = 0.1,B2 = −1 and the initial distances are h = 5, r⊥ = 10, and r∥ ={5, 4, 3, 2, 1, 0.2, 0} for figs. a to g, respectively. The points on thetrajectories show increments of t = 30 and the swimmers’ direc-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.18 Trajectories of different types of swimmers passing by each otherin vicinity of a wall with normal e∥. Results are for B1 = 0.1,B2 = −1, 0, 1 and the points on the trajectories show incrementsof t = 30 and the swimmers’ direction. Swimmers are initially athα = 5, hβ = 7, r∥ = 2, and r⊥ = 10. . . . . . . . . . . . . . . . . 683.19 a) Variation of r and h as a function of time t, b) Variation of ori-entation θ(p) and the direction of movement θ(U) as a function ofthe path taken l, and c) θ(p)-phase portrait of two pushers in fig.3.18a. The tick lines correspond to the swimmer initially closer tothe wall α, while the thin lines correspond to the other swimmer β. 693.20 Trajectories of two pushers passing by each other in vicinity ofa wall for different initial positions. Results are for B1 = 0.1,B2 = −1, and the initial distances are hα = 5, r⊥ = 10, andr∥ = {5, 3, 2, 1, 0.2, 0} for figs. a to f, respectively. The pointson the trajectories show increments of t = 30 and the swimmmers’direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.22 Spheres in a medium in different possible positions. . . . . . . . . 713.23 Comparison between three contact models of adhesive, rough, andsmooth, for a) passive particles and b) active particles. Particles areinitially located at x1 = −5e∥ + 0.1e⊥ and x2 = 5e∥ − 0.1e⊥. In a)external forces on the passive particles are Fext1 = e∥,Fext2 = −e∥,while in b) the first squirming mode is B1 = 1. Points on thetrajectories show increments of t/5 and the particle’s orientation. . 743.24 Trajectories of two pushers passing by each other near a wall witha) rough and b) smooth contact models implemented. Results arefor B1 = 0.1, B2 = 1. The points on the trajectories show incre-ments of t = 30 and the swimmmers’ direction . . . . . . . . . . . 76xiiiList of Figures3.25 a) Variation of r and h as a function of time t, and b) θ(p)-phaseportrait of two pushers for the conditions in fig. 3.24. Tick linescorrespond to the swimmer α, while thin dotted lines correspondto the swimmer β. Black lines correspond to the results beforethe swimmers’ initial touch. Red and blue lines correspond to theresults with rough, and smooth contact models, respectively. . . . 763.26 Coordinated migration of NC and placodal cells. a) Collisions be-tween NC cells (green) and placodal cells (red) in control condi-tions. b) NC cells are attracted to placodal cells owing to chemo-taxis. Then, contact between NC and placodal cells and promotingdirectional movement. NC cells are in green; placodal cells are inred. Reprinted from [204]. . . . . . . . . . . . . . . . . . . . . . 774.1 Schematic of a linked swimmer. Sphere α has surface activity us. . 814.2 Trajectories of a swimmer for different λ near a wall. Particle α isa neutral squirmer with first squirming modes of Bα1 = {−1, 1}. Re-sults are for L = hi = 2d¯, θi = 0 and the points on the trajectoriesshow increments of τ/5 and the direction of the swimmer p. Tra-jectories are plotted for a total time, τ, required for the swimmer toreach |x∥| = 30d¯ unless it hits the wall earlier. . . . . . . . . . . . 854.3 a) Trajectories of a swimmer for different link’s lengths L and λnear a wall. Results are for Bα1 = {−1, 1}, L/d¯ = {1.01, 2, 10},hi = 3d¯, θi = 0, and the points on the trajectories show incrementsof τ/5 and the direction of the swimmer p. Trajectories are plottedfor a total time, τ, required for the swimmer to reach |x∥| = 30d¯. b)Variation of angular velocity (left axis) and perpendicular velocity(right axis) as a function of length L. . . . . . . . . . . . . . . . . 874.4 Contour plots of the final pitch θ◦, and the height of the swimmerh/d¯ after it swims |x∥| = 100d¯ parallel to the wall as a function oflength L, and λ. Results are for initial values of θi = 0, hi = 3d¯.The area within the thick white line indicates that the swimmer hasalready crashed with the wall. . . . . . . . . . . . . . . . . . . . 88xivList of Figures4.5 a) Trajectories of a swimmer for different lengths L and λ near awall. The second squirming mode of sphere α is Bα2 = {−10, 10}corresponding to a pusher and a puller, respectively. Results arefor L/d¯ = {1.01, 2, 10}, θi = 0, hi = 3d¯ and the points on the trajec-tories show increments of τ/5 and the direction of the swimmer, p.Trajectories are plotted for a total time needed to reach |x∥| = 30d¯.Solid and dashed-lines are devoted to Bα1 = 0 and Bα1 = ±1, re-spectively. b) Variation of the angular velocity (left axis) and thevertical velocity (right axis) as a function of length L for hi = 3d¯,θi = 0. Thin lines correspond to the non-zero first squirming modeBα1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.6 Contour plots of the final pitch of the swimmer θ◦ after it swims|x∥| = 100d¯ parallel to the wall as a function of length L and λ.Results are for initial values of θi = 0, hi = 3d¯. The areas shownby thick arrows indicate that the swimmer has already fallen intothe wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.7 a) A swimmer with axial rotation Ωp due to the azimuthal squirm-ing mode of Cα1 , spins parallel to the wall normal e⊥ with angularvelocity ω⊥ at a point on the link distanced Lαrev from the centre ofthe active particle α. The particle α lies on a trajectory with radiusRαrev. b) Contour plots of Lαrev, and Rαrev and c) the axial rotation ofthe swimmer Ωp as a function of the swimmer pitch θ and λ. d)Variation of Ωp as a function of the swimmer’s height h on the bot-tom x-axis and swimmer’s length L on the above x-axis. Resultsare for Cα1 = 10, θi=0, and L = hi = 3d¯ unless mentioned otherwise. 944.8 Variation of the swimmer’s tumbling rate ω⊥, as a function of a)the pitch θ, b) the height of the swimmer h and c) the length ofthe swimmer L for different λ. Results are for Cα1 = 10, θi=0,L = hi = 3d¯, unless mentioned otherwise. . . . . . . . . . . . . . 954.9 Trajectories of a swimmer for different lengths and λ. Results arefor Bα1 = −1, Cα1 = 10, θi = 0, hi = 2d¯ and the points on the trajec-tories show increments of τ/5 and the direction of the swimmer p.Trajectories are plotted for a total time needed for the swimmer tocollide into the wall or reach the bounds of (x⊙, x∥) = (25d¯, 30d¯). 964.10 a) ϕ-phase portrait, b) variation of the minimum distance betweenthe swimmer and the wall ϵ as a function of the revolution angle ϕ,and c) variation of the vertical velocity U⊥ and the angular velocityof the swimmer ω⊥ as a function of ϵ. Results are for Bα1 = −1,Cα1 = 10, λ = 1, θi = 0, and L = hi = 2d¯. . . . . . . . . . . . . . . 96xvList of Figures4.11 a) ϕ-phase portrait, b) variation of minimum distance between theswimmer and the wall ϵ as a function of azimuthal angle ϕ, c) vari-ation of the vertical velocity U⊥ and angular velocity ω⊥ as a func-tion of ϵ, and d) θ-phase portrait. Results are for Bα1 = −1, Cα1 = 10,L = hi = 2d¯, and θi = 0. Dashed lines in (c) refer to the right axis. 984.12 Contour plots of the final azimuthal direction ϕ◦ for a) a pullerand b) a pusher swimmer, after it swims |x∥| = 100d¯, or |x⊙| =100d¯, whichever comes first, as a function of L and λ. The regioninside the thick white line in (a) and the empty region in (b) suggestthat the swimmer is already collapsed into the wall. Hashed areashows the region where the swimmer falls into an orbit close to thesurface. Figure c shows orbital trajectories for a rotating pusherin the hashed area. Results are obtained for Bα1 = −1, Cα1 = 10,θi = 0, and hi = 3d¯. . . . . . . . . . . . . . . . . . . . . . . . . . 995.1 Schematic of an elastic swimmer. . . . . . . . . . . . . . . . . . . 1015.2 Free diagram of spheres and rods in an elastic swimmer model. . . 1025.3 The rigid body motion of the swimmer. . . . . . . . . . . . . . . 1025.4 Trajectories of a swimmer for different values of stiffness k ={∞, 1.5, 1, 0.5, 0.1} near a wall. Particle α is a neutral squirmer withfirst squirming modes of Bα1 = {−1, 1}. Results are for L = hi = 3d¯,θi = 0, λ = 1 and the points on the trajectories show incrementsof τ/5 and the direction of the swimmer’s links eα, eβ. Trajectoriesare plotted for a total time, τ, required for the swimmer to reach|x∥| = 20d¯ unless it falls into an orbit or the spheres collide. Solidand dashed-lines correspond to δ ≤ 0, and δ > 0, respectively. . . 1055.5 Trajectories of a swimmer for different values of λ = {0.1, 0.5, 1, 2, 10}near a wall. Particle α is a neutral squirmer with first squirmingmodes of Bα1 = {−1, 1}. Results are for k = 1, L = hi = 3d¯, θi = 0and the points on the trajectories show increments of τ/5 and thedirection of the swimmer’s links eα, eβ. Trajectories are plotted fora total time, τ, required for the swimmer to reach |x∥| = 20d¯ unlessit falls into an orbit or the spheres collide. Solid and dashed-linescorrespond to δ ≤ 0, and δ > 0, respectively. Dotted lines corre-spond to a rigid swimmer. . . . . . . . . . . . . . . . . . . . . . 106xviList of Figures5.6 Trajectories of a swimmer for different values of initial orientationθi near a wall for a) puller, and b) a pusher. Increments of θi areπ/16. Particle α is a squirmer with the first squirming mode ofBα1 = {−1, 1}. Results are for λ = 1, L = 3d¯, hi = 20d¯, k = 1and the points on the trajectories show increments of τ/5 and thedirection of the swimmer’s links eα, eβ. Trajectories are plotted fora total time, τ, required for the swimmer to reach |x∥| = 20d¯, unlessit falls into an orbit or the swimmer collides with wall. Solid anddashed-lines correspond to δ ≤ 0, and δ > 0, respectively. Dottedlines correspond to a rigid swimmer. . . . . . . . . . . . . . . . . 1075.7 Trajectories of a swimmer for different values of length L/d¯ ={1.1, 2, 3, 5, 10} near a wall. Particle α is a neutral squirmer withfirst squirming modes of Bα1 = {−1, 1}. Results are for λ = 1, k =1, hi = 5d¯, θi = 0 and the points on the trajectories show incrementsof τ/5 and the direction of the swimmer’s links eα, eβ. Trajectoriesare plotted for a total time, τ, required for the swimmer to reach|x∥| = 20d¯ unless it falls into an orbit or the spheres collide. Solidand dashed-lines correspond to δ ≤ 0, and δ > 0, respectively.Dotted lines correspond to a rigid swimmer. . . . . . . . . . . . . 1085.8 Trajectories of a swimmer for different values of stiffness k near awall for a) λ = 0.5, b) λ = 1 and c) λ = 2. Particle α is a squirmerwith second squirming mode of Bα2 = {−10, 10}. Results are forBα1 = 0, L = hi = 3d¯, θi = 0, and the points on the trajectoriesshow increments of τ/5 and the direction of the swimmer’s linkseα, eβ. Trajectories are plotted for a total time, τ, required for theswimmer to reach |x∥| = 20d¯ or |x⊥| = 20d¯, or collapse to thewall, whichever happens first. Solid and dashed-lines correspondto δ ≤ 0, and δ > 0, respectively. . . . . . . . . . . . . . . . . . . 1105.9 Trajectories of a swimmer for different values of λ near a wall fora) k = 1, and b) k = 0.3. Particle α is a squirmer with secondsquirming mode of Bα2 = {−10, 10}. Results are for Bα1 = 0, L =hi = 3d¯, θi = 0 and the points on the trajectories show incrementsof τ/5 and the direction of the swimmer’s links eα, eβ. Trajectoriesare plotted for a total time, τ, required for the swimmer to reach|x∥| = 20d¯ or |x⊥| = 20d¯, or collapse to the wall, whichever happensfirst. Solid and dashed-lines correspond to δ ≤ 0, and δ > 0,respectively. Dotted lines correspond to a rigid swimmer. . . . . . 111xviiList of Figures5.10 Trajectories of a swimmer for different values of initial orientationθi near a wall for a) puller, and b) a pusher. Increments of θi areπ/16. Sphere α is a squirmer with the second squirming mode ofBα2 = {−10, 10}. Results are for Bα1 = 0, L = 3d¯, hi = 20d¯, k = 0.3and the points on the trajectories show increments of τ/5 and thedirection of the swimmer’s links eα, eβ. Trajectories are plotted fora total time, τ, required for the swimmer to reach |x∥| = 20d¯, unlessthe swimmer touches the wall. Solid and dashed-lines correspondto δ ≤ 0, and δ > 0, respectively. Dotted lines correspond to a rigidswimmer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.11 Trajectories of a swimmer for different values of length L/d¯ ={1.1, 2, 3, 5, 10} near a wall. Sphere α is a squirmer with the secondsquirming modes of Bα2 = {−10, 10}. Results are for λ = 1, k =0.3, hi = 5d¯, θi = 0 and the points on the trajectories show in-crements of τ/5 and the direction of the swimmer’s links eα, eβ.Trajectories are plotted for a total time, τ, required for the swim-mer to reach |x∥| = 20d¯ unless the swimmer collide. Solid anddashed-lines correspond to δ ≤ 0, and δ > 0, respectively. Dottedlines correspond to a rigid swimmer. . . . . . . . . . . . . . . . . 1136.1 Schematic of the elastic swimmer. . . . . . . . . . . . . . . . . . 1176.2 Free diagram of spheres and rods. . . . . . . . . . . . . . . . . . 1206.3 a) Trajectories and b) y phase portrait for different initial orienta-tions θi. Solid and dashed lines show δ < 0, δ > 0, respectively.In a) blue lines correspond to the position of the swimmer afterT = 1, 2, 3, 4, and 5 (scaled with H∗α/U∗). In b) black and blue linescorrespond to pushers and pullers, respectively, and red dots showthe initial position while blue dots across the curves demonstratethe position of the swimmer after increments of T = 1. Results areobtained for Hα = 1, FT = 1, k = 10−4, λ = 0.1, ϵ = ξ = 0.1 . . . . 1266.4 Trajectories for different a) spring stiffness k, b) ratio λ, c) α, andd) β. Blue lines correspond to the position of the swimmer afterT = 1, 2, 3, 4, and 5. Results are obtained for θi = 0,Hα = 1, FT =1, k = 10−4, λ = 0.1, ϵ = ξ = 0.1 unless otherwise mentioned. . . 1286.5 Variation of bending angle δ of a pusher as a function of λ fordifferent orientations θ. The right axis magnifies the results forhigher λ > 1. The Contour plot within the figure shows negativebending angles with solid lines and positive values with dashedlines. Same results are shown for a puller within the figure. Resultsare obtained for FT = 1, k = 10−4, ϵ = ξ = 0.1. . . . . . . . . . . . 129xviiiList of Figures6.6 Contour plots of normal velocity U · e⊥ = 0 (solid lines), angularvelocity Ω · (e∥ × e⊥) = 0 (dashed lines), and bending angle δ = 0(dotted lines) as a function of λ and initial orientation θi. Bluelines are for a rigid swimmer. Grey (white) area is the terminusattraction (escape) zone for a flexible swimmer, while the hatchedarea is the terminus attraction zone for a rigid swimmer. Resultsare obtained for FT = 1, k = 10−4, ϵ = ξ = 0.1. . . . . . . . . . . . 1306.7 Swimmers alongside their trajectories for different initial orien-tations θi. Blue and red links show δ < 0, δ > 0, respectively.Thin solid and dotted lines are the trajectories of the spheres Aand B, respectively. Results are obtained for Hα = 1, FT = 1, k =4 ∗ 10−5, λ = 0.1, ϵ = 0.2, ξ = 0.2 . . . . . . . . . . . . . . . . . . 1316.8 Schematic of the swimmer in an unbounded fluid. . . . . . . . . . 1326.9 Trajectories for different initial orientations −0.2 ⩽ θi ⩽ 0.2 andfor a,d,g) a rigid swimmer with k = ∞, b,e,h) a moderately flexibleswimmer and c,f,i) a very flexible swimmer. Also, the rows arefor λ = 0.1, 1, 3, respectively. Solid and dashed lines show δ ⩽0, δ > 0, respectively. Blue lines correspond to the position of theswimmer after T = 1, 2, 3, 4, and 5 (scaled with H∗α/U∗). Resultsare obtained for Hα = 1, FT = 1, ϵ = ξ = 0.1 . . . . . . . . . . . . 1366.10 The comparison of a far-field study of an elastic swimmer (solidlines) with the elastic swimmer investigated in chapter 5 (dashedlines). Results are obtained for θi = 0,Hα = 1, FT = 1, k = 4 ∗10−5, λ = 0.1, ϵ = 0.2, ξ = 0.2. . . . . . . . . . . . . . . . . . . . 1387.1 A prolate spheroid in simple shear flow. . . . . . . . . . . . . . . 1437.2 Trajectories of a prolate spheroid’s pole with aspect ratio 5 in alinear shear flow of Newtonian fluid for different initial positions. 1477.3 Modified orbits (red lines) in the presence of shear thinning andNewtonian orbits (blue lines) for a) different initial conditions ofθi = 0, π/12, π/4, π/3, 5π/12, 11π/24, π/2 and ϕi = 0, b) differ-ent initial conditions of ϕi = π/2, π/4, 0 on a specific Jeffery or-bit passing (θi, ϕi) = (5π/12, 0), c) different Carreau numbers ofCu = 0, 0.1, 0.12, 0.14, 0.16 for two different initial conditions of(θi, ϕi) = (π/3, 0), (0.33, π/2) belonging to a Jeffery orbit, and d)different values of aspect ratios λ = 1, 2, 3, 4, 5 for (θi, ϕi) = (π/3, 0).Calculations are carried out for Cu = 0.1, and λ = 5 unless other-wise stated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149xixList of Figures7.4 Phase portraits of a) θ and b) ϕ for different Cu numbers. Dashed-line shows the Newtonian case. Dotted-lines in a is for differentinitial condition of (θi, ϕi) = (0.33, π/2) following Fig.7.3c. Re-sults are for (θi, ϕi) = (π/3, 0), and λ = 5. . . . . . . . . . . . . . 1507.5 Phase portraits of a) θ and b) ϕ for different initial conditions of θi =π/12, π/4, π/3, 5π/12, 11π/24. Dashed-lines show the Newtoniancases. Thick, and thin lines correspond to Cu = 0.15, and Cu =0.1, respectively. Results are carried out for λ = 5. . . . . . . . . . 1517.6 Phase portraits of a) θ and b) ϕ for different aspect ratios of λ =1, 2, 3, 4, 5. Dashed-lines show the Newtonian cases. Thick, andthin lines correspond to Cu = 0.15, and Cu = 0.1, respectively.Results are carried out for (θi, ϕi) = (π/3, 0). . . . . . . . . . . . . 1528.1 a) Geometry inputs (Blue) and velocity outputs (red) of two inter-acting squirmers. b) The schematic diagram of a multilayer neuralnetwork for two interacting squirmers. . . . . . . . . . . . . . . . 1548.2 Interface of the software for two interacting squirmers. . . . . . . 1549.1 a) Puller and b) pusher suspensions with random initial orientationsafter T=10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156xxAcknowledgementsFirst and foremost, I would like to appreciate my advisor Prof. Gwynn. J. Elfringfor his continuous support of my study and research, for his patience, motivation,enthusiasm, and immense knowledge. His guidance helped me in all the time ofresearch and writing of this dissertation.Besides my advisor, I would like to thank the rest of my thesis committee:Prof. Neil J. Balmforth, Prof. Carl Ollivier-Gooch, and Prof. Kendal Bushe fortheir encouragement, and insightful comments.The members of Complex Fluids Lab at UBC have contributed immensely tomy personal and professional time at UBC. My sincere thanks also goes to mycolleagues, Dr. Babak Nasouri and Dr. Ronak Gupta, for their friendship andencouragement.I also want to thank my hard-working parents, Azar and Mehdi, who havedevoted their lives to make the best possible life for my brothers and I and providedunconditional love and care. I’d like to remember my brother Mehran too. He wasa postdoctoral fellow at UBC when he lost his life in PS752 tragedy. Mehran, westarted this journey together and I know how proud you are even though you won’tbe around to see me graduate. You left a void never to be filled in our lives. Thoughyour life was short, I will make sure your memory lives on as long as I shall live.My brothers, Mehran and Iman have been my best friends all my life and I thankthem for all their advice and supports.The best outcome of these past years at grad school is finding my better half.Yasaman, thank you for your unfailing love, help and support. I love you beyondwords.xxiDedicationTo my dearest Yasaman&In loving memory of MehranxxiiChapter 1IntroductionIt is as difficult for a microorganism to swim in water as it is for a goldfish to swimin coal tar. Two key parameters in the Reynolds number for microorganisms arelength and speed, which for a generic bacterium are of orders 1µm and 10µm/s,respectively and as a result Re ∼ 10−5. In this regime, friction dominates (creep-ing flow), and viscous effects quickly damp out inertial effects. In the limit thatRe → 0, the immediate consequence is the loss of non-linear terms in the govern-ing equations and Navier-Stokes equations reduce to linear time-invariant Stokesequations. In creeping flow, fluid motion can be completely reversed by revers-ing boundary kinematics. Thus, every fluid element near a scallop that is flappingback and forth returns to its original position and as a result the scallop cannot goanywhere (the Scallop theorem) [174]. In other terms, if a small swimmer in suchregime tries to swim by a reciprocal motion, in each stroke, the swimmer tracesback its trajectory and returns to its initial position.In general, the nature of swimming in creeping flow, is a periodic motion butnot a reciprocal one, which in sequence breaks the time-reversal symmetry. Oneway of swimming is adapted by ciliated cells like Paramecium, and Volvox. Eachcilium is driven by the internal forces and the cyclic motion shown in fig. 1.1 is notreciprocal and can generate net thrust. In each power stroke, the cilium generates ahigh perpendicular drag force while in a recovery stroke, it produces a low paralleldrag force leading to a net motion [157, 182].Figure 1.1: The asymmetric flexural motion exhibited by a single Volvox cilium.Reprinted from [182]1Chapter 1. IntroductionFlagellar propulsion is also another way of locomotion. Many microorganismssuch as spermatozoa, Escherichia coli, and Chlamydomonas use flagella to swimthrough the surrounding fluid. Few flagella are attached to the cell through separaterotary molecular motors. The rotation of motor causes a helical wave through theflagellum and propels the body [15]. In addition, a beating flexible flagellum candeform in a non-reciprocal way to create non-zero time-averaged propulsion [129].Figure 1.2 further illustrates the process of motion. Any short elements of thefilament is moving with velocity u. In case of isotropic drag, the force acting onthe filament has the same direction as the velocity of the filament. However draganisotropy in viscous fluids results in a component fprop that is perpendicular to thedirection of the local actuation [14]. Furthermore, in order to create non-zero forceson average, the periodic actuation of the filament must be non-time reversible.Figure 1.2: The drag anisotropy for a slender filament leads to a net motion.Reprinted from [129]Understanding the dynamical behaviour of active particles is imperative for in-vestigating the complex rheological behaviour of (biological) systems referred toas active matter [122, 176]. A microorganism, categorized as a sort of active parti-cle, is propelled by an internal energy source and dissipates it through movement.In order to model active particles, we employ spheres with a prescribed tangentialslip velocity representing cilia, flagella, phoretic slip, or chemical reactions. Theinduced tangential flows are strong enough to propel the swimmer forward. Thismodel first was proposed by Lighthill and referred to as a squirmer [137]. Theo-retically, based on the distribution of surface velocity, spherical active particles canbe categorized into three groups. An active particle that generates impetus fromits front end (e.g., Chlamydomonas that has flagella in front of the cell body) is re-ferred to as a puller swimmer. A puller induces a flow field attracted inwards alongthe propulsion axis and ejected to the sides. In the far field, the induced flow fieldis a negative force dipole. A force dipole is the flow field produced by two close2Chapter 1. Introductionpoint forces acting in opposing directions. A force dipole decays as 1/r2 where r isthe distance from the singular point. On the contrary, a pusher swimmer generatesthrust from the rear end (e.g., E. coli whose rear-mounted flagella push the cellbody forward), and the induced flow field is pushed away from the cell along thepropulsion axis and attracted inwards along the sides. A pusher generates flow pat-terns that in the far field resemble a positive force dipole. Finally, the third activeparticle is called a neutral swimmer, (e.g. Paramecium with thousands of tiny ciliaon its body) that is neither a pusher nor a puller. The flow field far away from theswimmer is that of a source dipole which is less long-ranged in comparison witha force dipole [65]. A source dipole is the flow field formed when a sink and asource are brought together and decays as 1/r3. In a sink (source), all streamlinespoint radially inward to (outward) the singularity at the origin. Figure 1.3 showsthe slip and associated fluid velocity fields for the squirmer model for differenttypes of active particles. The fluid mechanics of cilia and flagella are detailed inthe wonderful reviews in [28, 129].(a) (b) (c)Figure 1.3: Axisymmetric surface velocity profile vs and the analytical flow fieldsfor (a) a pusher (b) a neutral squirmer, and (c) a puller in the lab frame. Reprintedfrom [127, 224]Many microorganisms, such as bacteria and spermatozoa, accumulate near sur-faces as observed in a number of studies [68, 179, 192]. Microorganisms may bedrawn to surfaces due to a gradient of increasing or decreasing concentrations ofa particular substance near the surfaces, in quest for food or to avoid exposure totoxic substances [16, 211]. Microorganisms bias their movements to regions witha higher concentration of attractants by sensing the changes in their environment,which are mostly attributed to gradient of changes in concentration of chemicals3Chapter 1. Introductionin their vicinity [17]. The hydrodynamic interactions between an active particleand nearby surfaces also play an important role in the attraction of these swim-mers to surfaces. On that account, different behaviours of trapping, escaping, andnon-trivial steady trajectories above walls have been numerically observed usingthe squirmer model and bead-based swimmers [134, 158, 213]. Several of thesebehaviours for swimming organisms near a boundary have been seen in experi-ments, including wall-induced accumulation of bull spermatozoa, the circular or-bit of E. coli above a flat surface, and the so called ”dancing motion” of pairs ofVolvox near a solid boundary [66, 130, 179]. Therefore, near-wall hydrodynamicsfor swimmers needs to be addressed carefully in order to provide insights into theunderlying mechanisms of these behaviours. Comprehension of such mechanismsshall have various implications, ranging from a better understanding of the motionof cells in biological fluids [84], biofilm formation [45, 98, 123, 173], designingnanoswimmers to pick up and deliver drugs in realistic settings [168] to develop-ing methods to perform less invasive diagnostic and therapeutic procedures [156].A wall also causes particles to rotate because they encounter different drag on thewall side. Precise single-cell rotation is critical for several biotechnology applica-tions, including cell characterization [219], drug discovery [86], and cell imaging[148]. For example, a rotating cell is advantageous for cell imaging since it has noblind areas during cell observation. Moreover, it is critical that non-contact phys-ical fields, such as hydrodynamic fields, drive cell rotation rather than undesiredphysical or chemical disturbances [201].It has been shown both experimentally and theoretically that the hydrodynamicinteraction between the E. coli and a wall results in the attraction of the swimmer tothe wall, while reorienting the swimmer in the direction parallel to the wall. In con-trast, the algae Chlamydomonas reorients perpendicular to the wall and as a result,eventually crashes into the wall [19]. Another observation is the clockwise circularmotion of the E. coli in a plane parallel to the wall [18, 130]. The hydrodynamicmodel for the swimming of an individual pusher swimmer like an E. coli near a sur-face shows the mechanism for the previously observed circular motions of the cellclose to solid boundaries [118, 130]. Furthermore, a series of recent theoretical andexperimental studies illustrate that adjacent to a surface, some active particles tendto accumulate and rest in place for a longer period [13, 20, 25, 68, 69, 85, 135, 216],while some may slide off the surface [193, 200].Many qualitative dynamic behaviours of swimmers near a no slip wall can becaptured considering two-dimensional flow regimes. One advantage is the use ofvarious analytical methods to investigate the dynamics of a single disk swimmernear a wall. Results of complex variable methods for a disc swimmer show thatfor any initial positions and orientations, a force dipole swimmer will eventuallycollide with the wall in finite time, except for the cases of a parallel contractile4Chapter 1. Introduction(puller) and a perpendicular extensile (pusher) swimmer, which have a pure trans-lation away from the wall [53, 54]. Additionally, the general solution of the bihar-monic equation for the stream functions in the bipolar coordinates indicate that asquirmer will be able to escape a no-slip wall if the swimming orientation is awayfrom the interface. Pushers and pullers that are oriented towards the boundary, col-lide with the wall if they are sufficiently close to the wall. Outside of this attractionlayer, they eventually move away from the wall while they can have periodic tra-jectories [217]. The lattice Boltzman method for a two-dimensional disk squirmerin a channel shows that pushers and pullers slide along the channel walls while,weak pullers and pushers have periodic trajectories along the channel centrelineand spanning the channel width, respectively [4]. However, since the hydrody-namic interactions of cylindrical and spherical bodies differ, the results obtainedusing 2D methods do not entirely transfer to three-dimensional systems.Theoretical prediction shows that for a spherical squirmer swimming parallelto a flat wall, the attraction to the wall is controlled by the sign of stresslet: Apuller (pusher) escapes from (attracts to) the wall. In addition, the swimmer doesnot rotate and holds its orientation despite the presence of the wall [139]. Thestudy based on the far-field analytical approximations, demonstrates that the bodyrotates away from the wall regardless of its orientation except for the unstable caseof swimming directly towards the wall. For a squirmer initially oriented towardsthe wall, the rotational effect increases as it swims towards the wall, which resultsin an increase in the final pitching angle once it departs [190]. This behaviour issimilar to the dynamics of a Janus sphere, half catalytic, with uniform repulsion ofthe solute from the particle surface. When a Janus particle gets in close vicinityof a wall, it rotates its catalytic cap towards the wall, and escapes from the wallalmost parallel with the wall [208, 209, 223]. Dynamics of a rotary squirmer isonly modulated within the horizontal plane parallel to the wall and the height andthe orientation angle are unchanged [108].Non-flat surfaces may have varying impacts on the dynamics of micro swim-mers. The simulation of a squirmer inside a circular tube using the boundary ele-ment method shows a reduction in the locomotion speed of the swimmer parallelto the tube axis [221]. Neutral swimmers display a periodic wavelike trajectoryinside the tube. This is inline with the observations of Paramecium cells inside acapillary tube [111]. Additionally, pullers initially follow wavelike trajectories butend up swimming along the tube axis: Weak pullers swim in the centre of the tube,while strong pullers swim near the walls. In contrast, pushers are always unstablewith decreasing wavelengths and increasing amplitude, eventually crashing intothe walls of the tube.There is also further research on micro-swimmers in films between a wall and aliquid–air interface. Although pushers and pullers are attracted to both boundaries,5Chapter 1. Introductionthere is a long-ranged bias to accumulate at the no-slip wall [145]. In a recentendeavour, a new method is developed providing exact expressions for a pair ofactive particles, as well as an active particle squirming close to a surface using theLorentz reciprocal theorem [167].Modelling the behaviour of a suspension of active particles is significantlymore challenging due to the many-body interaction of the cells [99, 181]. In asuspension of N active particles, every combination of two active particles (in to-tal NC2) is interacting. It has been shown that any of the pairwise interactionsdoes not become negligible even at asymptotically large separations despite thedecay of their flow field with distance. The reason for this is that in a suspen-sion, no squirmer is too far away to have an impact, and the relative displacementof a squirmer is due to the accumulation of its interactions with all neighbouringsquirmers [44]. Each active particle in a suspension produces a spontaneous flowentraining other particles. A strong mutual entrainment can change the collectivebehaviour of the active matter not comparable with a suspension of passive par-ticles. Experiments and simulations show that active droplets near a plane wallform crystallites parallel to the wall with constant recirculation from the centre tothe edges [187]. In contrast, both theory and experiments show that in a chan-nel, droplets form stable bands that travel through each other [205]. The directnumerical simulation of a suspension of squirmers confined between two walls bya distributed Lagrange-multiplier-based finite-volume method reveals that even atsmall concentration of squirmers, around 70% of the squirmers are accumulatednear the walls [134] as previously observed in [19, 179]. However, the existingresearch has many problems in representing near wall hydrodynamics.The interaction between a swimmer with other swimmers or with a solid bound-ary can be mathematically classified into two categories. In one scenario, the swim-mer is very close to the surface or another swimmer, where near-field lubricationforces govern the motion [9, 152, 171]. In another scenario, the swimmer is at agreat distance from other swimmers and the solid boundary, where far-field hydro-dynamics dictate the dynamics of the swimmers [42, 190]. Asymptotic solutionsfor these two regions may be incorporated into a single expression that holds forall swimmer separations. The far-field solution can be represented as an inversepower series of the center-to-center separation. Then the asymptotic equation fornearly touching swimmers is combined with the series expansion to handle the sin-gular behavior caused by lubrication forces [115]. We can, however, use a sigmoidfunction to smoothly transition between solutions across a transition distance. Themethod of matched asymptotic expansions can also be used to estimate the domi-nant contribution in the intermediate range.We extend the asymptotic approaches developed for two passive spheres de-ployed at close distances to examine the near-field hydrodynamics of two unequal6Chapter 1. Introductionswimmers [47, 114, 116, 162, 163, 195]. This method may be modified to analyzethe interaction between a squirmer and a surface by including a reasonably largesize ratio while keeping the gap distance between the surfaces modest and finite.Furthermore, we may connect swim force and squirming modes in the same waythat the resistance tensor for two passive particles does for forces and velocities.As a result, the dynamics of two nearby active particles or an active particle neara wall are defined by combining the passive resistance tensor and the swimmingforce.To study dense active matter near a surface, we will leverage methods devel-oped for passive suspensions. The most popular method of simulating the hydro-dynamic interactions of a suspension of particles with full many-body system isthe Stokesian Dynamics (SD) method [27]. As a particle moves in a fluid medium,the created disturbance changes the drag forces exerted on other nearby particles.In this way, the motion of each particle is coupled with that of other particles. Inthe SD method, the far-field hydrodynamic forces for many-body interactions aresummed through a mobility tensor, constructed through a far-field expansion ofthe Boundary Integral (BI) equation for Stokes flow, and lubrication forces [71]are calculated pair wise for neighbouring particles through a resistance tensor wellknown in the literature. Then, these geometry-dependent tensors are used to builda solver for the motion of particles. The presence of a wall at a finite distance fromthe particles can be implanted in the SD method using the image representation ofsingular multipoles in the BI equations [119, 198]. In this technique, images of theproper strength on the exact location but on the opposite side of the wall yield theno-slip boundary condition on the wall. In the past few years, the SD method hasbeen used to simulate swimmers by constructing assemblies of spherical particles[88, 199]. Subsequently, the SD method can be adopted for the dynamics of activeparticles. The surface activity of particles contributes in either the velocity mo-ments [82, 106] or the force moments [146] of the system and the resistance tensorremains unchanged in comparison with passive particles.In this dissertation, we are interested in modifying the SD method for poly dis-perse suspension of active particles near a wall. The near-field hydrodynamics ofsquirmers is thoroughly investigated. The slip velocity can be extended into spher-ical harmonics and decomposed into squirming modes, but this is inaccurate whentwo squirmers are at close distance. In addition, we incorporate flexibility into as-semblies to capture the effect of flexibility on the swimmer’s dynamics near a wall[166, 215]. We use a minimal model consisting of two active spherical particleswith surface activities that are connected by two thin rods. These rods are also con-nected by a torsional spring allowing the swimmer to bend. Moreover, each sphereis self-propelled by surface activity [196]. Thereby, based on the direction of theimpetus generated by the spheres, we can model a puller, a pusher, or more com-7Chapter 1. Introductionplex swimmer. Additionally, our semi-analytical calculations allow us to explorethe area of lubrication for getting improved results (e.g., when the gap betweenthe swimmer and the wall is very small, or when the swimmer bends and spheresnearly touch).As noted above, the simplest squirmers are described as spherical colloids.However, bacteria such as E. coli are nonspherical [203]. For example, ellipsoidalsquirmers exhibit a variety of collective motions, including ordering, aggregation,and whirls, which are affected by the swimming mode and aspect ratio [128]. Fur-thermore, most previous research has concentrated on Newtonian fluids, but thefundamental building blocks determining rheology for single particles and sus-pensions in non-Newtonian fluids are still being discovered. In the case of a sin-gle microswimmer, the non-Newtonian environment has a considerable effect onits dynamical behaviour and frequently results in an increased swimming veloc-ity [84, 89, 191]. Moreover, non-Newtonian carrier fluids can cause qualitativechanges in the rheological behaviour of suspensions when compared to Newtonianfluids; For example, recent studies have calculated modifications of the Einsteinviscosity for dilute suspensions of spheres in weakly nonlinear viscoelastic flu-ids [77], and shear-thinning fluids [58]. Suspensions generally shear thin withincreasing the shear [210] while beyond a threshold exhibit shear thickening oreven jamming [212]. Phillips [170] extended the Stokesian dynamics method tolow Deborah number using the reciprocal theorem, and demonstrated that parti-cles prefer to cluster while sedimenting in a second-order fluid. Moreover, if theparticles are anisotropic, then even small deviations from spherical geometry cansignificantly impact the rheology even for dilute suspensions [133]. One reasonfor this is because elongated or rod-like particles tend to align with the flow thuschanging the properties of the fluid under shear. Therefore, understanding the dy-namics of even a single particle in shear is important to determine the rheology of asuspension of anisotropic particles. The dynamics of elongated particles, for exam-ple axisymmetric ellipsoids and spheroids, display significantly more complicatedbehaviour than spherical particles under shear [38, 73, 113, 159].We dedicate a chapter of this dissertation to the study of a neutrally buoyantthree-dimensional prolate spheroid in a shear flow of a weakly shear-thinning fluidas the first step in examining the physics of non-spherical active matter in non-Newtonian fluid. The particle is small enough that inertial forces are negligible,but big enough that Brownian forces do not play a role. To capture the leading-order effects of shear-thinning rheology on the dynamics, we solve for the motionof the particle for asymptotically weak shear thinning using the Carreau model [21,37, 58, 59, 83]. The derived mathematics can be implemented into the active SDapproach for non-spherical active matter in complex fluids in future investigations[170].81.1. Outline1.1 OutlineThis dissertation is organized as follows. In chapter 2, we investigate the hydrody-namics of two unequal swimmers at close proximity by considering the squirmingmodel. The transition between near-field and far-field solutions is also discussed.The final findings are supplied in the form of an active swim force, which maybe used in the active SD technique to calculate the dynamics of the suspension ofactive particles. In chapter 3, we develop the active SD algorithm for a suspen-sion of active particles near a wall. The combination of both far-field and near-field solutions provides a strong tool for the calculation of the dynamics of activeparticles in a bounded fluid at zero Reynold number. To model the far-field hy-drodynamic interactions among many active particles near a solid wall, we haveused the method of reflections in cooperation with BI equation for Stokes flow. Weexplore the dynamics of two active particles passing by each other near a wall fordifferent types of squirmers, namely pushers, pullers, and neutral squirmers. Toenhance the swimming model, in chapter 4, we propose separating the part of theswimmer responsible for generating the thrust (flagella) and the head (cell body)which experiences most of the drag. Chapters 4, 5 and 6 are devoted to the studyof different types of assemblies. Because we were interested in swimmers with ahead and a tail, we suggest a minimal toy model consisting of two spheres and arod. Also, the moments from the fluid on the flexible flagellar filament may leadto bending the swimmer, which leads to the reorientation of the body. Therefore,the flexibility dramatically affects orientation by a geometry symmetry-breaking.Thus, by adding a torsional spring in our model, we propose future research on theeffect of elasticity on the motion of a swimmer. While we use active SD methodin chapter 5 to model a flexible swimmer, in chapter 6 we use multipole singularsolutions to model the far-field hydrodynamics. As the last step, in chapter 7, weinvestigate the dynamics of a prolate spheroid in a shear flow of a shear-thinningCarreau fluid. The motion of a prolate particle is developed analytically for asymp-totically weak shear thinning and then integrated numerically. Finally in Chapter 8we use machine learning for the interactions of two squirmers, and in chapter 9 weconclude the dissertation with closing remarks.9Chapter 2Near-field hydrodynamics ofswimmers2.1 IntroductionThere is a rising demand for detailed information on the near-field interaction oftwo microorganisms in creeping flow of a Newtonian fluid. The far-field hydrody-namics of microorganisms has been one of the most popular ideas in active matterliterature for several decades. However, near-field hydrodynamics has been re-searched infrequently [142], while a number of works have shown that short-rangeforces are likely the governing determinant of the interactions between active par-ticles [68, 200] and their collective motion [222]. In order to model these microswimmers, we employ spheres with a prescribed tangential slip velocity, represent-ing cilia, flagella, phoretic slip, or chemical reactions. This first was proposed byLighthill [137] and referred to as a squirmer. Although the dynamics of catalyti-cally active colloids are richer and more complex [185], behaviour of swimmingbacteria is usually studied in the context of the self propelled squirmers that inter-act only via hydrodynamics [23, 105]. The pairwise interaction between two closesquirmers has a singular behaviour, and both numerical and analytical approachesto demonstrate the characteristics of their interactions are proved to be challenging[105, 107]. The interaction of two squirmers is computed analytically by Ishikawaet al. [105] for the limits of small and large separation, as well as numerically usinga boundary element method. In the near-field analysis, the leading-order terms forforce moments are weakly singular, and the next-order term is formed by fitting tothe numerical data. This study, however, does not provide all force moments upto the leading order and does not account for rotational squirming mode. Cortez[50], Ishimoto and Gaffney [109] numerically analysed near-field hydrodynamicsfor relatively dilute systems, however it is critical to design a simpler, yet com-putationally more efficient model capable of capturing the physics of near-fieldhydrodynamics. Such model constructs the grand mobility matrix for the suspen-sion of active matter [106], while alternative numerical models typically neglectthese imperative interactions [5, 10, 100, 190]. Existing models are reasonably102.1. Introductionaccurate when swimmers are well separated and less accurate at close proximity.The great advantage of the existing models focusing on far-field interactions is thesimplicity of studying a large number of swimmers [181, 218]. Recently, Zhanget al. [220] have proposed a solution for this problem by separating head and tail ofa bacterium instead of using squirmer model. In another attempt by Papavassiliouand Alexander [167], the exact solutions of two interacting squirmers at all sepa-rations are calculated by employing reciprocal theorem and the use of bisphericalcoordinates. However, such exact solution presents considerable limitations, as thenon-axisymmetric components of the motion have not been determined. In fact,for the non-axisymmetric Stokes drag of two spheres, there is currently no exactclosed-form solution.Furthermore, due to the existence of swimming organisms in constrained habi-tats, the hydrodynamics of organisms swimming near surfaces is arguably an es-sential subject to be addressed [129]. Li and Ardekani [134] performed numer-ical simulations for a single squirmer near a surface by a distributed Lagrange-multiplier-based finite-volume method. They demonstrated three distinct classesof trajectories based on the swimming mechanism: a) the squirmer escaping fromthe wall, (b) the squirmer swimming along the wall at a constant distance and orien-tation angle, and (c) the squirmer swimming near the wall in a periodic trajectory.A recent analytical study by Lintuvuori et al. [138] shows that repulsive interac-tions between swimmers and surfaces play a very significant role in the squirmer’sdynamics, while such interactions have been neglected in previous theories. K.et al. [121] coupled near-field hydrodynamic interactions with the already well-established far-field hydrodynamics using Lattice Boltzmann method to explainswimmer entrapment at boundaries as well as other classes of trajectories. Thutu-palli et al. [205] used a sphere model with a preset surface slip velocity in hisexperiments and simulations. The slip velocity has been designed to match ob-servations of the flow outside of droplets for various geometries. They show anordering mechanism for active droplets that is regulated by hydrodynamic bound-ary conditions such as no slip and free slip at the bounding surfaces, resulting indiverse characteristic flow fields and aggregation events, including the previouslyanticipated flow-induced phase separation, swirling and dense clustering. Theirsimulation, however, does not account for near-filed hydrodynamics.In this chapter, we extend the asymptotic methods developed for two unequalrigid spheres placed at close distances [47, 114, 116, 162, 163, 195] and adoptthem for the problem of squirmers. With some care, this approach can be adaptedto study the interaction between a squirmer and a surface through incorporating arelatively large size ratio while the gap distance between the surfaces remains smalland finite. Additionally, similar to the resistance tensor for two passive particleswhich relates forces and velocities linearly, we can relate swim force and squirming112.2. Problem formulation𝐏!𝐏" 𝐞!"(a)𝑧x, r𝛽1𝜖𝐏!𝜒𝑦(b)Figure 2.1: Squirmers α and β placed in creeping flow of a Newtonian fluid.modes. Therefore, dynamics of two close active particles, or an active particle neara wall, are determined through combination of passive resistance tensor and theswimming force. It is also worth noting that, unlike the study by Ishikawa et al.[105], our formulation is poly disperse with rotational modes.2.2 Problem formulationConsider two swimmers pα, and pβ, with the size ratio β placed in an otherwisequiescent creeping flow so that the closest distance between them is ϵ in eαβ direc-tion as shown in figure2.1a. Note that all lengths are non-dimensionalized using theradius of swimmer α. The slip velocity on each swimmer in spherical coordinatesystem, where the axis of rotation ϕ is aligned with the head direction p, isus = (B1 sin θ + B2 sin θ cos θ) eθ +C1 sin θ eϕ, (2.1)where θ is the polar angle. Also, B1, B2 and C1 are axisymmetric squirmingmodes. Mode B1 is the free squirming speed in direction of the axis of symme-try p, while C1 yields axial rotation of the squirmer and mode B2 defines the typeof the squirmer: For a neutral squirmer B2 = 0, while B2 < 0 and B2 > 0 define apusher and a puller type squirmer, respectively. The Stokes equations are governingthe dynamics of the flow field. Key properties of the Stokes equations are that ofno explicit time dependence and that the flow-field is linearly arising form bound-ary conditions. Therefore, dynamics of these swimmers are determined instanta-neously by the imposed boundary conditions. One immediate advantage would122.2. Problem formulationProblem p us1 pα = ez usα = Bα1 sin θ eθ2 pα = ez usα = Cα1 sin θ eϕ3 pα = ez usα = Bα2 cos θ sin θ eθ4 pα = ex usα = Bα1 sin θ eθ5 pα = ex usα = Cα1 sin θ eϕ6 pα = ex usα = Bα2 cos θ sin θ eθ7 pα = 1√2 (ex + ez) usα =14 Bα2 (3 + cos 2ϕ) sin 2θ eθ − 12 Bα2 sin θ sin 2ϕ eϕTable 2.1: Breaking the slip velocity on the surface of sphere α into seven cases.be breaking the boundary conditions of the whole problem into sub-problems thattheir solutions are feasible to tackle. For now, we only consider activity on sphereα while sphere β is non-active. We place a cylindrical coordinate system with zaligned with eαβ and pα · eαβ = cos χ as shown in fig.2.1b. We can split this prob-lem into seven sub-problems. These problems are summarized in table 2.1 and fig.2.2.From these boundary conditions, it is clear that problems 1 − 3 are axisym-metric and problems 4 − 7 are asymmetric. Ultimately, the full solution is a linearcombination of these sub-problems of the formu = (u1 + u2) cosχ + u3 cos2 χ + (u4 + u5) sinχ + u6 sin2 χ + u7 sin χ cos χ.(2.2)More details are shown in A.1.When the minimum clearance between the swimmers approaches to zero, theflow in inner region, compared to the flow in outer region, becomes strongly sheareddue to the large velocity gradients and pressure. In this chapter, we present theleading order solution to the lubrication in the inner region and show the singularnature of the hydrodynamic moments. Equations are solved in a stretched cylindri-cal coordinate system, with R = ϵ−1/2r and Z = ϵ−1z. The surfaces of swimmers areZα = 1+1/2 R2+1/8 ϵ R4+O(ϵ2), and Zβ = −1/2 β−1R2−1/8 ϵ β−3R4+O(ϵ2). Thehydrodynamic forces and torques are calculated up to O(1). Axisymmetric prob-lems are solved using stream functions ψ, while asymmetric problems are solveddirectly.132.2. Problem formulation(a) Problem 1 (b) Problem 2 (c) Problem 3(d) Problem 4 (e) Problem 5 (f) Problem 6 (g) Problem 7Figure 2.2: Sub-problems with boundary conditions summarized in table 2.1.142.3. Axisymmetric problems2.3 Axisymmetric problemsThe slip velocity in problems 1 and 3 in cylindrical coordinate system reduces tous = (usr , 0, usz) and we can solve them by defining stream function ψ so that thevelocity field is ur = r−1∂ψ/∂z and uz = −r−1∂ψ/∂r. Also, the governing equationsreduce to E2E2ψ = 0 (see A.2). We expand the stream function as ψ = ϵ2ψ(0) +ϵ3ψ(1)+O(ϵ4) consistent with the boundary conditions. Note that in this expansion,ψ(R,Z) diverges for touching swimmers ϵ → 0, as Z → 1/ϵ. The solutions for ψ(0),and ψ(1) are lengthy and not shown here. While the hydrodynamic torque is zerodue to symmetry, the vertical force on either swimmer isFz = πµ∫r3∂∂n(1r2E2ψ) ds (2.3)where ds is the arc-length element and the integral is taken on the surface of theswimmer in a direction making a positive right angle with the direction n that isnormal to the surface and outward to the flow [95]. For swimmer αwe get ds/ dR =ϵ1/2 + 1/2 ϵ3/2R2 + O(ϵ5/2) and ∂/∂n = −ϵ−1∂/∂Z + R2/2 ∂/∂Z + R ∂/∂R + O(ϵ).More details on the derivation of forces and matching inner lubrication and outersolutions are given in A.2. These forces are given asFαz 16πµBα1= − Fβz 16πµBα1= −4β3 + 2β2(β + 1)3log ϵ + O(1, ϵ log ϵ), (2.4)Fαz 36πµBα2= − Fβz 36πµBα2=4β3 + 2β2(β + 1)3log ϵ + O(1, ϵ log ϵ), (2.5)Note that problem 2 is not contributing in the forces up to O(log ϵ).2.4 Asymmetric problemsFor asymmetric problems of 4, 5, and 7 we expand the field as p(r, θ, z) = ϵ−3/2 p(0)+ϵ−1/2 p(1) +O(ϵ1/2), ur(r, θ, z) = u(0)r + ϵu(1)r +O(ϵ2), uθ(r, θ, z) = u(0)θ + ϵu(1)θ +O(ϵ2),and , uz(r, θ, z) = ϵ1/2u(0)z + ϵ3/2ϵu(1)z + O(ϵ5/2). However, the fields in problems 6are O(ϵ1/2) smaller. While ur, and uθ are driven as a function of p through Stokesequation, ∇p = µ∇2u, the challenging part is the derivation of pressure field. Thisis accomplished by integrating continuity along the shortest distance between twoswimmers, yieldingR2 p(0)RR +(R + 3H−1R3(1 + β−1))p(0)R + p(0)θθ = Q(R) (2.6)152.4. Asymmetric problemsProblem Q(R)4 −6µBα1 H−3R3(1 + β−1) cos θ5 6µCα1 H−3R3(1 + β−1) sin θ6 −3µBα2 H−3R2(2 + 2R2(1 + β−1) + R2(1 + β−1) cos 2θ)7 6µBα2 H−3R3(1 + β−1) cos θProblem p(0)4 6/5 µBα1 H−2R cos θ5 −6/5 µCα1 H−2R sin θ6 1/2 µBα2 H−2 (6β/(β + 1) + 3R2 + R2 cos 2θ)7 −6/5µBα2 H−2R cos θTable 2.2: The right hand side term in eq.2.6 and the particular solutions of p fordifferent problems.where H = 1 + 1/2 R2(1 + β−1) and Q(R) as well as the particular solutions fordifferent problems are tabulated in table 2.2. In fact, the particular solutions are theentire solution. Complementary solutions vanish as the pressure is bounded in theinner region R → 0, and decays at far-field R → ∞. More details are brought inA.3. Hydrodynamic moments on the swimmers areF =∫Sn · σ dS =∫S(−np + µn · (∇u + ∇uT ))dS ,T =∫Sa n × n · σ dS =∫Sa n ×(−np + µn · (∇u + ∇uT ))dS (2.7)where for sphere α we get n = {ϵ1/2R, 0,−1 + ϵ/2 R2}, and dS = ϵR dRdθ. Ulti-mately, forces and torques have the form ofFαx 46πµBα1= − Fβx46πµBα1= − 2β2 + 8β15(β + 1)2log ϵ + O(1, ϵ log ϵ),Fαy 56πµCα1= − Fβy 56πµCα1=2β2 + 8β15(β + 1)2log ϵ + O(1, ϵ log ϵ),Fαz 66πµBα2= − Fβz 66πµBα2= − β2(β + 1)2log ϵ + O(1, ϵ log ϵ),Fαx 76πµBα2= − Fβx76πµBα2=2β2 + 8β15(β + 1)2log ϵ + O(1, ϵ log ϵ),162.4. Asymmetric problemsTαy 48πµBα1=Tαx 58πµCα1= − Tαy 78πµBα2=2β5(β + 1)log ϵ + O(1, ϵ log ϵ),T βy 48πµBα1=T βx 58πµCα1= − Tβy 78πµBα2=β210(β + 1)log ϵ + O(1, ϵ log ϵ). (2.8)At this point we can compare our results with the ones derived by Ishikawa et al.[105]. They calculated the interaction of two squirmers for small and large sepa-ration limits analytically as well as numerically using a boundary element method.In their description for Fx at near-field (eq.3.39 in [105]), there is an extra term e.exwhich is equivalent to sin χ in our notation. Also, the coefficients of Fx for bothsolutions are identical for β = 1, but there is a factor of 4 difference for β→ ∞.IshikawaFx6πµ= − 415β(2β2 + β + 2)(1 + β)3sin χ(Bα1 sin χ −12Bα2 sin 2χ)log ϵAbtahiFx6πµ= − 215β(β + 4)(1 + β)2(Bα1 sin χ −12Bα2 sin 2χ)log ϵ (2.9)Unfortunately, the description for Fz has not been reported by Ishikawa et al. [105].They made reference to an equation in Kim and Karrila [124] that does not exist.Furthermore, Ty is reported solely on one of the spheres, asIshikawaTαy8πµ= − 110β(β + 4)(1 + β)2sin χ(Bα1 sin χ −12Bα2 sin 2χ)log ϵAbtahiTαy8πµ=25β1 + β(Bα1 sin χ −12Bα2 sin 2χ)log ϵ (2.10)which is different with our calculation in both sign and amplitude. Consider prob-lem number 4. For positive Bα1 we getIshikawa and Abtahi Fαx > 0IshikawaTαy8πµ= −18Bα1 log ϵ > 0AbtahiTαy8πµ=15Bα1 log ϵ < 0 (2.11)where Fx in the lubrication area is positive in both calculations. This traction leadsto a counterclockwise torque (Ty < 0) (see fig.2.3). Another reason to suspect thatthe sign of Ty > 0 in the formulation of Ishikawa et al. [105] is incorrect, is thatthe authors indicated that the coefficients of these moments are identical to the caseof rigid spheres given by Kim and Karrila [124] in section 9, where the sign of Tydoes not match. Finally, hydrodynamics of mode C1 have not been reported byIshikawa et al. [105].172.5. Interactive momentsFigure 2.3: Force and torque acting on the active swimmer of problem 4.2.5 Interactive momentsHaving these moments, we can construct the full solution using the summationin eq2.2. The evaluation of these moments needs to be in terms of directions.Recognizing that ex =(pα − (eαβ · pα)eαβ)/ sin χ, ey =(eαβ × pα)/ sin χ, and ez =eαβ, we obtain the hydrodynamics of two swimmers at close distance as a functionof the swimmers’ direction and the direction between their centres asFαα6πµ= −Fβα6πµ=β15 (1 + β)3[Cα1 (8 + 10β + 2β2) eαβ × pα− (8 + 10β + 2β2)(Bα1 − Bα2 eαβ · pα) pα+(− 15(β + β2)Bα2 + (8 − 20β − 58β2)Bα1 eαβ · pα+ (−8 + 35β + 73β2)Bα2 (eαβ · pα)2)eαβ]log ϵ + O(1, ϵ log ϵ),Tαα8πµ=2β5 (1 + β)[Cα1(pα − eαβeαβ · pα)+(Bα1 − Bα2 eαβ · pα)eαβ × pα]log ϵ+ O(1, ϵ log ϵ),Tβα8πµ=β210 (1 + β)[Cα1(pα − eαβeαβ · pα)+(Bα1 − Bα2 eαβ · pα)eαβ × pα]log ϵ+ O(1, ϵ log ϵ), (2.12)where Fβα is the force on sphere β due to the surface activity on sphere α. Keepin mind that Fαα + Fβα = 0 and Tαα + Tβα + (1 + β)eαβ × Fαα = 0. Furthermore,we can interchange labels of the swimmers to obtain hydrodynamics related to182.5. Interactive momentssurface activity of sphere β by Fαβ(β, eαβ) = Fβα(β−1,−eαβ), and Fββ(β, eαβ) =Fαα(β−1,−eαβ).Fββ6πµ= − Fαβ6πµ= − β15 (1 + β)3[Cβ1(8 + 10β + 2β2) eαβ × pβ+ (2 + 10β + 8β2)(Bβ1 + Bβ2 eαβ · pβ) pβ+(− 15(β + 1)Bβ2 + (58 + 20β − 8β2)Bβ1 eαβ · pβ+ (73 + 35β − 8β2)Bβ2(eαβ · pβ)2)eαβ]log ϵ + O(1, ϵ log ϵ),Tββ8πµ=2β25 (1 + β)[Cβ1(pβ − eαβeαβ · pβ)−(Bβ1 + Bβ2 eαβ · pβ)eαβ × pβ]log ϵ+ O(1, ϵ log ϵ),Tαβ8πµ=β10 (1 + β)[Cβ1(pβ − eαβeαβ · pβ)−(Bβ1 + Bβ2 eαβ · pβ)eαβ × pβ]log ϵ+ O(1, ϵ log ϵ). (2.13)We can see that these forces are weakly singular and O(1) is also important.We may also add the forces on an isolated squirmer that can be calculateddirectly as Fiso = −2/3B1p, and Tiso = C1p. Before O(1) terms become dominant,we gradually shift from near-field to far-field solutions. To simplify the transition,we can only add isolated individual forces to the solution. The total active force onswimmer α and β is Fα = Fαα +Fαβ +Fαiso and Fβ = Fββ +Fβα +Fβiso, respectively.We consider F lubs which contains all forces and torques as Flubs = [Fα,Fβ,Tα,Tβ]T .For suspended swimmers in fluid, the active forces/torques are balanced with thehydrodynamic passive forces/torques due to the force and torque-free conditionsF lubs + FD = 0. The passive forces/torques are linearly related to the translationaland angular velocities of the swimmers through a resistance matrix FD = −RFU ·Uwhere U = [Uα,Uβ,Ωα,Ωβ]T is the rigid body motion. Thus, dynamics of twobuoyant swimmers at close contact under rigid body motion have the form ofU = R−1FU · F lubs . (2.14)In the limit that the distance between the swimmers approaches zero, the genericform of the asymptotic expansion of passive forces are at the lowest order ofO(1/ϵ)and active forces are at the lowest order of O(log ϵ−1). Hence we can see that thelowest order of velocity moments are O(ϵ log ϵ−1) and as swimmers get closer toeach other, their relative motion decreases quasilinearly. At far-field the swim force192.6. Squirming near a wallis F f fs = −RFU ·U s−RFE : E s where U s contains velocities and E s contains strainrates of swimmers due to the surface activity [82]. To transit from these two solu-tions we use a tanh function from ϵ = 0.2 to ϵ = 0.01. For instance, consider twoidentical swimmers swimming towards each other, pβ = −pα = eαβ. The repulsiveswim force due to lubrication is F/6πµ = 3/2(B1 + B2) log ϵ−1 p while the dragforce is FD/6πµ = −1/2ϵ−1 U. This results in a rigid translation of the swimmerof the form U = 3(B1 + B2) ϵ log ϵ−1 p. Figure 2.4a shows the velocity for twoapproaching swimmers with only the first mode B1, where dashed-dotted lines cor-respond to lubrication results, dashed lines show the far-field results of U = 2/3B1pand solid lines show the transition between solutions. In another scenario, whentwo identical swimmers are passing by each other, where pβ = −pα, eαβ · pα = 0,the second mode contributes in the repulsion force F⊥/6πµ = 1/2B2 log ϵ−1 whilethe first mode contributes in the shear force F∥/6πµ = 1/3B1 log ϵ−1 and the torqueTy/8πµ = 1/4B1 log ϵ−1. The torque tries to reorient the swimmers away fromeach other. The passive resistance tensors RLU , and RLΩ are in order of O(log ϵ)which results angular velocity of O(1). This has been demonstrated in fig.2.4b.Note that in far-field solution, mode B1 doesn’t contribute in angular velocity. Ingeneral, modes B1 and C1 have no effect on the flow field in far-field formulation.Figure 2.5a shows the trajectories for a puller (red head), a pusher (blue head) anda neutral squirmer (gray head) moving towards a big obstacle. The dynamics ofsquirmers are dictated by the lubrication at near field. While pullers and neutralsquirmers reorient away from the obstacle, pushers tend to move backward andfall into an orbit around the obstacle. This can be seen in fig.2.5b where the flowdraws the pusher squirmer backward, while the angular velocity for pullers andneutral squirmers reorient the swimmer away from the obstacle. The backwardorbits recently has been reported for strong pushers [121].2.6 Squirming near a wallIt has been shown that the swim force for a swimmer near a wall is Fs = −RwFU ·U s − RwFE : Es yielding dynamics of the form U = −U s − RwFU−1 · RwFE : Es [82].As a result, modes B1 and C1 contribute linearly to the swimmer’s dynamics whilehaving no effect on the flow-field and nearby objects. These forces are shown inA.4. More accurately, to capture the effects of lubrication on the moments of aswimmer near a wall, we set β→ ∞ in F lubs . Then we can retrieve the solution for202.6. Squirming near a wall(a) (b)Figure 2.4: a) Swim force and velocity of two identical converging squirmers. b)Swim torque and angular velocity of two identical passing squirmers. The angularvelocity reorients the swimmers away. Results are for B1 = 1. Thick (thin) linesare corresponded to force (velocity) moments. Dashed-lines and dotted-dashed-lines show far-field and lubrication solutions, respectively, while solid lines showthe transition between two solutions.(a) (b)Figure 2.5: a) Trajectories for a neutral squirmer B2/B1 = 0 (gray head), a pullerB2/B1 = 20 (red head), and a pusher B2/B1 = −20 (blue head), near an obstaclewith β = 20. b) The inner region flow field for different head orientations. Thegreen arrows show transnational and angular velocities of the squirmers. Note thatthe size of arrows do not illustrate the velocity amplitude.212.7. Conclusionthe active force/torque on a swimmer at distance ϵ to a wall.F6πµ=115[2C1e⊥ × p − 2(B1 − B2 e⊥ · p) p+(−15B2 − 58B1 e⊥ · p + 73B2(e⊥ · p)2)e⊥]log ϵ + O(1, ϵ log ϵ),(2.15)T8πµ=25[C1(p − e⊥e⊥ · p) + (B1 − B2 e⊥ · p) e⊥ × p] log ϵ + O(1, ϵ log ϵ),(2.16)where e⊥ is normal to the wall. The equation for the torque shows that modeB1 always contribute in the rotation of the swimmer near a wall, except when theswimmer’s head is towards to or away from the wall. Mode B2 contributes when theswimmer’s head is neither parallel with nor perpendicular to the wall. A symmetryargument could also be used to explain previous conclusions. The force equationdemonstrates that a parallel swimmer near a wall senses an attraction or a repulsionforce only due to mode B2. The translational velocity of a swimmer swimmingtowards the wall is U = 4(B1 + B2) ϵ log ϵ−1 p. Results in fig.2.6 show the slowingdown of a collapsing swimmer and also the angular velocity of a swimmer parallelto the wall due to mode B1. Figure 2.7 shows the trajectory of different swimmersnear a wall for different orientations. Swimmers start from an initial height inthe far-filed region. A neutral squirmer doesn’t see the wall at far-field. At veryclose distances, the squirmer slows down and reorients away from the wall andeventually escapes. A puller squirmer oriented slightly towards the wall, swim ina cyclic motion. In each cycle, the swimmer bounces off the wall and re-orients[134]. A pusher squirmer, depending on the initial orientation, might escape thewall [190], oscillate near the wall [158], collapse into the wall, hover near the wall[208], or swim backward [121].2.7 ConclusionThis chapter deals with the hydrodynamics of two unequal swimmers at close prox-imity by considering the squirming model. A regular perturbation in the velocityand pressure field is employed to satisfy Stokes equations. Lubrication forces ona pair of squirmers with modes B1,B2, and C1 and arbitrary head directions arecalculated. A comparison between our calculations and those in [105] is provided.The transition between near-field and far-field solutions is also argued. In addi-tion, these findings provide additional information about lubrication forces on a222.7. Conclusion(a) (b)Figure 2.6: Velocity and force moments of a swimmer swimming a) towards a walland b) parallel to a wall. The angular velocity reorients the swimmer away fromthe wall. Results are for B1 = 1 and same conditions as in fig.2.4.squirmer near a solid surface. A relatively large size ratio leads to the hydrody-namics of a squirmer near a wall. Final results are presented in an active swimforce which can be incorporated in active Stokesian Dynamics method for the cal-culation of a suspension of active particles. In the near field, the pairwise ad-ditive lubrication forces between active particles are summed whereas in the farfield, many-body hydrodynamic reflections are calculated through inversion of ageometry-dependent mobility tensor (derived in chapter 3). Our asymptotic re-sults are broadly consistent with collective behaviours of active particles reportedin the literature. When approaching another swimmer or a surface, a swimmerslows down and reorients away. This results in the swimmers’ escape or cyclicmovements. The mechanism for strong pushers’ backward motions, which hadpreviously been reported in the literature, is also demonstrated. In the next chapter,we will investigate the hydrodynamics of a suspension of N active particles near asurface.232.7. ConclusionFigure 2.7: Trajectories of pushers (blue head), pullers (red head), and neutral(gray head) squirmers near a wall for different initial orientations χ. Narrow linesthroughout the trajectories show the direction of the swimmers. Results are forinitial height of 5 and B2/B1 = −20, 20, 024Chapter 3Active particles near a wall3.1 IntroductionThe attraction of active particles to surfaces is greatly influenced by hydrodynamicinteractions between active particles and neighbouring surfaces. As a result, vari-ous behaviours of trapping, escaping, and non-trivial steady trajectories above thewall have been experimentally and numerically observed [134]. Accordingly, near-wall hydrodynamics for swimmers must be thoroughly studied in order to giveinsights into the underlying mechanisms.In low Reynolds number flows, thanks to the linear nature of the governingequations, a full rank, symmetric, and positive-definite mobility tensor can lin-early couple the force moments on the particles to their velocity moments throughmultipole expansions of Boundary Integral (BI) equations [125]. Inclusion ofthe fundamental solution of the stokes equations e.g., the Stokeslet, the Stressletand other higher-order singularities in this tensor provides accurate results con-cerning the interactions of a particle with other particles [64, 71, 129] as well aswith the solid boundary using a multiple-reflection technique proposed by Blake[22, 24, 141, 192, 198]. In this technique, images of the proper strength on theexact location but on the opposite side of the wall satisfy the no-slip boundary con-dition on the wall. It should be noted that such interactions only depend on theinstantaneous geometry of the system, including the configuration of the particlesand the relative position of the solid boundary. Once the external forces exertedon the particles are known, the mobility tensor has all the hydrodynamic informa-tion to capture the trajectory of each individual particle in time. However, sincewe truncate the inclusion of multipole singularities in the construction of mobilitytensor, the solution is only valid for far-field interactions. Although the far-fieldapproximation appears to be reasonably accurate, it does not capture the contribu-tions of near-field lubrication, especially when a particle is almost touching anotherparticle or a surface [190]. The solution of two nearly touching spheres has beenstudied for various cases using bispherical coordinates and separation of variables[12, 90, 95, 112, 162, 195]. Lubrication approximations and asymptotic expan-sions are used to study the flow in the narrow gap between the spheres. Anothermethod called twin multipole expansions is developed [49, 116] employing the253.2. Active Stokesian Dynamicsmethod of reflections [40, 95]. Despite the fact that this method is not as accurateas bispherical coordinates method, it is found to be accurate enough to conve-niently be combined with lubrication solutions, that are singular when the distancebetween the pair spheres tends to zero, using matched asymptotic expansions tocover any separation of spheres [115, 117]. This method may be modified to anal-yse the interaction of a sphere and a surface by introducing a reasonably large radiiratio [26]. All these solutions are puzzle pieces for a resistance tensor in whichthe force, torque, and stresslet are related to the motion and the strain rate of thepaired spheres [31, 126]. Similar to the mobility tensor, the resistance tensor isonly geometry dependent, symmetric, and positive-definite. The combination ofboth far-field and near-field solutions provides a strong tool for the calculation ofparticulate fluids.This chapter is organized as follows. In section 3.2, we detail on active Stoke-sian dynamics method for a suspension of N active particles. The combinationof far-field and near-field solutions yields a powerful tool for computing activeparticles in a bounded fluid with zero Reynolds number. Section 3.3 is dedicatedto modelling the far-field hydrodynamic interactions among many active particlesnear a solid wall using the method of reflections in cooperation with boundaryintegral formulation for Stokes flow. Section 3.4 describes a model for the calcula-tion of the lubrication interactions among the spheres and the spheres and the wallwhose inner gap is relatively small. The development of an algorithm to obtainthe trajectory of the active particles near a solid surface is explained in section 3.5.Depending on the distance between the two active particles and also each particleand the wall in each time step, the algorithm uses either far-field or lubricationsolution. The algorithm switches over the solutions smoothly over a predefinedgap distance to prevent sharp turnovers. In section 3.6, we test and verify our al-gorithm by generating same results as presented in [71, 198] for sedimenting threeidentical spheres, and the co-rotation of a doublet of particles above a wall. Finally,section 3.7 is devoted to the study of different types of swimmers, namely pushers,pullers, and neutral squirmers near a flat wall followed by contact models proposedin section 3.9 to prevent swimmers from overlapping.3.2 Active Stokesian DynamicsConsider a system of N active particles in a bounded fluid with background flowof u∞ that are small enough to neglect inertia. The generated disturbance velocityfield by active particles isu′ = u − u∞. (3.1)263.2. Active Stokesian DynamicsThe motion of active particle α can be decomposed into the rigid body motion andthe surface activity of the formu(x ∈ ∂Dα) = U +Ω × r(x) + us(x), (3.2)where ∂Dα is the surface of particle α, r(x) = x − xc and xc is the centre of theparticle, U is the translational velocity, Ω is the rotational velocity, and us is thesurface activity assuming that they are not changing volume. An active spheregenerates swim force by means of surface activity, us, and is called a squirmer[137]. For the case of a squirmer with purely tangential surface activity, the slipvelocity can be described in a spherical coordinate system [23, 165]us = Σn≥1 − 2(Bnn(n + 1)P1n(p · n)eϕ +aCnn(n + 1)P1n(p · n)eθ), (3.3)where the axis of rotation ϕ is aligned with the head direction p, and θ is the polarangle. Also, n is the normal to the surface, P1n is the first associated Legendrepolynomial of degree n, and Bn and Cn are squirming modes. Furthermore, thesurface activity may be written as an expansion in tensorial spherical harmonics(see [82, 102]) of the formus(x ∈ ∂D) = Us + an(x) · Es + aΩs × n(x) + · · · , (3.4)where the zeroth and the first moments take the form ofUs =1S∫∂DusdS (y) = −23B1p,Ωs =12V∫∂Dn × usdS (y) = C1p,Es =12V∫∂D(nus + usn)dS (y) = − 35aB2(pp − 13I), (3.5)where B1 is the free squirming speed in direction of the axis of symmetry p, C1yields axial rotation of the squirmer, and B2 defines the type of the squirmer asdescribed in chapter 2: For a neutral squirmer B2 = 0, while B2 < 0 and B2 > 0define a pusher and a puller, respectively. Note that these modes are axisymmet-ric squirming modes. Here, the B2 mode has information about the propulsionmechanism and the type of the swimmer. At near-filed, all modes can have similarmagnitudes, and higher modes, such as B3, change the flow structure [110]. As aresult, we specify that higher modes are analytically zero for our purposes, Bn = 0for n ≥ 3. Furthermore, this model may raise concerns about the flow arisingfrom the periodic beating of flagella and cilia. We assume that the time-averagedvelocity moments dominates the overall dynamics [221].273.2. Active Stokesian DynamicsThe Stokes equations are governing the dynamics of the flow field. Key prop-erties of the Stokes equations are that of no explicit time dependence and the hy-drodynamic forces on the particles are coupled linearly to the particles’ velocities.Therefore, dynamics of active particles are determined instantaneously by the im-posed boundary conditions. Newton’s second law for particles in Stokes flow statesthat [FL]+[FextLext]=[00], (3.6)which means that the hydrodynamic forces and torques on particles are in equilib-rium with the external applied forces. The technique to calculate the interactionsis based on SD method which divides the hydrodynamic force into far-field (long-ranged) and near field (short-ranged) contributions of the formF = F f f + F n f , (3.7)where F is a 6N-dimensional vector of force F, and torque L combined of N activeparticles and F f f , and F n f are the far-field, and near-field hydrodynamic forces onthe active particles.In order to capture the dynamics of active particles, we need to relate the forcesacting on each active particle to its dynamics. In light of eq.3.2, we may decomposehydrodynamic force/torque into three separate components,F = F∞ + FD + Fs, (3.8)whereF∞ = RFU · U∞ +RFE : E∞ + · · · , (3.9)is the ambient force due to the background flow and RFU , and RFE are resistancetensors relating force and velocity moments. Also, U∞ and E∞ are moments of thebackground flow u∞, andFD = −RFU · U, (3.10)is the drag force due to the rigid-body motion, andFs = −RFU · U s −RFE : E s + · · · , (3.11)is the swim force due to the surface activity of a stationary particle. Also, U is a6N-dimensional vector of translational U and rotational Ω velocities combined of283.3. Far-field hydrodynamicsN active particles. Applied forces on active particles in an quiescent flow are inbalance with hydrodynamic forces, Fext + F = 0, which leads toFext = RFU · (U + U s − U∞) +RFE : (E s − E∞) + · · · , (3.12)thus, dynamics of buoyant active particles under rigid body motion have the formofU = U∞ − U s +R−1FU ·RFE : (E∞ − E s) +R−1FU · Fext + · · · , (3.13)where for force and torque free active particles, Fext = 0. Therefore, the goal is tocompute RFU , and RFE for all N active particles. In general, we may writeF = −R · U, (3.14)where the grand resistance tensor, R linearly maps velocity moments U to forcemoments F . The shorthand notation F consists of force moments, F, L, and Sof all active particles, and U includes the velocity moments, U, Ω, and E of allparticles. The hydrodynamic force and consequently the grand resistance tensor isdivided into far-field (long-ranged) and near-field (short-ranged) contributions ofthe formR = R f f + Rn f , (3.15)which are derived in the next two sections. For near-field contributions, Swanet al. [199] used lubrication results for interactions between near pairs of particlesgiven in [125], whereas we use the more complete results in [114–117] with someadjustments.3.3 Far-field hydrodynamicsIn the interest of Stokesian dynamics, we need to calculate a symmetric mobilitytensor for hydrodynamics of many active particles. We construct such tensor di-rectly from the Boundary Integral equation for a suspension of active particles inan unbounded flow field. Then we extend this study for a bounded flow field usingBlake’s solution [24].293.3. Far-field hydrodynamics3.3.1 Unbounded fluidSolution of a point force of magnitude F in an unbounded fluid in the limit ofStokes regime can be written as suchu(x) = G(x, y) · F,p(x) = GP(x, y) · F,σ(x) = GS (x, y) · F, (3.16)where x is a point within the fluid, and y is the point force. The Oseen tensor, andits pressure and stress fields areG(r) =18πµ( Ir+rrr3),GP(r) =14πrr3,GS (r) =−34πrrrr5, (3.17)where µ is the viscosity of the fluid, r = x − y, and r = |r|. The integral represen-tation of the velocity field due to presence of N active particles in the backgroundflow field of u∞(x) isx ∈ V u(x)x ∈ ∂D u(x)2x < V 0 = u∞(x) −NΣβ∫∂Dβ[G(x, y) · fβ(y) + (uβ(y)n(y)) : GS (x, y)]dS (y),(3.18)which consists of a single-layer and a double-layer potential and provides us witha representation of flow in terms of boundary distributions of Stokeslet G and itsstress field GS [150]. The normal vector n is directed into the control volume Vbounded by the surface ∂D. On the left side of Eq.3.18 we see a jump on the ve-locity field due to the discontinuity of double-layer terms at ∂D [172]. We proceedour formulation further for spherical active particles. The boundary condition onthe surface of the active particle β isuβ(y ∈ ∂Dβ) = Uβ + aβΩβ × n(y) + usβ, (3.19)where Uβ andΩβ are the tranlational and rotational velocities and usβ is the surfaceactivity. We write the force density and the slip velocity on the surface of the303.3. Far-field hydrodynamicsspherical active particle β in terms of an expansion in tensorial spherical harmonicsof the unit normal n as suchfβ(y) =FβS β+n(y) · S˜βVβ+Lβ × n(y)2Vβ+ · · · ,usβ(y) = Usβ + aβn(y) · Esβ + aβΩsβ × n(y) + · · · , (3.20)where S β and Vβ are the surface area and the volume of sphere β andF =∫∂DfdS (y), (3.21)L = a∫∂Dn × fdS (y), (3.22)S˜ = a∫∂D(nf + fn)dS (y), (3.23)Us =1S∫∂DusdS (y), (3.24)Ωs =12V∫∂Dn × usdS (y), (3.25)Es =12V∫∂D(nus + usn)dS (y). (3.26)The Taylor expansion of the Stokeslet and the associated stress field in y about thecentre of the active particle yβ isG(x, y) = G(x, yβ + aβn(y)) = Σn=0anβn!(nn(y) ⊙ ∇ny)G(x, yβ), (3.27)where nn =n times︷︸︸︷nn...n and symbol ⊙ is the proper contraction when tensors and vectorsare multiplied. Recasting nn in terms of the n-adic spherical harmonics yieldsG(x, y) = G(x, yβ + aβn(y)) = Σn=0anβn!(nn(y)⊙∇ny)(1 +a2β6 + 4n∇2y)G(x, yβ),n(y) ·GS (x, y) = (1 +a2β10∇2y)n(y) ·GS (x, yβ)+aβ(1 +a2β14∇2y)(n(y) · ∇y)n(y) ·GS (x, yβ) + · · · . (3.28)We substitute the expanded terms in y back into the BI equation and take the ad-vantage of divergence theorem to get the velocity of point x in presence of N active313.3. Far-field hydrodynamicsparticles.u(x) − u(x)∞ = −NΣβ[(1 +a2β6∇2y)G(x, yβ) · Fβ+ (1 +a2β10∇2y)∇yG(x, yβ) : S˜β+12(1 +a2β10∇2y)∇y ×G(x, yβ) · Lβ+ Vβ(1 +a2β10∇2y)GS (x, yβ) : Esβ + · · ·]. (3.29)We truncate our expansions at the level of the stresslet S˜ and the rate of strain Es.The velocity field over the surface of particle α is thenu(x ∈ ∂Dα) − u∞(x ∈ ∂Dα) = −((1 +a2α6∇2y)G(x, xα) · Fα+ (1 +a2α10∇2y)∇yG(x, xα) : S˜α+12(1 +a2α10∇2y)∇y ×G(x, xα) · Lα+ Vα(1 +a2α10∇2y)GS (x, xα) : Esα + · · ·)−NΣβ,α[(1 +a2β6∇2y)G(x, yβ) · Fβ+ (1 +a2β10∇2y)∇yG(x, yβ) : S˜β+12(1 +a2β10∇2y)∇y ×G(x, yβ) · Lβ+ Vβ(1 +a2β10∇2y)GS (x, yβ) : Esβ + · · ·], (3.30)323.3. Far-field hydrodynamicswhere self derivatives in above equation are(1 +a2α6∇2y)G(x ∈ ∂Dα, xα) =I6πµaα,(1 +a2α10∇2y)∇yG(x ∈ ∂Dα, xα) =3n(x)I20πµa2α,(1 +a2α10∇2y)∇y ×G(x ∈ ∂Dα, xα) =3n(x) · ϵ8πµa2α. (3.31)The velocity filed on the surface of the active particle α in terms of an expansionin tensorial spherical harmonics isu(x ∈ ∂Dα) − u∞(x ∈ ∂Dα) = (Uα + Usα − U∞α) + aα(Ωα +Ωsα −Ω∞α) × n(x)+ αn(x) · (Esα − E∞α) + · · · . (3.32)We Taylor expand the Stokeslet and its derivatives about xα, visG(x ∈ ∂Dα, yβ) = G(xα + aαn(x), yβ) = Σn=0anαn!(nn(x)⊙∇nx)(1 +a2α6 + 4n∇2x)G(xα, yβ).(3.33)One can use the identity belowGS (x, y) = −µ(∇yG(x, y) + ∇yG(x, y)T ) − (x − y)I4π|x − y|5 , (3.34)to absorb the double-layer contribution into the stresslet for an active particle ofthe formSβ = S˜β − 2µVβEsβ. (3.35)Substituting the expanded terms and the new stresslet into eq.3.30 lead to the cal-culation of dynamics of particle α of the form(Uα+Usα − U∞α) + aα(Ωα +Ωsα −Ω∞α) × n(x) + αn(x) · (Esα − E∞α) =− I6πµaα· Fα + n(x)8πµa2α× Lα − 3In(x)20πµa2α: Sα−NΣβ,α[(1 +a2β6∇2y)((1 +a2α6∇2x) + aα(1 +a2α10∇2x)n(x) · ∇x)G(xα, yβ) · Fβ+12(1 +a2β10∇2y)((1 +a2α6∇2x) + aα(1 +a2α10∇2x)n(x) · ∇x)∇y ×G(xα, yβ) · Lβ+ (1 +a2β10∇2y)((1 +a2α6∇2x) + aα(1 +a2α10∇2x)n(x) · ∇x)∇yG(xα, yβ) : Sβ].(3.36)333.3. Far-field hydrodynamicsSimilarly, dynamics of other active particles (here denoted by β) can be derived.The mobility formulation for N active particles isUα+Usα−U∞αUβ+Usβ−U∞β...Ωα+Ωsα−Ω∞αΩβ+Ωsβ−Ω∞β...Esα−E∞αEsβ−E∞β...=MααUF MαβUF ··· MααΩF MαβΩF ··· MααEF MαβEF ···MβαUF MββUF ··· MβαΩF MββΩF ··· MβαEF MββEF ···....... . ........ . ........ . .MααUL MαβUL ··· MααΩL MαβΩL ··· MααEL MαβEL ···MβαUL MββUL ··· MβαΩL MββΩL ··· MβαEL MββEL ···....... . ........ . ........ . .MααUS MαβUS ··· MααΩS MαβΩS ··· MααES MαβES ···MβαUS MββUS ··· MβαΩS MββΩS ··· MβαES MββES ···....... . ........ . ........ . .⊙FαFβ...LαLβ...SαSβ.... (3.37)In general, we can writeU +Us −U∞ = −M · F , (3.38)where M is the grand mobility tensor that relates all force moments and velocitymoments of all active particles with two major properties: being symmetric andpositive-definite. In the shorthand notation F consists of hydrodynamic forces F,torques L and stresslets S of all active particles. Moreover, U consists of the rigidbody translation U, rotation Ω and strain rates E of all the particles. In this anal-ysis, the active particles are considered to be rigid spheres r = an with constantradius a and surface normal n and as a result E = 0. Likewise,Us contains velocitymoments due to the surface activity. Finally, U∞ contains the velocity moments ofthe ambient flow u∞ at the centre of each particle. In case of a linear backgroundflow,Ω∞ and E∞ are constants spatially but can change with time. We can observethat activity only affects the velocity and force moments, while M remains un-changed and same as the grand mobility tensor in SD method. For completeness,the truncated M f f is given here [71, 186]. We can match these mobilities from theright hand side of eq.3.36 since these tensorial series are irreducible and orthogo-nal. More details on the projection (by integration onto the tensorial harmonics)can be found in B.3. Finally, the mobility coefficients are calculated asMααUF =I6πµaα,MααΩL =I8πµa3α,MααES =3I20πµa3α,343.3. Far-field hydrodynamicsMαβUF =(1 +a2β6∇2y)(1 +a2α6∇2x)G(xα, yβ),MαβUL =12(1 +a2α6∇2x)∇y ×G(xα, yβ),MαβUS =(1 +a2β10∇2y)(1 +a2α6∇2x)Y(xα, yβ),MαβΩF =12(1 +a2β6∇2y)∇x ×G(xα, yβ),MαβΩL =14∇y × ∇x ×G(xα, yβ),MαβΩS =12(1 +a2β10∇2y)∇x × Y(xα, yβ),MαβEF =(1 +a2β6∇2y)(1 +a2α10∇2x)T(xα, yβ),MαβEL =12(1 +a2α10∇2x)∇y × T(xα, yβ),MαβES =(1 +a2β10∇2y)(1 +a2α10∇2x)TY(xα, yβ), (3.39)whereIi jkl = δilδ jk,Y(x, y) =12(∇yG + ∇yGT ),T(x, y) =12(∇xG + ∇xGT ),TY(x, y) =12(∇xY + ∇xYT ). (3.40)Later explained we use an equivalent tensor I asIi jkl = 12(δilδ jk + δikδ jl −23δi jδkl), (3.41)to fit M f f into a 2-dimensional diagonal matrix. The rest of mobilities, Mβα,can be computed directly by taking a transposition. So far we have calculated thedynamics of a suspension of active particle in an unbounded flow. Next to study, isthe effect of a solid boundary on the hydrodynamics of active particles.353.3. Far-field hydrodynamicsrer<latexit sha1_base64="pFaqFnm+PU0bdDsGUD5XzVkTqk0=">AAAB83icbVBNS8NAEJ34WetX1aOX [binary]</latexit>ReR<latexit sha1_base64="y+THHNQWqwJypCuwA2Ivs8AtsSU=">AAAB83icbVBNS8NAEN34WetX1aOX [binary]</latexit>h↵<latexit sha1_base64="fCUfaZ6DGDuu02/8U8 u3HB2whwU=">[binary]</latexit>h<latexit sha1_base64="PqV6NyPlksTRcRd43Vm7JzM3TnY=">AAAB8HicbVBNS8NAEJ3Ur1q/qh69 [binary]</latexit>x↵<latexit sha1_base64="sj68VrcmXwtjZ8N9+soZ6M6km6U=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXo [binary]</latexit>y0↵<latexit sha1_base64="FtahSVDmkm25CX6zrPRYEDRUjNY=">AAAB+XicbVBNS8NAEN34WetX1KOX [binary]</latexit>y0<latexit sha1_base64="FwTRxxHjwyHv8SS1irnjBgt0yqk=">AAAB+HicbVDLSsNAFJ3UV62PRl26 [binary]</latexit>y<latexit sha1_base64="Ycxhl5dXJCnOm+Jbi2UkGRLwq/k=">AAAB9XicbVBNS8NAEN34WetX1aOX [binary]</latexit>O<latexit sha1_base64="0U0GBEJkwIwjbffHLZ w8sbukW6c=">[binary]</latexit>H<latexit sha1_base64="FGmZKUrH8v0GgMBjV4 wk868lgtY=">[binary]</latexit>e?<latexit sha1_base64="uWqqTNuDKA379eVgLPp8D5U54UE=">AAAB9HicbZBNS8NAEIYnftb6VfXo [binary]</latexit>2h↵e?<latexit sha1_base64="c/hD24gBIPeXFXBzbF 4QlUpzy/8=">[binary] los=</latexit>Figure 3.1: Active particles above a stationary wall.3.3.2 Bounded fluidThe hydrodynamics of active particles near a stationary wall can be similarly con-structed using Blake’s solution for the image of Stokeslet above the wall [22, 24].The solution is an image flow under the wall that satisfies the Stokes equations andcancels the flow owing to the point force, exactly on the wall. For a Stokeslet or aStresslet near a wall, a finite number of reflections is required at the image positionto satisfy the wall’s boundary condition. Figure 3.1 shows two particles, α and β,above a wall located at height −H in Stokes flow with the normal vector e⊥ and theparallel vector e∥. The images of the fundamental solutions of the Stokes equationsabove the wall are as followsGw(x, y,H) = −G(x, y′) + 2(y⊥ − H)(P · ∇y′G(x, y′) · e⊥)T+ (y⊥ − H)2∇2y′G(x, y′) · P,GPw(x, y,H) = −GP(x, y′) + 2(y⊥ − H)P · ∇y′GP(x, y′) · e⊥,GS w(x, y,H) = −GS (x, y′) + 2(y⊥ − H)(P · ∇y′GS (x, y′) · e⊥)T+ (y⊥ − H)2∇2y′GS (x, y′) · P, (3.42)which are the images of Oseen tensor, its pressure, and stress fields, respectively.Also, P = I − 2e⊥e⊥, y′ = y − 2y⊥e⊥, and y⊥ = e⊥ · y. The superscript T denotestransposition. Now the total Oseen tensor and its stress field take the form ofGtotal(x, y,H) = G(x, y) + Gw(x, y,H),GS total(x, y,H) = GS (x, y) + GS w(x, y,H), (3.43)363.3. Far-field hydrodynamicswhere can be used in BI equation to obtain the wall hydrodynamics.u′(x) = −NΣβ[(1 +a2β6∇2y)Gtotal(x, yβ,H) · Fβ+ (1 +a2β10∇2y)∇yGtotal(x, yβ,H) : S˜β+12(1 +a2β10∇2y)∇y ×Gtotal(x, yβ,H) · Lβ+ Vβ(1 +a2β10∇2y)GStotal(x, yβ,H) : Esβ + · · ·]. (3.44)Note that y-derivatives of Blake’s solution for the wall induced velocity field isquite complex and not trivial as f is distributed on the surface rather than a centrepoint [197, 198]. Derivatives of Gw are given in the appendix B.1.Each component in the mobility tensor is decomposed of force multipoles inthe disturbance velocity field. The dynamics of particle α is a superposition ofself interaction, Mαα, self-wall interaction Mαα′ , the interaction with the adjacentparticles Mαβ, and the interaction with the wall Mαβ′ . As an example, MUF isconstructed asMααUF =I6πµaαMαα′UF =(1 +a2α6∇2y)(1 +a2α6∇2x)Gw(xα, xα, hα)MαβUF =(1 +a2β6∇2y)(1 +a2α6∇2x)G(xα, yβ)Mαβ′UF =(1 +a2β6∇2y)(1 +a2α6∇2x)Gw(xα, yβ, hβ) (3.45)Full expressions for non-dimensional mobilities, using 6πµanα, and λ = aα/aβ, aresummarized in the appendix B.1. However, as a consequence of truncating theexpansions after the stresslet and the strain-rate, the mobility tensor is only a far-field approximation and incapable of capturing the near-field interactions. In thenext section, we look into the near-field hydrodynamics.373.4. Near-field hydrodynamics3.4 Near-field hydrodynamicsIn this section we start with derivation of the grand resistance tensor for two passiveparticles that are at an arbitrary distance from one another in an unbounded fluid.The calculated resistance tensor is also valid for active particles whose volumeis preserved through the description explained in section 3.2. In order to obtainresults for arbitrary spacing, Jeffrey and Onishi [114, 115, 117] have consideredtwo regions of interest: Widely separated, and nearly touching spheres. They haveused a method of reflections which uses Faxe´n laws that are exact mathematicalexpressions about the response of a particle to an ambient flow, u∞(x). Theseexpressions cannot tell the difference between an actual ambient flow and the dis-turbance caused by another particle. This is the base for calculation of Stokes flowin a suspension using the reflection method. Then the equations are expressed asa series expansion in the separation distance of the spheres using the method oftwin multipole expansions for widely separated spheres while for nearly touchingspheres, lubrication theory [48, 114] has been employed. Combining these resultsproperly and using matched asymptotic expansions, give expressions valid for anyseparations. Then we use asymptotic methods for when the particles are close totouching to find the near-field approximation including order unity corrections tothe expressions. In the limit of one sphere near a fairly large fixed sphere, we findthe relations for the interactions of a sphere close to a wall.In the case that there are more than two particles, the force F or the torque Lexerted on the particle is obtained by summing the contribution of, on one hand,the self and the individual pair interactions and on the other hand, sphere-wallinteraction. For example, the force on particle α isF α =N∑iF α,i + F α,w, (3.46)where F α,i is associated with the force/torque on particle α due to its interactionwith particle i, and F α,w is associated with its interaction with the wall. Further-more, the near-field hydrodynamics are constructed by pairwise additive approxi-mations. In following sections, we study the interaction between two spheres andthen between a sphere and a wall, respectively.3.4.1 Pairwise particle–particle interactionThe general formulation of resistance relations between two particles of α and β,where the spheres’ size ratio is λ = aα/aβ has the form ofF = −Rpp · U. (3.47)383.4. Near-field hydrodynamicsFurther on, we absorb the effect of the surface activity and the ambient field intoU. Also, the grand resistance tensor Rpp containsRpp =RααFU RαβFU RααFΩ RαβFΩ RααFE RαβFERβαFU RββFU RβαFΩ RββFΩ RβαFE RββFERααLU RαβLU RααLΩ RαβLΩ RααLE RαβLERβαLU RββLU RβαLΩ RββLΩ RβαLE RββLERααSU RαβSU RααSΩ RαβSΩ RααSE RαβSERβαSU RββSU RβαSΩ RββSΩ RβαSE RββSE, (3.48)or in compact form FS = −RFU RFERSU RSE ⊙UE . (3.49)The grand resistance tensor contains second-rank tensors RFU , third-rank tensorsRFE and RSU , and forth-rank tensors RSE . Following Lorenz reciprocal theorem[140], the symmetry relations for the components areRαβFUi j = RβαFU ji,RαβLΩi j = RβαLΩ ji,RαβSEi jkl = RβαSEkli j,RαβLUi j = RβαFΩ ji,RαβSUi jk = RβαFEki j,RαβSΩi jk = RβαLEki j, (3.50)which can be applied to particles with arbitrary shapes. Besides, for two sphereswhose geometries can be specified by a separation distance of r and the size ratioλ, an arbitrary tensor R in R follows these symmetrical rulesRαβ(r, λ) = Rβα(−r, λ−1),Rαα(r, λ) = Rββ(−r, λ−1). (3.51)The components in the resistance tensor for two-sphere-geometry are axisymmetricabout r = rer and consequently can be written as a function of scalar resistance393.4. Near-field hydrodynamicsfunctions X,Y,Z and directional vectors along the axis er [29, 30, 117], visRαβFUi j = XαβFUd1i j + YαβFUd2i j,RαβLUi j = YαβLUd3i j,RαβLΩi j = XαβLΩd1i j + YαβLΩd2i j,RαβSUi jk = XαβSUd4i j + YαβSUd5i j,RαβSΩi jk = YαβSΩd6i jk,RαβSEi jkl = XαβSEd7i jkl + YαβSEd8i jkl + ZαβSEd9i jkl, (3.52)where the geometrical tensors di(er) are given in appendix B.2, and X,Y,Z-functionsare associated with flows axisymmetric about, transverse to, and perpendicular tothe line of centres er, respectively. These scalar functions depend on two variables:centre to centre separation r, and the size ratio of the pair spheres λ, are tabulatedin [115, 117] for widely separated spheres (in power series of r), nearly touchingspheres (up to O(ϵ ln ϵ)), and arbitrary separated spheres.In this study we have used a¯ as a characteristic length scale and have used6πµa¯n to non-dimensionalize moments and have adopted the equations for ournotation. In the case of interaction between two particles of α and β, we havea¯ = (aα + aβ)/2 while we have used a¯ = aα (a¯ = aβ) for the self interaction ofparticle α (β). Therefore, the physical dimensions of the individual resistance ten-sors in R are not all the same. For the sake of consistency, we adopted and keptthe formulation for scalar functions in this paper in a notably compact form: Wedefine length ψ where 0 ≤ ψ ≤ 1 for 1 + λ ≤ r ≤ ∞, and parameter Λ asψ2 = 1 − (1 + λ)2r−2, Λ = 1 + λ−1. (3.53)Formulations are derived and discussed individually through out two papers byJeffrey [115], Jeffrey and Onishi [117] for each function. We detected a numberof typographical errors in those formulations as well as omissions of equationsrequired to generate values for different λ independently. We also could derivea universal equation covering all the scalar functions to save space which fixesthe typographical errors in the original papers and serves as a reference. These403.4. Near-field hydrodynamicsfunctions for self and paired interactions areXααFU = CXFUe , XαβFU = −2Λ−1CXFUo , YααFU = CYFUe , YαβFU = −2Λ−1CYFUo ,YααLU = −23CYLUo , YαβLU =83Λ−2CYLUe ,XααLΩ =43CXLΩe , XαβLΩ = −323Λ−3CXLΩo , YααLΩ =43CYLΩe , YαβLΩ = −323Λ−3CYLΩo ,XααSU = −23CXSUo , XαβSU =83Λ−2CXSUe , YααSU = −23CYSUo , YαβSU =83Λ−2CYSUe ,YααSΩ =43CYSΩe , YαβSΩ =323Λ−3CYSΩo ,XααSE =109CXSEe , XαβSE =809Λ−3CXSEo , YααSE =109CYSEe , YαβSE =809Λ−3CYSEo ,ZααSE =109CZSEe , ZαβSE = −809Λ−3CZSEo ,(3.54)where these scalar functions for arbitrary separation between two spheres, throughcareful rearrangement, areCXe = f X0 + gX1 + gX21ψ2+ gX3√1 − ψ2 ln1 + √1 − ψ21 − √1 − ψ2+(gX4 + gX5ψ21 − ψ2 + gX6ψ2)ln(ψ2)+M→∞∑m=1(1 − ψ2)m(f X2m + gX1 − gX5 −gX32m2 − m +gX4 − gX3m− gX6(1 − δm1)m2 − m +gX7m2 + m),CXo =(gX81ψ2+ gX10 ln(ψ2) − 2gX1111 − ψ2 + gX12) √1 − ψ2+(gX9 + gX11ψ21 − ψ2 + gX13ψ2)ln(1 +√1 − ψ21 − √1 − ψ2 )+M→∞∑m=1(1 − ψ2)m− 12 ( f X2m−1 − gX8 + gX10 − gX9m − 12 +gX102m2 + m(2δm1 − 3) + 1 − δm1+gX11m2 − 14+gX13m2 − 2m + 34). (3.55)413.4. Near-field hydrodynamicsIn above, Ce (Co) contains a series only of terms in which k in f Xk is even (odd).Functions f Xk are series in λ−1 and along with other coefficients are shown in ap-pendix B.2.In the limit that the distance between the particles approaches zero, the genericform of the asymptotic expansion of these functions is A1ϵ−1 + A2 ln(ϵ−1) + A3 +A4ϵ ln(ϵ−1) +O(ϵ) where ϵ = Λ(r − 2)/2 is the dimensionless distance between theparticles’ surfaces. Note that to non-dimensionalize r, and ϵ, the lengths (aα+aβ)/2and aα are used, respectively. More specifically, we have relations below that canbe plugged in eq.3.54 to calculate the asymptotic expressions which are only validwhen the particles are nearly touching, ϵ ≤ 0.1. Scalar functions for nearly touch-ing unequal spheres areXααFU =1(λ + 1)2ϵ−1 +λ2 + 7λ + 15(λ + 1)3ln ϵ−1 + O(1)+λ4 + 18λ3 − 29λ2 + 18λ + 121(λ + 1)4ϵ ln ϵ−1,XαβFU = −2λ(λ + 1)3ϵ−1 −2λ(λ2 + 7λ + 1)5(λ + 1)4ln ϵ−1 + O(1)−2λ(λ4 + 18λ3 − 29λ2 + 18λ + 1)21(λ + 1)5ϵ ln ϵ−1,YααFU =4(2λ2 + λ + 2)15(λ + 1)3ln ϵ−1 + O(1)+4(λ − 1)2(16λ2 − 13λ + 16)375(λ + 1)4ϵ ln ϵ−1,YαβFU = −8λ(2λ2 + λ + 2)15(λ + 1)4ln ϵ−1 + O(1)−8(λ − 1)2λ(16λ2 − 13λ + 16)375(λ + 1)5ϵ ln ϵ−1,YααLU =2(1 + 4λ)15(λ + 1)2ln ϵ−1 + O(1)+2(32λ3 − 33λ2 + 83λ + 43)375(λ + 1)3ϵ ln ϵ−1,YαβLU = −8λ2(1 + 4λ)15(λ + 1)4ln ϵ−1 + O(1)423.4. Near-field hydrodynamics−8λ2(32λ3 − 33λ2 + 83λ + 43)375(λ + 1)5ϵ ln ϵ−1,XααLΩ = O(1) −23(λ + 1)2ϵ ln ϵ−1,XαβLΩ = O(1) +16λ33(λ + 1)5ϵ ln ϵ−1,YααLΩ =815(λ + 1)ln ϵ−1 + O(1) +8(8λ2 + 6λ + 33)375(λ + 1)2ϵ ln ϵ−1,YαβLΩ =16λ215(λ + 1)4ln ϵ−1 + O(1) +16λ2(43λ2 − 24λ + 43)375(λ + 1)5ϵ ln ϵ−1,XααSU = −1(λ + 1)2ϵ−1 − λ2 + 12λ − 45(λ + 1)3ln ϵ−1 + O(1)− 5λ4 + 181λ3 − 453λ2 + 566λ − 65105(λ + 1)4ϵ ln ϵ−1,XαβSU =4λ2(λ + 1)4ϵ−1 +4λ2(λ2 + 12λ − 4)5(λ + 1)5ln ϵ−1 + O(1)+4λ2(5λ4 + 181λ3 − 453λ2 + 566λ − 65)105(λ + 1)6ϵ ln ϵ−1,YααSU = −4λ2 − λ + 715(λ + 1)3ln ϵ−1 + O(1)− 32λ4 − 179λ3 + 532λ2 − 356λ + 221375(λ + 1)4ϵ ln ϵ−1,YαβSU =4λ2(4λ2 − λ + 7)15(λ + 1)5ln ϵ−1 + O(1)+4λ2(32λ4 − 179λ3 + 532λ2 − 356β + 221)375(λ + 1)6ϵ ln ϵ−1,YααSΩ =4λ − 215(λ + 1)2ln ϵ−1 + O(1)+2(16λ3 − 61λ2 + 180λ + 2)375(λ + 1)3ϵ ln ϵ−1,YαβSΩ =8λ2(λ + 7)15(λ + 1)5ln ϵ−1 + O(1)433.4. Near-field hydrodynamics+8λ2(43λ3 + 147λ2 − 185λ + 221)375(λ + 1)6ϵ ln ϵ−1,XααSE =23(λ + 1)2ϵ−1 +2(λ2 + 17λ − 9)15(λ + 1)3ln ϵ−1 + O(1)+2(5λ4 + 272λ3 − 831λ2 + 1322λ − 415)315(λ + 1)4ϵ ln ϵ−1,XαβSE =16λ23(λ + 1)5ϵ−1 −16λ2(4λ2 − 17λ + 4)15(λ + 1)6ln ϵ−1 + O(1)−16λ2(65λ4 − 832λ3 + 1041λ2 − 832λ + 65)315(λ + 1)7ϵ ln ϵ−1,YααSE =4(λ2 − λ + 4)15(λ + 1)3ln ϵ−1 + O(1)+4(8λ4 − 67λ3 + 274λ2 − 394λ + 197)375(λ + 1)4ϵ ln ϵ−1,YαβSE =8λ2(7λ2 − 10λ + 7)15(λ + 1)6ln ϵ−1 + O(1)+8λ2(221λ4 − 748λ3 + 1902λ2 − 748λ + 221)375(λ + 1)7ϵ ln ϵ−1,ZααSE = O(1) −2(λ2 + 1)3(λ + 1)4ϵ ln ϵ−1,ZαβSE = O(1) +16λ3(λ2 + 1)3(λ + 1)7ϵ ln ϵ−1, (3.56)where the order unity correction O(1) for each scalar function is a function of λ,independent of ϵ and can be evaluated numerically by calculating the series forhundreds of terms to get accuracy of O(ϵ). Otherwise, for a not touching regime,the series in the eq.3.55 can be truncated after few orders.O(1)(for CXe ) = f X0 + gX1 +34gX2 + gX3 ln (Λ) − gX4 ln (Λ/2)+M→∞∑m=1(f X2m + gX1 − gX5 −gX32m2 − m +gX4 − gX3m− gX6(1 − δm1)m2 − m +gX7m2 + m),443.4. Near-field hydrodynamicsO(1)(for CXo ) =14gX8 + gX9 ln (Λ) − gX10 ln (Λ/2) − 2gX11 + gX12+M→∞∑m=1(f X2m−1 − gX8 +gX10 − gX9m − 12+gX102m2 + m(2δm1 − 3) + 1 − δm1+gX11m2 − 14+gX13m2 − 2m + 34). (3.57)Figure 3.2a shows the response of the order unity correction to up to 300 terms inthe series, for XααFU as a function of 1/M for λ = 1. Then, the order unity correctioncan be calculated as explained using a linear extrapolation to M → ∞ shown by adotted line in fig. 3.2b.0 0.02 0.04 0.06 0.08 0.10.9920.9930.9940.9950.9960.9970.9980.99911.001(a)0 1 2 3 4 5 610-30.9953450.995350.9953550.995360.9953650.995370.9953750.995380.995385(b)Figure 3.2: a) The order O(1) correction for XααFU plotted against 1/M as M → ∞for λ = 1 demonstrated by dot marks for up to M = 300 and dashed-dotted line forup to M = 500. b) A closer look at M−1 → 0. Solid, dashed, and dotted lines areMR, DS, and linear fits to the series data for M = 300, respectively.However, for some scalar functions, the linear extrapolation cannot providesufficiently good accuracy and we need other means to estimate the value of theseseries. The series can be written asM→∞∑m=0Cm(λ)(ϵλ1 + λ+ 1)−m (3.58)where for nearly touching spheres it reduces to∑∞m=0 Cm. As m increases, thesecoefficients converge to a linear asymptote. Fig. 3.3a shows the trend of thesecoefficients as they settle into zero for λ = 1. In order to accurately estimate the453.4. Near-field hydrodynamicsradius of convergence, we use two techniques. Domb-Sykes plot can estimate theradius of convergence when the coefficients share a common sign or alternatingsigns [63, 72, 183]. The convergence ratio isrDSm =CmCm−1, (3.59)where the reciprocal radius of convergence is 1/r = limm→∞ rDSm and can be esti-mated by plotting rDSm versus 1/M and linearly extrapolate to 1/M = 0. Fig. 3.3bshows the Domb-Sykes plot, nonetheless the coefficients have a more complexpattern and this approach fails. Mercer-Roberts [147] estimator can accuratelyestimate the radius of convergence when the pattern of signs is not simple. Theconvergence ratio in this method has the form ofrMRm =Cm+1Cm−1 −C2mCm−2Cm −C2m−1. (3.60)Then the coefficients can be calculated using the linear fit belowrMRm =1r(1 − 1 + am)(3.61)where the radius of convergence 1/r = limm→∞ rMRm can be estimated by plottingthe rMRm versus 1/M, and extrapolating to 1/M = 0 via a linear fit. Simultaneously,the slope gives the parameter a in above equation. Fig. 3.3c shows the extrapo-lation of Mercer-Roberts estimator. Having the coefficients, we can estimate theorder one correction as shown in fig. 3.2b. The solid (dashed) line shows the esti-mation using Mercer-Roberts (Domb-Sykes) estimator while the dotted-line showsa graphically linear extrapolation. We also continued calculating the series up toM = 500 and plotted the results using dashed-dotted lines in fig. 3.2b. Resultsprove the agreement of Mercer-Roberts estimator with series data.463.4. Near-field hydrodynamics0 0.01 0.02 0.03 0.04 0.05-1-0.500.511.5 10-3(a)0 0.01 0.02 0.03 0.04 0.05-505101520(b)0 0.01 0.02 0.03 0.04 0.050.30.40.50.60.70.80.91(c)Figure 3.3: Coefficients Cm, Domb–Sykes rDSm , and Mercer-Roberts rMRm plots forthe series in the scalar function XααFU as a function of M−1. The solid lines in (b) and(c) are linear asymptotes, which intercepts the vertical axis at 1/r and has a slopeof a.The convergence of O(1) contributions for different scalar functions for twoequal spheres, λ = 1 are displayed in fig. 3.4 for M = 500 terms and are tabulatedin table 3.1. In each case, we use a proper method of extrapolation depending onthe pattern of coefficients.XααFU YααFU YααLU XααLΩ YααLΩ XααSU0.9954 0.9983 -0.1593 1.4024 0.9371 0.3128YααSU YααSΩ XααSE YααSE ZααSE0.0940 -0.0988 0.7970 0.9829 1.1390XαβFU YαβFU YαβLU XαβLΩ YαβLΩ XαβSU-0.3502 -0.2737 0.0011 -0.2003 -0.0366 -0.1303YαβSU YαβSΩ XαβSE YαβSE ZαβSE-0.0683 -0.0391 -0.1622 -0.2307 -0.0803Table 3.1: Order unity corrections of scalar functions for two nearly touching equalspheres, λ = 1.Now that we have the solution for two unequal spheres, we can derive theasymptotic lubrications for sphere-wall interactions by taking one of the radii toinfinity.473.4. Near-field hydrodynamics0 0.002 0.004 0.006 0.008 0.010.995320.995340.995360.995380.99540.995420 0.002 0.004 0.006 0.008 0.01-0.35026-0.35024-0.35022-0.3502-0.35018-0.35016-0.350140 0.002 0.004 0.006 0.008 0.010.998310.9983150.998320.9983250.998330.9983350.998340.9983450 0.002 0.004 0.006 0.008 0.01-0.273685-0.27368-0.273675-0.27367-0.273665-0.27366-0.273655-0.273650 0.002 0.004 0.006 0.008 0.01-0.15935-0.159345-0.15934-0.159335-0.15933-0.159325-0.15932-0.159315-0.159310 0.002 0.004 0.006 0.008 0.011.11.111.121.131.141.15 10-30 0.002 0.004 0.006 0.008 0.011.40241.402411.402421.402431.402441.402451.402460 0.002 0.004 0.006 0.008 0.01-0.20041-0.2004-0.20039-0.20038-0.20037-0.20036-0.20035-0.200340 0.002 0.004 0.006 0.008 0.010.9370.937020.937040.937060.937080.93710.937120.937140 0.002 0.004 0.006 0.008 0.01-0.03666-0.03665-0.03664-0.03663-0.03662-0.03661-0.0366-0.03659-0.036580 0.002 0.004 0.006 0.008 0.010.31260.312650.31270.312750.31280.312850.31290.312950 0.002 0.004 0.006 0.008 0.01-0.1305-0.13045-0.1304-0.13035-0.1303-0.13025-0.1302-0.130150 0.002 0.004 0.006 0.008 0.010.093980.0940.094020.094040.094060.094080.09410.094120 0.002 0.004 0.006 0.008 0.01-0.06834-0.06832-0.0683-0.06828-0.06826-0.06824-0.068220 0.002 0.004 0.006 0.008 0.01-0.09882-0.0988-0.09878-0.09876-0.09874-0.09872-0.0987-0.098680 0.002 0.004 0.006 0.008 0.01-0.03918-0.03916-0.03914-0.03912-0.0391-0.03908-0.03906-0.039040 0.002 0.004 0.006 0.008 0.010.79640.79660.79680.7970.79720.79740 0.002 0.004 0.006 0.008 0.01-0.1626-0.1624-0.1622-0.162-0.1618-0.1616-0.16140 0.002 0.004 0.006 0.008 0.010.98260.98270.98280.98290.9830.98310.98320.98330 0.002 0.004 0.006 0.008 0.01-0.2312-0.2311-0.231-0.2309-0.2308-0.2307-0.2306-0.23050 0.002 0.004 0.006 0.008 0.011.138941.138951.138961.138971.138981.138991.1391.139011.139020 0.002 0.004 0.006 0.008 0.01-0.08037-0.08036-0.08035-0.08034-0.08033-0.08032-0.08031-0.0803-0.08029Figure 3.4: The O(1) contributions for different scalar functions plotted against1/M as M → ∞ for λ = 1 demonstrated by dot marks. Solid, dashed, and dottedlines are MR, DS, and linear fits to the series data, respectively.483.4. Near-field hydrodynamics3.4.2 Pairwise particle–wall interactionSupposing λ = aα/aβ approaches zero, while the gap distance between the surfacesϵ remains small and finite, the results for the general formulation of resistancerelations between two unequal spheres become those for a particle (namely α) nextto a wall. In general, the resistance relation has the form ofF pw = −Rpw · U, (3.62)where Rpw containsRpw =RαwFU RαwFΩ RαwFERαwLU RαwLΩ RαwLERαwSU RαwSΩ RαwSE. (3.63)The scalar functions in the limit of λ→ 0 and ϵ → 0 areXαwFU = ϵ−1 +15ln ϵ−1 + O(1) + 121ϵ ln ϵ−1,YαwFU =815ln ϵ−1 + O(1) + 64375ϵ ln ϵ−1,YαwLU =215ln ϵ−1 + O(1) + 86375ϵ ln ϵ−1,XαwLΩ = O(1) −23ϵ ln ϵ−1,YαwLΩ =815ln ϵ−1 + O(1) + 88125ϵ ln ϵ−1,XαwSU = −ϵ−1 +45ln ϵ−1 + O(1) + 1321ϵ ln ϵ−1,YαwSU = −715ln ϵ−1 + O(1) − 221375ϵ ln ϵ−1,YαwSΩ = −215ln ϵ−1 + O(1) + 4375ϵ ln ϵ−1,XαwSE =23ϵ−1 − 65ln ϵ−1 + O(1) − 16663ϵ ln ϵ−1,YαwSE =1615ln ϵ−1 + O(1) + 788375ϵ ln ϵ−1,ZαwSE = O(1) −23ϵ ln ϵ−1, (3.64)with order unities calculated for 500 terms and λ = 0.005 which are tabulated intable 3.2. Figure 3.5 shows the convergence of these series for couple of small λalong with linear fits to extrapolate the O(1) contributions.493.4. Near-field hydrodynamics0 0.005 0.01 0.015 0.020.90.920.940.960.9810 0.005 0.01 0.015 0.020.880.90.920.940.960.980 0.005 0.01 0.015 0.02-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.20 0.005 0.01 0.015 0.021.522.533.540 0.005 0.01 0.015 0.02-1-0.500.50 0.005 0.01 0.015 0.02-1.5-1-0.500.50 0.005 0.01 0.015 0.020.40.60.811.21.41.61.820 0.005 0.01 0.015 0.02-0.2-0.15-0.1-0.0500.050.10.150 0.005 0.01 0.015 0.0202468100 0.005 0.01 0.015 0.02-6-5-4-3-2-100 0.005 0.01 0.015 0.0211.522.533.54Figure 3.5: The O(1) contributions for different scalar functions plotted against1/M as M → ∞ for λ = 0.1, 0.01, 0.0067, 0.005 showed by dots, stars, circles, andtriangles, respectively. Dotted lines are linear fits to the series data.503.5. AlgorithmXαwFU YαwFU YαwLU XαwLΩ YαwLΩ XαwSU YαwSU YαwSΩ XαwSE YαwSE ZααSE0.97 0.95 -0.26 1.61 0.49 -0.30 0.61 0.12 2.62 -0.60 1.40Table 3.2: Order unity corrections of scalar functions for a sphere near a wall λ→ 0For a wall with normal e⊥, we substitute er = e⊥ into eq. 3.52 which givesRαwFU33 = XαwFU, RαwFUii = YαwFU,RαwLUi j = YαwLUϵi j3,RαwLΩ33 = XαwLΩ, RαwLΩii = YαwLΩ,RαwSU333 =23XαwSU , RαwSUii3 = −13XαwSU , RαwSU3ii = RαwSUi3i = YαwSU ,RαwSΩ3i j = RαwSΩi3 j = YαwSΩϵi j3,RαwSE3333 =23XαwSE + ZαwSE , RαwSE ii33 = RαwSE33ii = −13XαwSE ,RαwSE3ii3 = RαwSE i33i = RαwSE3i3i = RαwSE i3i3 =12YαwSE ,RαwSE ii j j =16XαwSE −12ZαwSE , RαwSE i ji j = RαwSE i j ji =12ZαwSE , RαwSE iiii =16XαwSE +12ZαwSE ,(3.65)where {i, j} = {1, 2} and i , j. Now that the formulation for scalar resistancefunctions is complete, we solve the series valid for arbitrary spacing for a few λonce and then interpolation enables us to calculate the trajectories of the swimmerswithout solving the computationally expensive series again. Figure 3.6 demon-strates how the values of scalar functions are changed as a function of spacing ϵ,for two interacting spheres with λ = 1, 0.1 and for a sphere in vicinity of a wall(we chose λ = 0.005).3.5 AlgorithmThe next task is to build an efficient solver to overcome the complexity of theproblem of active particles near a flat wall. So far we have formulated the terms inthe grand mobility M f f , and resistance Rn f tensors. Tensors R, and M containR =RFU RFERSU RSE , M =MUF MEFMUS MES , (3.66)513.5. Algorithm10-5 10-3 10-1 10110010210410610-5 10-3 10-1 10110-210010210410610-5 10-3 10-1 101012345678910-5 10-3 10-1 101-2.5-2-1.5-1-0.5010-5 10-3 10-1 10100.511.5210-5 10-3 10-1 101-2-1.5-1-0.5010-5 10-3 10-1 1011.31.351.41.451.51.551.61.651.710-5 10-3 10-1 101-0.25-0.2-0.15-0.1-0.05010-5 10-3 10-1 10101234567810-5 10-3 10-1 10100.20.40.60.8110-5 10-3 10-1 10110-210010210410610-5 10-3 10-1 10110-210010210410610-5 10-3 10-1 101-6-5-4-3-2-1010-5 10-3 10-1 10100.20.40.60.8110-5 10-3 10-1 101-2-1.5-1-0.500.510-5 10-3 10-1 10100.511.5210-5 10-3 10-1 10110010210410610-5 10-3 10-1 10110-410-210010210410610-5 10-3 10-1 1010246810121410-5 10-3 10-1 101-0.2-0.100.10.20.310-5 10-3 10-1 1011.11.21.31.41.510-5 10-3 10-1 101-0.08-0.06-0.04-0.020Figure 3.6: Values of scalar functions as a function of spacing ϵ, for two identicalspheres λ = 1 shown by dots, two spheres with λ = 0.1 shown by stars, and for asphere near a wall λ = 0.005 shown by circles. Asymptotes for λ = 1, 0.005 areshown with solid lines that are accurate for ϵ < 10−2 .523.5. Algorithmwhere entries RFU , and MUF are second rank tensors, RFE , RSU , MEF , and MUSare third rank tensors, and RSE , and MES are forth rank tensors. It is important tonote that RFU , M−1UF but each inner tensor of R is a result of the inversion of theentire grand mobility tensor M:RFU RFERSU RSE =MUF MEFMUS MES−1. (3.67)In order to fit these tensors inside a large matrix, we use symmetric and trace-less properties of E, and S. Each of these matrices only have five independentterms and instead of matrix representation, we can use their contracted form, e.g.,S [m], where m = 1, 2, ..., 5 asS [m] = Ψ[m]i jS i j. (3.68)where the contractor Ψ[] is defined in [206]. The force on a particle in strainingflow is Fi = RFEi jkE jk = RFEi[m]E[m] hence similarlyRFE = RFE i[m] = Ψ[m] jkRFE i jk,RSU = RSU [m]k = Ψ[m]i jRSU i jk,RSE = RSE [m][n] = Ψ[m]i jΨ[n]klRSE i jkl,MUS = MUS i[m] = Ψ[m] jkMUS i jk,MEF = MEF [m]k = Ψ[m]i jMEF i jk,MES = MES [m][n] = Ψ[m]i jΨ[n]klMES i jkl. (3.69)Therefore the Rn f can be fitted inside a big matrix displayed in B.2. The selfinteraction resistance tensor for sphere α isFαLαSα=I 0¯0¯0¯43 I 0¯0¯0¯109 J⊙UαΩαEα, (3.70)where J = δ[m][n] and m, n ∈ {1, 2, 3, 4, 5} for the choice of contractorΨ of the formΨ[1]i j =√3 + 12δi1δ j1 +√3 − 12δi2δ j2, Ψ[2]i j =√3 − 12δi1δ j1 +√3 + 12δi2δ j2,Ψ[3]i j =√2δi1δ j2, Ψ[4]i j =√2δi1δ j3, Ψ[5]i j =√2δi2δ j3.(3.71)533.5. AlgorithmAlso, the 4th order mobility tensor MααESi jkl =910δilδ jk is alternatively defined asMααESi jkl =920(δilδ jk + δikδ jl − 23δi jδkl), (3.72)so contracting it with Ψ gives a diagonal matrix to avoid singular inversion.Note that for terms in M f f we have used 6πµanα, while for Rn f , we have used6πµa¯n, to non-dimensionalize the moments where a¯ is the average radius of inter-action. For example, a¯ = (aα+aβ)/2 forRαβ and a¯ = aα forRαα. This presentationof results was necessary for symmetry analysis of the Rn f . Now that we are fittingall the resistance functions within a matrix and the forces for each interaction aresummed, we need to use a universal characteristic length, same as the one used forM f f . We chose aα to be the characteristic length and correct the resistance termsasRαα, (1 + λ2λ)nRαβ, (1 + λ2λ)nRβα, (1λ)nRββ.(3.73)Next step is to analyze the symmetry features of Rn f and M f f which can speedup their calculation. The component matrices on the diagonal of Rn f and M f f aresymmetric due to the reciprocal theorem. For Rn f we haveRααFUi j = RααFU ji, RββFUi j = RββFU ji, RαβFUi j = RαβFU ji = RβαFU ji,RααLΩi j = RααLΩ ji, RββLΩi j = RββLΩ ji, RαβLΩi j = RαβLΩ ji = RβαLΩ ji,RααSE[m][n] = RααSE[n][m], RββSE[m][n] = RββSE[n][m], RαβSE[m][n] = RαβSE[n][m] = RβαSE[n][m],(3.74)and for M f f we haveMααUFi j = MααUF ji, MββUFi j = MββUF ji, MαβUFi j = MβαUF ji,MααΩLi j = MααΩL ji, MββΩLi j = MββΩL ji, MαβΩLi j = MβαΩL ji,MααES[m][n] = MααES[n][m], MββES[m][n] = MββES[n][m], MαβES[m][n] = MβαES[n][m].(3.75)Off diagonal component matrices are also symmetric in the sense that we can con-struct two problems and apply the reciprocal theorem [31].RααLUi j = RααFΩ ji, RββLUi j = RββFΩ ji, RαβLUi j = RβαFΩ ji,RααSU[m]i = RααFEi[m], RββSU[m]i = RββFEi[m], RαβSU[m]i = RβαFEi[m],RααSΩ[m]i = RααLEi[m], RββSΩ[m]i = RββLEi[m], RαβSΩ[m]i = RβαLEi[m], (3.76)543.6. Test casesMααULi j = MααΩF ji, MββULi j = MββΩF ji, MαβULi j = MβαΩF ji,MααUS[m]i = MααEFi[m], MββUS[m]i = MββEFi[m], MαβUS[m]i = MβαEFi[m],MααΩS[m]i = MααELi[m], MββΩS[m]i = MββELi[m], MαβΩS[m]i = MβαELi[m],(3.77)which lead to symmetric matrices of: Rn f = Rn f T and M f f =M f f T .Eventually R for a number of active particles near a wall based on solely thefar-field solution is R = M f f −1. However, a combination of both far-field andnear-field solutions is possible by using far-field asymptotic solutions for summingmany-body hydrodynamic interactions and pairwise exact analytical solutions forparticles close to each other or particles close to the wall such thatR =M f f −1 +(Rn f ,pp −M f f ,pp−1)+(Rn f ,pw −M f f ,pw−1). (3.78)Since we have already included N-body mobility interactions in M f f , we haveremoved the two-particle and particle-wall interactions, M f f ,pp−1, and M f f ,pw−1,to avoid double counting. Moreover, the exact lubrication solutions Rn f ,pp, andRn f ,pw, only capture close pairs interactions only if, r < r∗ = 4, and ϵ < ϵ∗ = 2,respectively. In order to keep transition between the two solutions smooth, we usea linear transition of the formR =M f f −1+min(max(0,r∗ − r0.1r∗), 1) (Rn f ,pp −M f f ,pp−1)+min(max(0,ϵ∗ − ϵ0.1ϵ∗), 1) (Rn f ,pw −M f f ,pw−1), (3.79)where the transition happens in the 10% of the defined length for solutions to shift.In each time step, we calculate R, and use eq.3.13 to find the dynamics ofswimmers. An RK4 method is used for the time derivative of each swimmer’sposition and orientation. Next, we update the geometry and the solver and repeatthe process. The time-step adaptively changes and has a direct and indirect relationwith the gap distance between two surfaces, and their relative velocity, respectively.We shall use a contact model once two surfaces overlap (see section 3.8).3.6 Test casesTo verify the accuracy of the Stokesian dynamics method we chose an originaltest by Durlofsky et al. [71] of sedimenting three identical passive spheres wheninitially they are 5 and 7 radii apart. Figure 3.7a shows a slight deviation in the553.6. Test casestrajectories just after particles interact at close distances. The reason is eliminationof lubrication behaviours in the used F-T-S method in [71], which includes far-field interactions between force/torque/stresslet and translational velocity/angularvelocity/rate of strain. There is a perfect match when we ignore the lubricationsolutions. In addition, to confirm the precision of the developed solver for the in-teraction of two spheres near a solid surface, we reproduce the results for a doubletof spherical particles presented by Swan and Brady [198]. Consider a pair of iden-tical spheres with the radius of a that their centres are distanced by r, and oppositetorques are applied along their line of symmetry. In an unbounded flow, spheresrotate due to the applied torque but remain still at their position. Adding a surfaceparallel to their line of symmetry at distance h leads to the formation of a couplingbetween the applied torque and translation and the spheres move with the velocityU, while rotating along the direction of applied torques. Thus, the doublet spinsabout its centre of mass, Ω = 2U/r. The rate of rotation as a function of r and his given in fig.3.7b and matches with [198]. Maximum of this rotation happens ath = 1.01a and r = 2.1a.(a)-4 -3 -2 -1 0-4-3.5-3-2.5-2-1.5-1-0.500.51(b)Figure 3.7: a) Trajectories of the sedimentation of three passive spheres with initialpositions of x⊥ = 0, and x∥ = −5, 0, 7. Red dots are the full F-T-S solution ofDurlofsky et al. [71]. b) The rotation rate of a pair of spheres above a surface.563.7. Discussion of results3.7 Discussion of resultsWhat drives the swimmer in quiescent flow u∞ = 0, is the surface activity of theswimmer us. Consider a sphere that its surface undergoes a tangential activity ofus = uϕ(ϕ)eϕ, where ϕ is the polar angle, and p is the axis of symmetry. Suchactive particle is so called an axisymmetric squirmer with dominant modes of B1and B2 with the tangential velocity on the sphere of the form us = (B1 sin ϕ +B2/2 sin 2ϕ)eϕ. Then the velocity moments take the form of Us = − 23 B1p, Ωs = 0,and Es = − 35a B2 pp, where pp = pp − 13 I. The first mode B1, sets the swimmingspeed in an unbounded medium while the contribution of the second mode B2 inswimming speed is apparent only in the presence of other bodies or surfaces inthe medium. It becomes clear that the leading-order hydrodynamic interactions areas a consequence of B2. To model a neutral squirmer, a pusher, and a puller, thesecond squirming mode is set to B2 = 0, B2 < 0 and B2 > 0, respectively. Incase that B2 = 0, the leading-order term attributes to higher squirming modes (orhigher-order moments of the surface activity). More details are in [82].After understanding the simpler cases we can begin to build theories that dealwith a suspension of active particles near a boundary. Thus, we start with the prob-lem associated with one active particle near a wall. Figure 3.8a shows the trajectoryof a swimmer with only the second squirming mode B2 = ±1 near a wall for dif-ferent initial orientations p. Such swimmers, B1 = 0, B2 , 0 are called shakersin the literature [175] and would not move in an unbounded medium. However,its interaction with the wall results in a net motion. We see that for a shaker, par-allel or perpendicular orientations |p| = e⊥, e∥ result in a pure translation towardsor away from the wall, while other initial orientations lead to reorientation of theshaker as well. Note that a neutral squirmer, maintains its distance from the wall. Apusher-shaker, shown with blue heads, located perpendicular to the wall p = ±e⊥,and a puller-shaker, shown with red heads, located parallel to the wall p = ±e∥,escape the wall. All other initial orientations result in the attraction of the swim-mer to the wall. The immediate outcome is that p = ±e⊥ for pusher-shakers andp = ±e∥ for puller-shakers are unstable orientations, and any variation from theseorientations results in the swimmer colliding with the wall. Pusher-shakers hit thewall from the sides, whereas puller-shakers hit the wall from the heads. Thus,pusher-shakers, reorient themselves towards p = ±e∥, and puller-shakers, reorientthemselves towards p = ±e⊥ before crashing into the wall. Figure.3.8b shows theorientation-height phase portrait of a shaker. The most obvious shortcoming of ourapproach is that the solution does not exhibit periodic orbits in the (h, θ(p)) previ-ously proved in [54, 209]. The reason is the truncation of the slip velocity at thefirst moment near the wall. In future studies, we propose employing the near-fieldswim force calculated in chapter 2.573.7. Discussion of resultsA further finding is that a pusher-shaker tends to repel other objects locatedat its front or rear, while draws at its sides as expected considering the generatedforce-dipole velocity field by squirmers. On the other hand, a puller-shaker, repelsfrom the sides and draws along its axis of symmetry. It is also worth mentioningthat the second squirming mode, B2 is front-back symmetric and thus the missingtrajectories for ±θi(p) in fig.3.8a can be retrieved and are left and right symmetric.These results are inline with the results discussed in [19] and presented in fig.3a byCrowdy and Or [54]. When compared to the results in Chapter 2, fig.3.8b showsthe limitation of using multipole expansions for slip velocity at near-fields. Figure3.8c details the h-phase portraits of the swimmers parallel or perpendicular to thewall. The effect of hydrodynamics decays at h2 rate when it escapes the wall andaccelerates nearly at ln(h − 1)−1 as it approaches the wall. Results also suggestthat the hydrodynamic interactions are twice strong along the axis of symmetrythan perpendicular to the p: A puller-shaker, oriented towards the wall, reaches thewall in half time as a pusher-shaker parallel to the wall. Dotted lines on fig.3.8cdisplay merely far-field solutions and emphasize the importance of the inclusion oflubrication behaviours.In order to test the effects of the first squirming mode B1, we show the trajec-tories of a single swimmer above a wall for different scenarios in fig.3.9. The firstmode B1 does not contribute in the wall hydrodynamics at least up to this orderof calculations. For parallel swimming, the attraction to the wall is controlled bythe second mode B2 as discussed by Llopis and Pagonabarraga [139]. A puller(pusher) oriented parallel to the wall θ(p) = 0, escapes from (attracts to) the walldue to B2. In addition, the swimmer holds its orientation as the translation and rota-tion are not coupled in the case of an axisymmetric setting. For other orientations,depending on the ratio β = B2/B1 different results may obtain. When the swimmeris oriented away from the wall θ(p) > 0, the swimmer might be able to escapes thewall due to B1 and the contribution of B2 fades out. When θ(p) < 0, both modesare in favour to bring the swimmer towards the wall except for the case of a pusherwith p = −e⊥ which is explained later. A detailed comparison of fig.3.9 and fig.2.7demonstrates the distinction between using multipole expansions of slip velocity atnear-field and its exact form. A pusher’s backward motion, a puller’s periodic mo-tion, and re-orientation of a neutral squirmer near a wall cannot be captured usingmultipole expansions of the slip velocity.The second mode B2 reorients a pusher to its sides and a puller to its head nearthe wall. Figure 3.10 shows the contribution of the second mode in (h, θ(p)) plane.The blue and red circles indicate saddle points. For higher values of β, this pointhappens at higher heights.Consider a pusher swimming towards the wall with θ(p) = −π/2. A compe-tition between the first squirming mode B1 and the second mode B2 occurs and583.7. Discussion of results(a)(b)1 2 3 4 5 610-210-1(c)Figure 3.8: a) Trajectories of non-interacting shakers near a wall with initial posi-tion of h = 5 and different initial orientations θi(p) for a pusher-shaker B2 = −1shown by blue heads, and a puller-shaker B2 = 1 shown by red heads. Points onthe trajectories show increments of t = 100 as well as the swimmer’s orientation.b) (h, θ(p)) - phase portraits of a shaker. c) (h,U⊥)-phase portraits of shakers withinitial orientations of θi(p) = {0, π}. Dashed lines correspond to asymptotes, whiledotted-lines are merely far-field solutions. Transition length is showed betweentwo dashed lines corresponding to the change of far-field and near-field solutions.593.7. Discussion of results(a)(b)Figure 3.9: Trajectories of non-interacting swimmers near a wall (located at x⊥ =0) at initial position of h = 5 for different initial orientations p. Results are for B1 =0.1, B2 = ±1,±10, and thin dashed lines are for the case of a neutral swimmer B2 =0. Points on the trajectories show increments of t = 30 as well as the swimmer’sorientation.603.7. Discussion of results(a) (b)(c) (d)Figure 3.10: Phase portrait of trajectories in (θ(p), h) plane for squirmers withdifferent β. Thick blue lines show pushers and thin red lines show pullers.613.7. Discussion of results0 0.1 0.2 0.3 0.4 0.5 0.6-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.21.011.011.051.11.21.31.523410Repulsion10AttractionFigure 3.11: Contour plot of hc as a function of B1 and B2 for a pusher orientedparallel to the wall, p = −e⊥.finally the pusher freezes near the wall at the saddle point where U(hc) = 0. Thisalso hold for a puller swimming away from the wall θ(p) = π/2. The relation be-tween the ratio of squirming modes and geometry is B1B2 = 0.9e⊥ ·R−1FU ·RFE : e⊥e⊥,and in near-field and far-field regions can be approximated as(B1B2)n f=−1.83hc + 2.4(hc − 1) ln(hc − 1) + 4.83−4.85hc + (hc − 1) ln(hc − 1) − 0.15 ,(B1B2) f f= − 3128h2c+271024h3c, (3.80)where limhc→1 B1/B2 = 0.6 and limhc→∞ B1/B2 = 0.Next to experiment, is when a swimmer passes a non-motile particle, or a deadswimmer (see fig.3.12a). Initially, a pusher repels the particle away whilst it is atthe swimmer’s frontal region. As the pusher exceeds the particle into the attractionzone, the particle moves towards the swimmer, but shortly before they touch, thepusher outpaces and the particle falls again into the repulsion region. A similarscenario can be argued for a puller passing by a non-motile particle. Figure 3.12bdetails the variation of angular velocity Ω of the non-motile particle along with itsdistance with the swimmer r as a function of its orientation θ(p). Although theparticle is non-motile (B1 = B2 = 0), we marked an initial orientation to study its623.7. Discussion of results(a)- /4 0 /4 /200.0510.1501234567(b)Figure 3.12: a) Trajectories of a buoyant non-motile particle when a pusher B2 =−1, a neutral squirmer B2 = 0, and a puller B2 = 1, passes by. b) θ-phase portrait(on the left axis) and the variation of r as a function of θ (on the right axis) for thenon-motile particle. Results are for B1 = 0.1, λ = 2, and the total non-dimensionaltime is t = 150. Points on the trajectories show increments of t/5 and the particle’sorientation. Transition length is showed between two dashed lines correspondingto the change of far-field and near-field solutions.rotation during the run. As one would expect, the rotation rate is maximum whenthe distance is minimum.Given two active particles passing by each other in an unbounded medium,we study different settings. Trajectories plotted in fig.3.13 indicate qualitativelydifferent behaviours for different types of swimmers. Two pushers fall into an orbit(see fig.3.13a), while two pullers initially accelerate towards each other, but as theyproceed, they reorient and eventually escape from one another (see fig.3.13c) aspresented in fig.19 in [105]. We expect the pullers to collide as their axis becomecloser to one another. Wherein a pusher approaches either a puller or a neutralsquirmer, even though the pusher orients away, the other swimmer attracts to theside of the pusher until eventually they touch figs.3.13b,3.13d. Lastly, a neutralsquirmer re-orients and runs away from a puller fig.3.13f. These motion rangesare also found in two interacting Quadroar swimmers. Unlike the squirming modelthat tries to recreate the swimming mechanism of specific microorganisms, theQuadroar is intended to generate an oscillating flow field with anterior, side, andposterior vortices in its surroundings. [149].The orbit arises in the case of two pushers requires more investigation. One keyfeature necessary to allow an orbit to be formed, is the symmetry of the setting.We keep the size ratio and squirming modes identical for two pushers and the633.7. Discussion of results(a) (b) (c) (d) (e) (f)Figure 3.13: Trajectories of different types of swimmers, when they are passing byeach other. Results are for B1 = 0.1, B2 = −1, 0, 1 and the points on the trajectoriesshow increments of t = 30 and the swimmer’ direction. Swimmers are initially atr∥ = 2, and r⊥ = 10.only varying geometry parameter would be their initial perpendicular distance r∥shown in fig.3.14. Two primarily distanced pushers in fig.3.14a begin with gettingcloser at a constant rate followed by a quick turn over until they escape fasterthan their approaching speed (see fig. 3.15c). We place the pushers a bit closerin the direction perpendicular to their axis of symmetry as shown in fig.3.14b.On this occasion, pushers touch at their sides. Bringing them closer, lead to theformation of an orbital trajectory, and pushers spin about the midpoint of theirline of centres. As mentioned earlier, pushers repel along their axis of symmetryand attract along their sides. If the axis of the swimmer, p, forms the right anglewith the axis linking two pushers together, er, interactions are balanced in such away that pushers maintain their distance with one another and rotate on an orbitfor a long time. However this orbit cannot be stable as a consequence of time-reversibility of Stokes flow. Figure 3.15b suggests that this angle should be aboutπ/6. To justify this angle, one can argue that the repulsion along p is twice strongerthan the attraction perpendicular to this axis. As evidenced in figs.3.15, the radiusof the orbit as well as the spin rate are not a function of the initial distancing andis only a matter of time until they reach that orbit. Lastly, when two pushers arealigned along their line of centres as is evident in fig. 3.14g, we produce equivalentresults as in the case of a pusher swimming towards a wall.Ultimately, we address the problem of two active particles in the vicinity of awall. Accordingly, we repeat the setting in fig.3.13 but with the addition of a solidboundary initially placed parallel to the plane of swimmers, with normal e⊙, and643.7. Discussion of results(a) (b) (c) (d) (e) (f) (g)Figure 3.14: Trajectories of two pushers passing by each other for different initialpositions r∥. Results are for B1 = 0.1, B2 = −1, and the initial distance is r⊥ = 10,and r∥ = {5, 4, 3, 2, 1, 0.2, 0} for figs. a to g, respectively.perpendicular to the plane of swimmers, with normal e∥. The former setting yieldsqualitatively the same results as for active particles in an unbounded fluid, althoughthe pushers attract towards the wall while pullers move away. The dynamics of theswimmers are dictated by the nearby object whether it’s a swimmer, or a boundaryas demonstrated in fig. 3.16. Pushers in fig.3.16a no longer adopt an orbit aspreviously observed. These pushers repel each other along their frontal region,while their sides are faced to the wall leading to an attraction. Depending on howthese pushers are distanced vertically, illustrated in fig.3.17, they might eventuallytouch each other or collide with the wall. In either case, they touch from their sidesrather than along their axis of symmetry.We continue with placing the wall perpendicular to the plane of swimmers.This configuration guarantees the two-dimensionality of the problem. Apart fromthe two pushers, other results reflected in fig.3.18 are comparable with the previouswall configuration. To gain a better understanding of the trajectories, we name theswimmer closer to the wall α, and the other one β. In the event of two pullers,swimmer α in an attempt of running away from β reorients and eventually hits thewall by its head, fig.3.18c. On the contrary, as can be seen in fig.3.18a, two pushersavoid falling into the wall, instead, they conduct a chase-and-run movement untilthey touch. This behaviour was seen in the experimental investigation of Theve-neau et al. [204] between neural crest (NC) cells, a highly migratory cell popu-lation, and placodal cells, an epithelial tissue that contributes to sensory organs.They discovered that NC cells chemotactically chase placodal cells, whereas pla-codal cells run when contacted by NC cells. As recorded in fig.3.19a, the distancebetween the swimmers, r, decreases followed by an oscillation prior collision. Ineach stroke of r, angular velocity reaches a maximum as elucidated in fig.3.19c.Each time the rear swimmer decreases its distance, it slows down, reorients, and653.7. Discussion of results0 50 100 150 200 250 500 550 600234567891011egab cdf(a)0 100 200 300 400 500 600-2-7 /4-3 /2-5 /4--3 /4- /2 geafdcb(b)2 3 4 5 6 7 8 900.050.10.150.20.25-0.3-0.25-0.2-0.15-0.1-0.0500.050.10.150.2cgfeadbbgfdeca(c)0/6/3/22 /35 /67 /64 /33 /25 /311 /600.010.020.03g fe dcba0.040.06(d)Figure 3.15: a) Variation of r and b) θ(p), θ(er) as a function of t, c) U and r-phaseportrait, and d) θ(er)-phase portrait for two pushers shown in fig.3.14.663.7. Discussion of results(a) (b) (c) (d) (e) (f)Figure 3.16: Trajectories of different types of swimmers when they are passingby each other in vicinity of a wall with normal e⊙. Results are for B1 = 0.1,B2 = −1, 0, 1 and the points on the trajectories show increments of t = 30 and theswimmers’ direction. Swimmers are initially at hα = hβ = 5, r∥ = 2, and r⊥ = 10.(a) (b) (c) (d) (e) (f) (g)Figure 3.17: Trajectories of two pushers passing by each other in vicinity of awall for different initial positions. Results are for B1 = 0.1, B2 = −1 and theinitial distances are h = 5, r⊥ = 10, and r∥ = {5, 4, 3, 2, 1, 0.2, 0} for figs. a tog, respectively. The points on the trajectories show increments of t = 30 and theswimmers’ direction.673.7. Discussion of results(a) (b) (c) (d) (e) (f)Figure 3.18: Trajectories of different types of swimmers passing by each other invicinity of a wall with normal e∥. Results are for B1 = 0.1, B2 = −1, 0, 1 and thepoints on the trajectories show increments of t = 30 and the swimmers’ direction.Swimmers are initially at hα = 5, hβ = 7, r∥ = 2, and r⊥ = 10.continues chasing the front swimmer. Besides, a closer look at the trajectory re-veals that the front swimmer is drifted from its side, while the rear swimmer ismoving along its axis p. Further details in fig.3.19b show that the phase differencebetween the direction of translation and the axis of symmetry is about π/2, and 0,for the front, and the rear swimmer, respectively.Finally we expand our analysis for two pushers at different initial positions r∥.In the case of pushers relatively distanced, swimmer α reorients towards the walldue to its interaction with β and eventually hit the wall from its side as displayedin fig.3.20a. Placing the swimmers a bit closer in r∥ direction, see fig.3.20b, leadsto the collision of them as the interaction between them is stronger than the effectof the wall. In another scenario, that is shown in figs.3.20c,3.20d, swimmer αis drawn by the wall a bit before it meets swimmer β. Eventually, pushing side(head) of α encounters the pulling part (side) of β and thereby a run-and-chasemotion happens. Supposing the pushers are aligned, as in fig.3.20f, they maintaina distance from each other while both are drawn to the wall.Up to this point, we would stop calculating the trajectories once the swimmersoverlap with each other or with the wall. Clearly, the algorithm suffers lack of acontact model. Next section tries to overcome the matter of particle overlappingand ways to prevent it.683.7. Discussion of results0 100 200 300 4002345678910112468101214(a)0 5 10 15 20 25 30-3-5 /2-2-3 /2-- /20/2(b)0/6/3/22 /35 /67 /64 /33 /25 /311 /6-0.08-0.040-0.15(c)Figure 3.19: a) Variation of r and h as a function of time t, b) Variation of orienta-tion θ(p) and the direction of movement θ(U) as a function of the path taken l, andc) θ(p)-phase portrait of two pushers in fig. 3.18a. The tick lines correspond to theswimmer initially closer to the wall α, while the thin lines correspond to the otherswimmer β.(a) (b) (c) (d) (e) (f)Figure 3.20: Trajectories of two pushers passing by each other in vicinity of awall for different initial positions. Results are for B1 = 0.1, B2 = −1, and theinitial distances are hα = 5, r⊥ = 10, and r∥ = {5, 3, 2, 1, 0.2, 0} for figs. a tof, respectively. The points on the trajectories show increments of t = 30 and theswimmmers’ direction.693.8. Contact models3.8 Contact modelsGenerally, lubrication forces slow down the approaching non-active pairs. For in-stance, sedimenting particles under finite forces can never come into contact. How-ever, to avoid overlapping, the timestep must get infinitesimally small which causesthe simulation to stagnate. Plus, in the case of active particles, swimmers are force-free and the surface moments set to be constant, e.g., squirming modes B1 and B2.A swimmer at close contact must use more energy to generate a stronger thrust toovercome the drag. Consequently, lubrication forces do not necessarily slow downthe swimmer and prevent it from overlapping. For example a puller accelerates to-wards nearby surfaces/particles aligned with its axis of symmetry. In either cases,the need for a universal contact model is fueled by concern about particles overlap-ping with each other or a surface. Previous contact models are based on a repulsionforce FP (interparticle force) between two hard spheres. A potential-free contactmodel is employed by Heyes and Melrose [103] for the overlapping particles alongtheir lines of centres. However, in this approach, the lubrication forces are absentand the hydrodynamic interactions between the particles are neglected. Other ap-proaches use stick/slide friction model employing springs and dashpots as in Mariet al. [143]. In this approach, the strength of repulsion is a parameter of choiceand an extra care needs to incorporate it. As a sphere gets closer to a surface oranother sphere, the amount of the force required to stop the pairs from overlapping,increases due to lubrication [206]. Instead of introducing a repulsive force betweenparticles, Ardekani and Rangel [6] numerically calculated the contact force usinga finite-volume distributed Lagrange multiplier algorithm. In this approach, themagnitude of the contact force is the criteria that defines the end of the contact.Also, these contact models are problematic when particles are sandwiched.In this chapter, a different approach is employed to develop three contact mod-els. First, we develop a contact model for sticky swimmers called adhesive contactmodel. When these swimmers come to contact, they stick to each other and formassemblies and never separate. Second contact model concerns rough swimmers.They, at the contact point, have tangential and normal forces. Two swimmers areattached together as long as the normal contact force remains repulsive. Finally, thethird contact model is suitable for softer swimmers which do not have tangentialforces at the contact point and can roll on one another. In the following we studythese contact models in details.3.8.1 Adhesive contact modelConsider a setting where there are a couple of spheres attached to the wall, namelyA, several spheres are attached together and formed clusters, C1,C2, ..., while the703.8. Contact modelsrest of spheres are unattached, B. Figure 3.21 qualitatively demonstrates possibleconfiguration of a suspension of spheres. The union of all particles are p = {A ∪B ∪Cn}.A<latexit sha1_base64="Lihtv2jYSe0R aYbwwPdS8141boc=">[binary]</la texit>B<latexit sha1_base64="wW6OXYFoJg3RvlrYIdlxqPzQzSE=">AAAB6HicbVDLTgJ [binary] fnxXl3PpatOSebOYU/cD5/AJWvjMo=</latexit> Cn<latexit sha1_base64="RDfKswpdOeGGSj1aiIgAvjdVvbM=">AAAB6nicbVBNS8N [binary] Ok8+K8Ox+L1jUnnzmBP3A+fwAYtI2s</latexit>C2<latexit sha1_base64="KNWi9tJFLpH6A67xd3t7x9D5nfc=">AAAB6ni [binary]</latexit>C1<latexit sha1_base64="ia4Xhr/CiI0sUHUTYcF6cpsSKv8=">AAAB6ni [binary]</latexit>i<latexit sha1_base64="lSkYyoM8vP8I4CTyN8mY2YBiMwA=">AAAB6HicbVB [binary]</latexit>j<latexit sha1_base64="Cp8PIPHRO0KWts/i1syr+1Pavx8=">AAAB6HicbVD [binary]</latexit>k<latexit sha1_base64="tgIj26ZWcA3amFO+MGSWQOuZURM=">AAAB6HicbVB [binary]</latexit>l<latexit sha1_base64="0C8sfUJYbrjC43r1LRBZ8OgM3fg=">AAAB [binary]</latex it>m<latexit sha1_base64="hgrbGyfUs1b/Mr5+M32Ck4D9zmA=">AAAB6HicbV [binary]</latexit>Figure 3.21Figure 3.22: Spheres in a medium in different possible positions.The velocity of the system as explained before isU = GUF · (F P + F ) − U s − GUF ·RFE : E s, (3.81)where GUF = R−1FU . Spheres attached to the wall cannot move UA = 0, whilespheres in a cluster move as a rigid body of the formUiU jUk...ΩiΩ jΩk...=I 0I ri j·ϵI rik ·ϵ......0 I0 I0 I......·[UiΩi], (3.82)or in a compact form UC1 = ΣUC1 · Ui. Note that ϵ is the Levi-Civita tensor. In-terparticle forces among spheres in each cluster are also satisfying the zero overallforce and torque relation of the form[I I I ··· 0 0 00 −Ri j·ϵ −Rik ·ϵ ··· I I I]· F PC1 = 0, (3.83)713.8. Contact modelsor compactly ΣFC1 · F PC1 = 0. Ultimately, the velocity of the system isUA = 0,UB =(I − GUF Bp ·RpBFU + GBBUF ·RBBFU +RBCnFU−1 ·RCnBFU)−1·(− U sB + (GBpUF ·RFU − GBBUF ·RBpFU) · U s − GBBUF ·RBpFE · E s+ GBBUF · FB −RBCnFU−1 · (−FCn +RCn pFU · U s +RCn pFE · E s)),UCn = −RCnCnFU−1 ·(− FCn +RCnBFU · UB +RCn pFU · U s +RCn pFE · E s), (3.84)whereRCnCnFU−1= ΣUCn ·(ΣFCn ·RCnCnFU · ΣUCn)−1 · ΣFCn ,RBCnFU−1=(GBpUF·RpCnFU− GBBUF ·RBCnFU) ·RCnCnFU−1. (3.85)In this model, the algorithm in each time step labels all spheres A, B,Cn dependingon their relative positions. If the gap between two particles gets smaller than acritical distance (usually ϵ < 10−4), they stick together and will never separateeven if the interparticle forces are not repulsive. This model also can be used toconstruct more complex geometries using small spheres.3.8.2 Rough contact modelOne beneficial addition to the adhesive model is to calculate the interparticle forceson the spheres of the formF PCn = −FCn +RCnBFU · UB +RCnCnFU · UCn +RCn pFU · U s +RCn pFE · E s,F PA = −FA +RABFU · UB +RACnFU · UCn +RApFU · U s +RApFE · E s, (3.86)which is the case for rough swimmers with both tangential and normal contactforces. To find the normal and tangential components at the contacts, we use the723.8. Contact modelsgeneral form ofFPiFPjFPkFPlFPmFPn...LPiLPjLPkLPlLPmLPn...=0 0 ··· I 0 0 ···0 0 ··· −I I 0 ···0 0 ··· 0 −I 0 ···0 0 ··· 0 0 I ···I 0 ··· 0 0 −I ···0 I ··· 0 0 0 ···....... . ........... . .0 0 ··· −Ri j·ϵ 0 0 ···0 0 ··· −Ri j·ϵ −R jk ·ϵ 0 ···0 0 ··· 0 −R jk ·ϵ 0 ···0 0 ··· 0 0 −Rlm·ϵ ···amew·ϵ 0 ··· 0 0 −Rlm·ϵ ···0 anew·ϵ ··· 0 0 0 ···....... . ........... . .·FcontmwFcontnw...Fconti jFcontjkFcontlm..., (3.87)for the setting shown in fig.3.21. Note that eq.3.87 can be generalized for allpossible settings. For instance when a sphere is attached to multiple spheres:FPi =∑j Fconti j +Fcontiw , and LPi =∑jRi j×Fconti j −ew×Fcontiw , where sphere j is attachedto sphere i. Another point worth mentioning is that the coefficient matrix in eq.3.87is a full-rank matrix and a unique solution can be obtained using either Moore-Penrose inverse or the least-squares method. Eventually, the normal and tangentialcomponents of the contact forces can be evaluated as Fnconti j = eri jeri j · Fconti j , andFtconti j = (I − eri jeri j) · Fconti j . In each time-step, the algorithm examines these nor-mal contact forces. In case of any attraction force Fnconti j ,Fncontiw ≥ 0, instead ofrepulsion contact force, the contact line is cut and the spheres get relabeled and thesolution iterates.3.8.3 Smooth contact modelSoft swimmers at contact do not have tangential forces. To implement this con-straint, we make individual swimmers in the clusters torque free, LPCn = LPA = 0.Eventually we getFPCn = ΣFCn ·(ΣUCn · GCCUF · ΣFCn)−1 · ΣUCn · [GCpUF · F − UsC − GCpUF ·RFE · E s],FPA = −GAAUF−1 ·[GApUF · F − UsA − GApUF ·RFE · E s], (3.88)where ΣUCn · UCn = 0, and ΣFCn · Fcontn = FpCn .Figure 3.23 shows the differences of these contact models for passive and activeparticles. In fig.3.23a two passive particles are placed a bit off from horizontal733.8. Contact models(a) (b)Figure 3.23: Comparison between three contact models of adhesive, rough, andsmooth, for a) passive particles and b) active particles. Particles are initially locatedat x1 = −5e∥ + 0.1e⊥ and x2 = 5e∥ − 0.1e⊥. In a) external forces on the passiveparticles are Fext1 = e∥,Fext2 = −e∥, while in b) the first squirming mode is B1 = 1.Points on the trajectories show increments of t/5 and the particle’s orientation.743.8. Contact modelsaxis. There is a fixed external force on the left (right) particle to the right (left).They move towards each other and when the distance between them gets smallerthan ϵ < 10−4, they form an assembly. While the forces on the assembly cancelout each other, their moment leads to the spin of the assembly about its centreof mass until it turns around, and stops moving. At this moment the assemble isforce/torque free. However, in the other two contact models, the particles separatewhen the normal interparticle force at their contact point is not repulsive anymore.In fig.3.23b, active particles move towards each other due to mode B1. They comeinto contact at a point that p1 and p2 are not aligned with er which leads to theendless spin of the assembly. However, in smooth contact model, particles rollon each other and separate once they can. Now let’s again review the problemof two passing pushers near a wall. Figure3.24 shows the trajectories of thesetwo swimmers after implementing each of the rough and smooth contact models.In adhesive or rough contact mode, the swimmers form an assembly after theytouch and move in a circular trajectory with a nearly constant angular velocity, seefig.3.25b. Two pushers with smooth body surfaces don’t have tangential frictionforce at the contact point. As a result, they can slide on each other. Results infigs.3.25 show that when swimmer β touches swimmer α, α rotates promptly tillthey repel each other again and continue their chase-and-run motion.Figure 3.26 is reprinted from [204]. Placodes create a chemoattractant for NCcells. Placode cells are chased by NC cells. They only make temporary physicalcontact on collision, which regulates the direction of placodal cell displacementby locally inhibiting cell protrusions, and placodal cells run as NC cells contactthem. Following a collision, the new directionality is skewed away from the pointof contact (see feg.3.26a for NC response: green; placode response: red). Then,the source of the attractant displaces, increasing the distance between NC and pla-codes. This is a mechanism that assures persistent directional migration by modi-fying the source of the chemoattractant (placodes) by the attracted cells (NC). Weproved that hydrodynamics may also cause chase-and-run motion in cells. Thesame chase-and-run behaviour could be a more general mechanism explaining thecoordinated migration of cells, from embryo development to cancer metastasis.753.8. Contact models(a) (b)Figure 3.24: Trajectories of two pushers passing by each other near a wall with a)rough and b) smooth contact models implemented. Results are for B1 = 0.1, B2 =1. The points on the trajectories show increments of t = 30 and the swimmmers’direction0 100 200 300 400 500 60023456789246810121416RoughSmoothWithout contact modelContact model(a)0/6/3/22 /35 /67 /64 /33 /25 /311 /6-0.15-0.1-0.050(b)Figure 3.25: a) Variation of r and h as a function of time t, and b) θ(p)-phaseportrait of two pushers for the conditions in fig. 3.24. Tick lines correspond tothe swimmer α, while thin dotted lines correspond to the swimmer β. Black linescorrespond to the results before the swimmers’ initial touch. Red and blue linescorrespond to the results with rough, and smooth contact models, respectively.763.9. Conclusions (a) (b)Figure 3.26: Coordinated migration of NC and placodal cells. a) Collisions be-tween NC cells (green) and placodal cells (red) in control conditions. b) NC cellsare attracted to placodal cells owing to chemotaxis. Then, contact between NCand placodal cells and promoting directional movement. NC cells are in green;placodal cells are in red. Reprinted from [204].3.9 ConclusionsIn this chapter, an algorithm is developed that calculates the behaviour of a polydisperse suspension of small swimmers moving in a viscous fluid in the vicinity ofa rigid boundary. These swimmers are close enough to each other and the bound-ary to hydrodynamically interact. The methods of reflection and lubrication theoryare employed to obtain the motion of swimmers in the low-Reynolds-number flowof Newtonian fluid. Each swimmer is modelled by a self-propelled sphere whichmoves by incompressible surface distortions. Moreover, three contact models areproposed to prevent swimmers from overlapping. Pusher-like swimmers attractother objects or swimmers from their sides while repelling along their front orback. Puller swimmers, on the other hand, reject from the sides and drag otherobjects along their head. Also, the hydrodynamics along the head axis are twice aspowerful as the hydrodynamics along the sides. This causes a cluster of pullers in asuspension of pullers, while pushers scatter in suspensions. Furthermore, a chase-and-run motion is observed for two pushers as previously been observed by [204]between neural crest (NC) cells, and placodal cells. Cells’ chase-and-run motioncould be a more general mechanism describing coordinated cell migration, fromembryo development to cancer metastasis. Nonetheless, up to this degree of com-putations, neutral swimmers do not interact with neighbouring objects/swimmers.However, the results in chapter 2 may be integrated into the algorithm for near-field contribution of swimmers’ activity us. Finally, this algorithm can be used toinvestigate the isolation or collective motion of swimmers near a wall, as well as773.9. Conclusionsthe assemblies of active particles. In the next chapter, we will look at an assemblyto distinguish the head and the tail hydrodynamics.78Chapter 4A linked-swimmer near a wall4.1 IntroductionSwimming microorganisms are routinely found near surfaces. Confining surfacescan dramatically alter the trajectories of microswimmers such as bacteria or sper-matozoa [85, 130] and the interaction between swimmers and a nearby surface maylead to the formation of a biofilm through the attachment of colonist of bacteria tothe surface [164].The combination of both far-field and near-field solutions provides a strongtool for calculation of active particles’ microhydrodynamics. We have so far onlylooked at spherical active particles but more complex swimmers may be con-structed as assemblies of spheres. A number of different assembly designs havebeen proposed in literature [129]. A well-established design is Purcell’s three-linkswimmer consisting of three rigid links that are connected by two rotary joints[14, 174]. Another design is a three-sphere design linked by rigid rods with theirlengths can change in time [152, 153]. An alternative and efficient swimmer con-sists of only two linked spheres with different radii. The spheres can expand orcontract in the radial direction leading to non-reciprocal kinematics [8]. Recently,a two-sphere swimmer is proposed, wherein one sphere is rigid while the other oneis elastic and the linking rod can change in length [155]. A more realistic model forE. coli bacteria exists in the literature that considers each bacterium as two beadslinked by a rigid rod, and represents stress distribution as two point forces situatedat the centres of these two beads. The tail bead in this two-bead model is drivenby a force generated by flagella that are not explicitly described. The forces ap-plied on the fluid are then easily calculated using the force balance on each bead[70, 94, 100, 101, 136]. A simple three-bead-spring model depicts the body andtwo flagella, inspired by recent experimental measurements of the flow field aroundalgae C. reinhardtii [46, 56, 67, 92]. The dynamics of a biflagellate swimmer ap-proaching a wall demonstrate scattering of the puller swimmer as a result of hy-drodynamic interaction with the surface [141]. However, the stress distributionson bacterial surfaces change drastically in the near-field limit that using two/threefixed point forces becomes an oversimplification inadequate of expressing the cruxof the hydrodynamic interactions [220]. In this chapter, we are particularly inter-794.2. Dynamics of a linked swimmerested in modelling a swimmer capable of capturing the effects of a tail and a headindependently when the swimmer is close to a wall and extending our methodologyto be able to simulate many such swimmers. We use a minimal model consistedof two active spherical particles with surface activities that are connected by a thinrod. Moreover, each sphere is self-propelled by incompressible surface activity[196]. Thereby, based on the direction of the impetus generated by the spheres, wecan model a puller (e.g., Chlamydomonas) or a pusher (e.g., E. coli). Addition-ally, our semi-analytical calculations allow us exploring the area of lubrication forgetting improved results and understanding the extreme situations when the gapbetween swimmers or a swimmer and a wall is very small.This chapter is organized as follows. In section 4.2, we look at the dynamics ofa linked swimmer. We use the mobility and resistance tensor calculated in chapter3 to calculate the link force and torque. Section 4.3 shows the results due to eachmode of squirming. One of the spheres in the toy model generates the impetus. Ifthe motor sphere is at the front, the toy model depicts a puller, whereas the impetusin a pusher comes from behind. Finally, section 4.4 concludes the remarks of thischapter.4.2 Dynamics of a linked swimmerTo comprehend and quantify the influence of separating head and tail on a swim-mer’s motion near a wall, we propose a simple toy model made up of two spheresand a rod. Figure 4.1 shows a two-sphere model to capture the physics of the head(experiencing a portion of the hydrodynamic drag) and the tail (generating the hy-drodynamic thrust) with radii aβ and aα, respectively, and the ratio of λ = aα/aβ.Sphere α propels forward by generating thrust Fαs ,Lαs by surface activity us. Thecentres of these two spheres are connected by a hydrodynamically insignificant thinrod with length L. The centre of the link, O, is at distance h from the wall and werefer to the angle between the symmetry axis p and the wall e⊥ as the swimmingangle, π/2 − θ, where θ > 0, θ = 0 and θ < 0 are orientations away from, paral-lel to, and towards the wall, respectively. Regardless of the squirming type of thepropulsive sphere, when sphere α generates impetus towards the head, it is referredto as a pusher, while the opposite direction of the impetus resembles a puller.804.2. Dynamics of a linked swimmerL<latexit sha1_base64="VCNK7cI/fdvFoN9GUKtcrbqC7MM=">[binary]</latexit>F<latexit sha1_base64="KpZF/NXivpiks6dHJWbfF+RQ1nM=">[binary]</latexit>F↵<latexit sha1_base64="0ta0moKlkefepoAY2ykheZXT9AM=">[binary]</latexit>L↵<latexit sha1_base64="eQJCoAdyknDM0CWE4MLXlHBFswk=">[binary]</latexit>L/2<latexit sha1_base64="3NanTwtpFdeMit8rBkuRqDTuzmI=">[binary]</latexit>L/2<latexit sha1_base64="3NanTwtpFdeMit8rBkuRqDTuzmI=">[binary]</latexit>✓<latexit sha1_base64="UCRbW5+F1hj3DYIftSlmAFwhi1I=">[binary]</latexit>O<latexit sha1_base64="cF4V042zCXH1radUW2idv8wJRHI=">[binary]</latexit>h↵<latexit sha1_base64="aW+vnMnYtBYRWYLxUf5VJjDSLJE=">[binary]</latexit>h<latexit sha1_base64="eCFCG60tXMttpB/PSNQCvW/9GY0=">[binary]</latexit>h<latexit sha1_base64="bQUPoZ9huxJ97SswNNGUC/Yk+v8=">[binary]</latexit>p<latexit sha1_base64="zebRLbjgKq4nQ84+yD/ISqIaMxQ=">[binary]</latexit>e?<latexit sha1_base64="QtFSpWJOXu1I/vy/2YXf2mMr8z0=">[binary]</latexit>✏↵<latexit sha1_base64="f3zNoi+ibtQHjRPr6CM+jbi76ZI=">[binary]</latexit>Figure 4.1: Schematic of a linked swimmer. Sphere α has surface activity us.Analysis of the swimmer is based on the framework proposed in chapter3 forhydrodynamics of active particles near a wall. The methodology for the near wallhydrodynamics of spheres is taken from [198] and assembly of spheres is takenfrom [199]. In this approach, a swimmer is represented as an assembly of sphericalparticles. The particles in such an assembly must be constrained so that it behaveslike a rigid body. Furthermore, there is no time dependency in Stokes flow andthe hydrodynamic response to forcing is instantaneous, and a body appears rigidfrom the fluid’s perspective at each instant in time. Consider a system of one activeparticle, α, and one passive particle, β, in a bounded quiescent flow that is smallenough to neglect inertia. The motion of the particles can be decomposed into therigid body motion and the surface activity of the formu(x ∈ α) = Uα +Ωα × r + us,u(x ∈ β) = Uβ +Ωβ × r, (4.1)where U is the translational velocity, Ω is the rotational velocity, and us is the sur-face activity assuming that the active particle is not changing volume. The dynam-ics of the flow field are governed by the Stokes equations. The Stokes equationshave no explicit time dependency, and the hydrodynamic forces on the particles arerelated linearly to their velocities. Newton’s second law for each sphere in Stokesflow states thatF + Fext = 0, L + Lext = 0, (4.2)814.2. Dynamics of a linked swimmerwhich means that the hydrodynamic forces and torques on each sphere are in equi-librium with the external applied forces from the connecting link. The hydrody-namic force can be divided into far-field and near-field contributions, that are thefar-field hydrodynamic force among the particles and the wall, the near-field forceamong the particles if they are in close distance to one another, and the near-fieldforce among the particle α or β and the wall. A proper superposition of these forcesgives the hydrodynamic force acted on each sphere. The hydrodynamics of thesetwo spheres near a wall are summarized in a single resistance tensor, that is onlydependent on the geometry of the particles, we may continue with the externalforce due to the connecting rod, vizFext = −F , (4.3)where Fext = [Fαext,Fβext,Lαext,Lβext]T is a 12-dimensional vector of external forcesand torques combined due to the link. The hydrodynamic forces are decompositionof two separate components,F = FD + Fs, (4.4)where FD = [FαD,FβD,LαD,LβD]T is drag forces and torques combined due to therigid-body motion andFD = −RFU · U, (4.5)whereU = [Uα,Uβ,Ωα,Ωβ]T is a 12-dimensional vector of translation and rotationof the spheres combines. Also, Fs = [Fs, Ls]T is the swim force due to the surfaceactivity of sphere α and has the form ofFs = −RFU · U s −RFE : E s + h.o.t., (4.6)where U s = [Us,Ωs]T . The applied force on the swimmer due to the connectingrod is thenFext = RFU · U +RFU · U s +RFE : E s + h.o.t.. (4.7)The force balance on the assembly givesFαext + Fβext = 0,Lαext + Lβext +L2p × Fαext −L2p × Fβext = 0, (4.8)824.3. Discussion of resultsor in a compact formΣF · Fext = 0, (4.9)whereΣF =[I I 0 0− L2 p · ϵ L2 p · ϵ I I]. (4.10)It is to be expected that, the dynamics of sphere β under the rigid body motion isrelated to sphere α and has the form ofUβ = Uα − LΩα × p, (4.11)Ωβ = Ωα. (4.12)or more compactlyU = ΣUUα · Uα, ΣUUα =I 0I −Lp · ϵ0 I0 I . (4.13)From above, the dynamics of the assembly finally is to beU = −ΣUUα · (ΣF ·RFU · ΣUUα)−1 · (ΣF ·RFU · U s + ΣF ·RFE : E s) . (4.14)Calculation of the dynamics of the assembly gets feasible by properly summingthe resistance tensors of the formR =M f f −1 +(Rαβ,n f −Mαβ, f f −1)+(Rαw,n f −Mαw, f f −1)+(Rβw,n f −Mβw, f f −1).(4.15)The far-field asymptotic solution captures the interactions between the particles andthe wall, while the exact lubrication solution captures the dynamics when the pairsor each of the particles and the wall are separated by a short distance. Since wehave already included two-body mobility interactions in M f f , we have removedthe two-particle and particle-wall interactions to avoid double counting.4.3 Discussion of resultsThis chapter contributes to a thorough analysis of the effects of surface activityon a swimmer’s mobility near a wall. Sphere α in the assembly generates swim834.3. Discussion of resultsforce by means of surface activity, us, which decomposed into squirming modes:Bα1 , Cα1 , and Bα2 . In order to observe the motion of a linked-swimmer near a wall,we investigate the role of each squirming mode on the motion of the swimmer indetail. Throughout this chapter, we report the location of the swimmer at the centreof the link and its orientation by θ and ϕ. Angle θ is shown in fig.4.1 and angle ϕis the azimuthal angle about e⊥.4.3.1 First squirming mode Bα1This section provides insights into the first squirming mode Bα1 . We learned inChapter 3 that the first mode has no influence on hydrodynamic interactions uptothe first moment level, but this two-sphere swimmer model can capture the effectsassociated with the first mode. In an unbounded fluid, a swimmer with L → ∞moves with the velocity ofU∞ =23λ1 + λBα1 p, (4.16)where positive and negative values of Bα1 represent a puller, and a pusher, respec-tively. The interaction of the spheres and the wall modifies the trajectory of theswimmer. Figure 4.2 shows the trajectories of swimmers with different λ, normal-ized with the average diameter of d¯ = aα + aβ. The active sphere α is a neutralsquirmer with only the first squirming mode, Bα1 . For negative Bα1 , the swimmerpropels in −p direction towards sphere β and mimics a pusher squirmer. This canbe seen on the right half of the fig.4.2. While for positive Bα1 , the swimmer swimstowards sphere α in p direction, similar to a puller squirmer. Pullers move in nega-tive x∥ direction and can be seen in the left side of the fig.4.2. When λ = 1, a pusherattracts to the wall, while a puller repels the wall. Due to the symmetry, no reorien-tation occurs while moving. When the active particle is larger than the non-activeparticle, λ > 1, the torque dipole reorients both a puller and a pusher towards thewall. This re-orientation is in favor of a pusher and against a puller. For smallerλ < 1, the torque dipole reorients the swimmer away from the wall. A Pusher isfirst drawn to the wall because its orientation is not in line with its velocity, but asthe wall’s effects get greater, it reorients away from the wall and finally escapes. Apuller, on the other hand, reorients away from the wall and exits in a straight line.Figure 4.3a displays how the link’s length modifies the dynamics of the swim-mer. In particular, each sphere in this model serves as a point force and higher-order terms. When the link gets smaller, the swimmer resembles that of a forcedipole. For smaller λ, the effect of length is stronger on the swimmer’s dynamicsthan larger λ. For λ < 1, the smaller the length, the stronger the torque dipoleand ultimately the swimmer reorients further away from the wall. However, this844.3. Discussion of resultsFigure 4.2: Trajectories of a swimmer for different λ near a wall. Particle α isa neutral squirmer with first squirming modes of Bα1 = {−1, 1}. Results are forL = hi = 2d¯, θi = 0 and the points on the trajectories show increments of τ/5 andthe direction of the swimmer p. Trajectories are plotted for a total time, τ, requiredfor the swimmer to reach |x∥| = 30d¯ unless it hits the wall earlier.854.3. Discussion of resultsrelative relation between the torque dipole and the length does not exist for λ = 1.Due to symmetry, no torque dipole exists and when the length gets fairly small,the vertical velocity gets weaker so does when the length gets very large. As amatter of fact, maximum vertical velocity happens for medium lengths L ≈ 2d¯ (seefig.4.3b).Details of the final pitch of the swimmer, θ, along with the height of the swim-mer, h, as a function of λ and length, after the swimmer swims 100d¯ parallel tothe wall, are displayed in fig.4.4. A puller with λ < 0, escapes and remains ori-ented away from the wall, h > 3d¯, θ < 0. In contrast, for λ > 0, the puller’sorientation is always towards the wall θ > 0. The most interesting aspect of thesegraphs is that the force dipole is the dominant escape mechanism for pullers withlonger lengths, allowing them to escape the wall, whereas the torque dipole is thedominant attraction mechanism for pullers with moderate lengths, causing them tocollide with the wall. The puller scarcely reorients or changes height for quite largeratios λ > 5, and either close touchingL < 1.2d¯, or far separated spheres L > 5d¯.Figures.4.4a,4.4b further illustrate the above claims. A pusher, unlike a puller, hasa lower probability of escaping the wall. Pushers with small λ and lengths with-stand the wall-attraction, as demonstrated in fig.4.4c,4.4d. Pushers with longerlengths approach the wall gradually and without significant reorientation. Finally,pushers with λ > 1 and lengths in the middle range are more likely to collide withthe wall.4.3.2 Second squirming mode Bα2The next section is concerned with the study of the second squirming mode, Bα2 ,and how it propels a swimmer. In general, an active particle with a positive secondmode of squirming, Bα2 > 0, is called a puller. A puller attracts items parallel to itsaxis of symmetry, here p, and repels objects perpendicular to its axis of symmetryat its sides. On top of that, the attraction is twice stronger than the repulsion. Unlikea puller, a pusher repels along its axis of symmetry and attracts towards its sides.Note that this time, the repulsion is twice stronger than the attraction. A pullersquirmer placed at our toy model, pulls the sphere β inward, driving the swimmerin the direction of p. Therefore, the swimmer also serves as a puller. In addition,sphere α repels the wall, causing the swimmer to re-orient away θ < 0, even in thecase of a symmetric swimmer λ = 1. Further impacts of λ and length are exhibitedat the left side of fig.4.5a. The torque dipole grows so large as the length rises thatthe swimmer with λ = 1, L = 10d¯ crashes into the wall in an attempt to rotate.The increase in length, on the other hand, diminishes particle interactions and, asa result, the vertical swim force falls, as seen in fig.4.5b. It is worth noting thatwhen λ = 1, the free velocity of the active particle transfers immediately via the864.3. Discussion of results(a)100 101-0.00500.0050.010.0150.020.025-0.12-0.1-0.08-0.06-0.04-0.020(b)Figure 4.3: a) Trajectories of a swimmer for different link’s lengths L and λ neara wall. Results are for Bα1 = {−1, 1}, L/d¯ = {1.01, 2, 10}, hi = 3d¯, θi = 0, and thepoints on the trajectories show increments of τ/5 and the direction of the swimmerp. Trajectories are plotted for a total time, τ, required for the swimmer to reach|x∥| = 30d¯. b) Variation of angular velocity (left axis) and perpendicular velocity(right axis) as a function of length L.874.3. Discussion of results(a) (b)(c) (d)Figure 4.4: Contour plots of the final pitch θ◦, and the height of the swimmer h/d¯after it swims |x∥| = 100d¯ parallel to the wall as a function of length L, and λ.Results are for initial values of θi = 0, hi = 3d¯. The area within the thick white lineindicates that the swimmer has already crashed with the wall.884.3. Discussion of resultslink in the case of the first squirming mode, Bα1 . This is depicted in the figurewithin the fig.4.5b. However, in the case of the second squirming mode, Bα2 , theswimmer shifts forward due to the interaction of the particles which decreases bythe increase of the length as L−2. Furthermore, the greater λ, the greater the impactof the wall. Ultimately, a swimmer with a larger ratio λ, escapes the wall furtheraway. A similar argument can be made for the case of a pusher squirmer linkedto a non-active particle. The pusher pushed the other sphere away, allowing theswimmer to act as a pusher. Furthermore, the active sphere α is drawn to thewall by the straining flow caused by its surface activity, while the forehead of theswimmer, sphere β, misses out. As a result, while the swimmer is first drawn to thewall, it reorients and flees. Trajectories for pushers with different λ and lengths aredisplayed at the right side of fig.4.5a. Pushers with longer lengths and higher ratiosλ, like pullers, re-orient greater. Nonetheless, for higher ratios λ, the attractiongrows and the swimmer may collide with the wall.Further explanatory results are shown in figs.4.6a,4.6c for pullers and pusherswith Bα2 = ±10. Each of the two kinds is oriented away from the wall, θ < 0 forpullers, and θ > 0 for pushers. It is noteworthy to mention that the maximum pitchfor a puller, away from the wall, happens at the boundary of the escape/collisionregion. The reason is that of the swimmer’s ability to rotate. In the collision region,the swimmer’s length is larger than its height from the wall. Therefore, there is notenough room for the swimmer to re-orient, and eventually sphere β touches thewall. The addition of the first squirming mode, Bα1 = 1, to a puller swimmer,guides the swimmer away from the wall as illustrated in figs.4.5,4.6b. On thesame grounds as for a puller, fig.4.6c reveals that the maximum pitch happens atthe boundary owing to the geometry of the swimmer and has no relevant physicalrationale. Sphere α touches the wall before the swimmer has the chance to re-orientand escape the wall. A boost by the first squirming mode Bα1 = −1, postpones thecollision region further to larger ratios λ and lengths (see figs4.5,4.6d). Thesefindings suggest that the second mode induces the swimmer, whether a pusher or apuller, to run away from the wall. For a pusher, the torque dipole, and for a puller,both the torque dipole and the vertical force are accountable for the swimmer’sescape. In another scenario for long swimmers, one of the spheres may collidewith the wall as the swimmer reorients.4.3.3 First azimuthal squirming mode Cα1Let us now look at a swimmer which is rotating owing to the first azimuthal modeCα1 . The surface activity of the active particle results in the rotation of the swimmeralong its axis p with the angular velocity of Ωp = Ω · p. Neglecting the effects ofthe wall and the interaction of the spheres, h, L → ∞, the axial angular velocity894.3. Discussion of results(a)100 10100.511.522.533.54-16-14-12-10-8-6-4-2010-2 10-1 100 10102468 10-310-210-1100(b)Figure 4.5: a) Trajectories of a swimmer for different lengths L and λ near a wall.The second squirming mode of sphere α is Bα2 = {−10, 10} corresponding to apusher and a puller, respectively. Results are for L/d¯ = {1.01, 2, 10}, θi = 0, hi = 3d¯and the points on the trajectories show increments of τ/5 and the direction of theswimmer, p. Trajectories are plotted for a total time needed to reach |x∥| = 30d¯.Solid and dashed-lines are devoted to Bα1 = 0 and Bα1 = ±1, respectively. b)Variation of the angular velocity (left axis) and the vertical velocity (right axis) asa function of length L for hi = 3d¯, θi = 0. Thin lines correspond to the non-zerofirst squirming mode Bα1 .904.3. Discussion of results(a) (b)(c) (d)Figure 4.6: Contour plots of the final pitch of the swimmer θ◦ after it swims |x∥| =100d¯ parallel to the wall as a function of length L and λ. Results are for initialvalues of θi = 0, hi = 3d¯. The areas shown by thick arrows indicate that theswimmer has already fallen into the wall.914.3. Discussion of resultswould beΩ∞p = −Cα1λ31 + λ3, (4.17)where the slip angular velocity of the active particle Cα1 , and the axial angularvelocity of the swimmer Ω∞p , are proportional to the volume of only the activeparticle, λ3, and the swimmer as a whole 1+λ3. The inclusion of the hydrodynamicinteractions causes a deviation of Ωp from Ω∞p . It is apparent that the swimmeritself revolves about a point, on its link, distanced Lαrev from the centre of the activeparticle and falls into a circular trajectory with radius Rαrev. We deduct the axialangular velocity to study just the rigid motion of the swimmer such thatω = Ω − Ωpp, (4.18)where Ω = Ωα = Ωβ since the swimmer’s angular velocity is identical to thatof each sphere. Figure4.7a shows a swimmer with axial rotation Ωp near a wall,revolving about e⊥ with angular velocity ω⊥ = ω · e⊥. The variations of Lαrev,Rαrev,as a function of the pitch angle and λ are illustrated in fig.4.7b. A symmetricalswimmer parallel to the wall λ = 1, θ = 0, spins about its centre. As the ratio λchanges, the centre of the spin gets closer to the larger particle. For instance, if theactive particle is just a point force, λ → 0, the swimmer spins at the centre of theparticle β. In the event that the swimmer is vertical, θ = ±π, the swimmer onlyspins about its axis and doesn’t tumble Rαrev = 0. Besides, when we measure theradius of the trajectory of the active particle, Rαrev, the maximum tumbling occursfor θ ≈ π/8. The axial rotation of the swimmer, Ωp, and how it differs from apassive swimmer in an unbounded medium, Ω∞p , are examined as a function ofpitch angle and λ, as shown in fig.4.7c. For λ > 1, the axial angular velocity doesnot deviate very much from the ambient angular velocity Ω∞p , even though the ratiois bigger than unity Ωp/Ω∞p ≥ 1. In contrast, the ratio is considerably sensitive forsmaller ratios, Ωp/Ω∞p < 1. The effects of the pitch angle on this deviation can beexplained considering the geometry of the swimmer. For smaller λ, when the pitchangle is negative, the passive particle, which also happens to be the larger sphere,is closer to the wall. As a result, the wall’s influence is stronger than the oppositescenario of a positive pitch angle. Figure 4.7d shows that the length and height ofthe swimmer have no effect on Ωp/Ω∞p for a symmetric swimmer with λ = 1, θ =0. The inner plot helps us comprehend the influence of length and height on theswimmer’s dynamics. Instead of plotting results for L, and h, we plot the ratio forL− d¯, and ϵ which are the minimum distance between two spheres in the swimmermodel and the minimum distance of the swimmer and the wall, respectively. Theaxial angular velocity of a swimmer with contacting spheres, L → d¯, achieves924.3. Discussion of resultsa saturation point and is independent to additional decreases in L. This impliesthat the swimmer’s dynamics are those of two connected spheres. In contrast tothe length, when the swimmer approaches the wall, ϵ → 0, depending on λ, wemay encounter an unphysical condition. The swimmer with λ = 1, does not stoprotating when it touches the wall. These results show that our technique has ashortcoming in that there is no model for when the swimmer comes into contactwith the wall. Lubrication forces must slow down the swimmer in real cases. Forλ < 1, the wall slows down the swimmer, while for λ > 1, the swimmer slightlyaccelerates. While this analysis does not account for contact forces, to overcomethis issue, we consider a swimmer which is stationary at a specific distance fromthe wall.Now that we have analyzed the swimmer’s axial rotation, it’s time to discussabout the rigid swimmer angular velocity ω shown by the data in fig.4.8. Figure4.8a, shows that the swimmer does not tumble while it is upright. Otherwise, thetumbling rate is maximum when the pitch angle favours the larger sphere placedcloser to the wall, θ = −π/4 for λ < 1,and θ = π/4 for λ > 1. In addition, whenλ decreases, the swimmer tumbles more with respect to the ambient axial angularvelocity of the swimmer, Ω∞p . The squirming mode Cα1 can be used to normalisethe tumbling angular velocity. The results reveal that when the ratio λ deviatesfrom unity for a given squirming mode, the tumbling diminishes (See the innerfigure in fig.4.8a). Also, the results are geometrically symmetric in the sense thatω⊥(λ, θ) = ω⊥(λ−1,−θ). Figure 4.8b shows that as the swimmer’s height increases,the tumbling rate reduces proportionally to h−4. However, as previously shown, awall does not prevent the swimmer from tumbling. In fact, as ϵ drops, the tumblingrate reaches a saturation threshold. Finally, asymptotic findings shown in fig.4.8cdemonstrate that ω⊥ decreases when swimmer length increases proportionally toL−1. The tumbling rate for contacting spheres, on the other hand, remains constantfor L < 1.1d¯. It is worth noting that the maximum of ω⊥ occurs at a moderatelength L ≈ 3d¯.The last section covers the contributions made by Cα1 to the previously studiedpusher and puller-type swimmers as a result of the first squirming mode Bα1 . Asevidenced in fig.4.9, a revolving pusher deviates from an otherwise straight path.Pushers that were intended to collide with the wall, as seen in fig.4.2, now travela spiral course before collapsing. A revolving pusher with λ > 1 reaches the wallbefore deviating much. The angular velocity rises as the height falls for λ = 1.The swimmer eventually spins in a spiral course until it hits the wall. By extendingthe length, the swimmer swims a longer distance before touching down. For lowerratios λ < 1, the swimmer first draws to the wall, resulting in a higher wall effectand azimuthal deviation, before reorienting away from the wall, θ > 0, and escap-ing. As the swimmer swims away from the wall, the hydrodynamic interactions934.3. Discussion of results(a)0.10.20.20.30.40.50.60.70.80.80.90.10.10.20.20.30.40.50.60.70.810-1 100 101- /2- /40/4/2(b)0.950.960.970.980.99 11.00210-1 100 101- /2- /40/4/2(c)100 1010.20.30.40.50.60.70.80.911.1100 101 10210-3 10-2 10-1 100 1010.20.30.40.50.60.70.80.911.110-3 10-2 10-1 100 101(d)Figure 4.7: a) A swimmer with axial rotation Ωp due to the azimuthal squirmingmode of Cα1 , spins parallel to the wall normal e⊥ with angular velocity ω⊥ at a pointon the link distanced Lαrev from the centre of the active particle α. The particle αlies on a trajectory with radius Rαrev. b) Contour plots of Lαrev, and Rαrev and c)the axial rotation of the swimmer Ωp as a function of the swimmer pitch θ and λ.d) Variation of Ωp as a function of the swimmer’s height h on the bottom x-axisand swimmer’s length L on the above x-axis. Results are for Cα1 = 10, θi=0, andL = hi = 3d¯ unless mentioned otherwise.944.3. Discussion of results- /2 - /4 0 /4 /210-710-610-510-410-310-210-1- /2 - /4 0 /4 /210-810-710-610-510-4(a)100 10110-810-710-610-510-410-310-210-110-3 10-2 10-1 100 10110-810-610-410-2100(b)100 101 10210-810-710-610-510-410-310-210-2 100 10210-810-610-410-2(c)Figure 4.8: Variation of the swimmer’s tumbling rate ω⊥, as a function of a) thepitch θ, b) the height of the swimmer h and c) the length of the swimmer L for dif-ferent λ. Results are for Cα1 = 10, θi=0, L = hi = 3d¯, unless mentioned otherwise.fade, and the swimmer does not deviate further and maintains on a straight path.The trajectory of the swimmer with λ = 0.2 shows a clear picture of the indicatedargument after it escapes the wall, aligned perpendicular to its initial orientationϕ = −π/2.Figure 4.10 quantifies a swimmer with λ = 1 and L = 2d¯, initially located atthe height of hi = 2d¯, and parallel with the wall θi = 0. The angular velocity of thespinning pusher grows to the saturation value of ω⊥ = ϕ˙ ≈= −0.5 as it gets closerto the wall. Conversely, the swimmer’s vertical velocity U⊥ slows down near thewall proportional to ϵ. The results show that the technique suffers from a lack ofa contact model. Future research might perhaps investigate this topic further byincluding an appropriate model to overcome the contact issue.The results obtained for other values of λ are summarized in fig.4.11. Theimpact of λ on a pusher is already well determined. A rotating pusher with amoderate values of λ, on the other hand, slows down vertically while spinningfaster as it approaches the wall until it touches down. According to the data shownin fig.4.11c, when λ decreases, the swimmer’s downward velocity, U⊥, also reducesuntil the positive pitch angle, θ, and its positive rate, θ˙, could outweigh the wall-attraction and allow the swimmer to escape. Both parallel and vertical angularvelocities, θ˙, ϕ˙, which are mostly due to Bα1 and Cα1 , respectively, reach zero as theswimmer runs away from the wall. A rotating pusher advances in a straight lineaway from the wall in the direction of the ultimate pitch and azimuthal angles θ, ϕ(see fig.4.11b and fig.4.11d for λ = 0.1, 0.2). For example, suppose λ = 0.3, thepusher moves towards the wall while first orienting away from it and then towardsit until it touches the wall. Thumbling rises up to a saturation point during motion954.3. Discussion of resultsFigure 4.9: Trajectories of a swimmer for different lengths and λ. Results arefor Bα1 = −1, Cα1 = 10, θi = 0, hi = 2d¯ and the points on the trajectories showincrements of τ/5 and the direction of the swimmer p. Trajectories are plotted fora total time needed for the swimmer to collide into the wall or reach the bounds of(x⊙, x∥) = (25d¯, 30d¯).0/6/3/22 /35 /67 /64 /33 /25 /311 /6-0.5-0.4-0.3-0.2-0.10(a)0/6/3/22 /35 /67 /64 /33 /25 /311 /610-8 10-610-410-2100(b)10-6 10-4 10-2 10010-810-710-610-510-410-310-210-110-310-210-1100(c)Figure 4.10: a) ϕ-phase portrait, b) variation of the minimum distance between theswimmer and the wall ϵ as a function of the revolution angle ϕ, and c) variation ofthe vertical velocity U⊥ and the angular velocity of the swimmer ω⊥ as a functionof ϵ. Results are for Bα1 = −1, Cα1 = 10, λ = 1, θi = 0, and L = hi = 2d¯.964.4. Conclusionswhile the vertical velocity decreases. In the instance of λ = 0.2, the pusher isinitially drawn to the wall while orienting away from it. At a turning point, thepusher escapes the wall with constant θ, ϕ values. When the results are comparedto a non-rotating pusher, we can observe that attraction and repulsion are generallycontrolled by mode B1 rather than C1.Figure 4.12 compares the azimuthal angles of a rotating pusher and a rotat-ing puller after they swim |x∥| = 100d¯, or |x⊙| = 100d¯, whichever happens first.The pitch angle is comparable to those for non-rotating swimmers seen in fig.4.4.As explained before, a puller is most likely to escape the wall. A rotating puller,however, deviates slightly from its initial azimuthal angle ϕ. This angle is greatestfor shorter lengths and size ratio λ. Note that in fig.4.12a, results inside the thickboundary show the collision region. Figure 4.12b is highly instructive in variousways. To begin, unlike a puller, a rotating pusher will almost probably collapseinto the wall. The collision region is highlighted by the empty zone in this figure.The azimuthal angle for this location is not shown since the swimmer travels a cir-cular path before striking the wall, therefore it is meaningless. Second, inside thecollision zone, a hashed region is reported in which the swimmer falls into a cir-cular trajectory fairly near to the wall illustrated in fig.4.12c. Third, like a puller,the azimuthal angle of an escaping pusher is greatest for smaller λ and lengths.The high-value azimuthal angles around the collision boundary are somewhat de-ceptive when compared to the height of the swimmer described in fig.4.4d. Theswimmer in that area is still close to the wall and will most likely crash with it overtime. Taken together, these findings show that the azimuthal squirming mode Cα1largely modifies the swimmer’s azimuthal orientation and contributes less to theattraction/repulsion process than the first squirming mode Bα1 .4.4 ConclusionsWe have used a basic toy model of two spheres and a rod to see how a wall in-fluences a swimmer’s mobility. The centres of these two spheres are linked by athin, hydrodynamically insignificant rod. The swimmer’s head is represented by apassive sphere, while the tail is represented by a squirmer, which is responsible forthe swimmer’s propulsion. For the first squirming mode, the swimmer imitates apusher (puller) for negative (positive) B1. A pusher, as opposed to a puller, has adecreased chance of escaping the wall. Pushers with small λ and lengths resist wallattraction. In the first squirming mode, B1, the tail’s push is transmitted instantlyover the link. In the second squirming mode B2, however, the swimmer pushesforward owing to spheres’ interaction. As a result, the interactions fade away asthe length increases. Finally, the azimuthal squirming mode C1 significantly alters974.4. Conclusions0/23 /2-0.03-0.02-0.010 0/23 /2-5 10-3-3 10-3-1 10-300/23 /2-6 10-4-4 10-4-2 10-40 0/23 /2-1 10-4-0.5 10-40(a)0/23 /210-810-610-410-21000/23 /210-810-610-410-21000/23 /211.522.50/23 /2246(b)(c)0 5 10 15 20-2-101234510-4(d)Figure 4.11: a) ϕ-phase portrait, b) variation of minimum distance between theswimmer and the wall ϵ as a function of azimuthal angle ϕ, c) variation of thevertical velocity U⊥ and angular velocity ω⊥ as a function of ϵ, and d) θ-phaseportrait. Results are for Bα1 = −1, Cα1 = 10, L = hi = 2d¯, and θi = 0. Dashed linesin (c) refer to the right axis.984.4. Conclusions(a) (b) (c)Figure 4.12: Contour plots of the final azimuthal direction ϕ◦ for a) a puller andb) a pusher swimmer, after it swims |x∥| = 100d¯, or |x⊙| = 100d¯, whichever comesfirst, as a function of L and λ. The region inside the thick white line in (a) and theempty region in (b) suggest that the swimmer is already collapsed into the wall.Hashed area shows the region where the swimmer falls into an orbit close to thesurface. Figure c shows orbital trajectories for a rotating pusher in the hashed area.Results are obtained for Bα1 = −1, Cα1 = 10, θi = 0, and hi = 3d¯.the swimmer’s orientation while contributing little to the attraction/repulsion pro-cess. The azimuthal mode explains the mechanism behind the rotation of an E. coliabove a surface. It should be noted that this mode has no effect on the hydrody-namic interactions of a suspension of squirmers. The separation of the swimmer’shead and tail is essential for rotation.In the following chapter, we will modify our toy model to account for swim-mer’s flexibility. To do this, we place a torsional spring in the centre of the toymodel, allowing the swimmer to bend due to torque dipole and higher order inter-actions.99Chapter 5An elastic swimmer near a wall(Active Stokesian Dynamicsmethod)5.1 IntroductionIn the previous chapter we looked at the dynamics of a linked swimmer near an ob-stacle. We discovered that some swimmers could become entrapped at the surface,swim beside it, or escape. Depending on the geometry of the wall and the swim-mer, as well as the propulsion mechanism, the swimmer-wall interaction can resultin attraction or repulsion [19, 192]. However, this model may fail to account for theeffect of swimmer flexibility on its dynamics. In this chapter, we are particularlyinterested in modelling a swimmer capable of capturing the impact of flexibilityon its dynamics when the swimmer is close to a wall [166, 215]. For example, aflagellum is elastic, and its bending can drastically affect its hydrodynamic inter-actions [214, 215]. Montenegro-Johnson et al. [151] have computationally shownthat an elastic filament is dispersed by 5 to 10 degrees while swimming througha back-step in a microchannel. They discovered that depending on the amountsof viscous and elastic forces, the swimmer may disperse toward or away from theback-step. In this case, we employ a simple model made up of an active and a pas-sive spherical particle linked by two thin rods. A torsional spring connects theserods, allowing the swimmer to bend. Furthermore, the active sphere propels itselfdue to surface activity [196]. Thereby, we may model a flexible puller or a pusherdependent on the direction of the impetus provided by active spheres. Additionally,our semi-analytical calculations allow us to explore the lubrication region in orderto enhance findings and grasp extreme circumstances when the distance betweenspheres or spheres and the wall is very narrow.The following is how this chapter is organised. The dynamics of an elasticswimmer is modelled in section 5.2. We utilise the mobility and resistance tensorsacquired in Chapter 3 to compute the forces and torques of the links and the spring.The section 5.3 displays the outcomes of each squirming mode B1 and B2. Finally,1005.2. Dynamics of a elastic swimmer✓<latexit sha1_base64="69v2qD6cDV8GFw1uLq12ExGzvqs=">[binary]</latexit><latexit sha1_base64="ZDQr0NKF+P5vpwv6T5PeHKyaWkU=">[binary]</latexit><latexit sha1_base64="ZDQr0NKF+P5vpwv6T5PeHKyaWkU=">[binary]</latexit>a↵<latexit sha1_base64="CQfvJejMQ/sluAcnxsy2oBBiwGk=">[binary]</latexit>a<latexit sha1_base64="RyW1o+mK15I0UQZ0PCf9k4OQk/s=">[binary]</latexit>e↵<latexit sha1_base64="h50z48J+LMZ1LFRk+RWCCnDhJvs=">[binary]</latexit>e<latexit sha1_base64="f6k4gPEQS5YW7O580H5TScq5weU=">[binary]</latexit>eO<latexit sha1_base64="ASBW3enL3ZwjmExu+q2SFXnzvDA=">[binary]</latexit>L/2<latexit sha1_base64="3NanTwtpFdeMit8rBkuRqDTuzmI=">[binary]</latexit>L/2<latexit sha1_base64="3NanTwtpFdeMit8rBkuRqDTuzmI=">[binary]</latexit>h<latexit sha1_base64="eCFCG60tXMttpB/PSNQCvW/9GY0=">[binary]</latexit>h↵<latexit sha1_base64="aW+vnMnYtBYRWYLxUf5VJjDSLJE=">[binary]</latexit>e?<latexit sha1_base64="QtFSpWJOXu1I/vy/2YXf2mMr8z0=">[binary]</latexit>pFigure 5.1: Schematic of an elastic swimmer.section 5.4 ends this chapter’s observations.5.2 Dynamics of a elastic swimmerFigure 5.1 shows the modified toy model for an elastic swimmer to quantify the ef-fect of flexibility on hydrodynamic interactions with a wall. The model is identicalto the rigid swimmer model, with the exception that the spheres are linked by tworods (each length L/2) via two bearings that allow the spheres to spin and a lineartorsional spring (stiffness k) in the middle. The angle between p and each rod eα,eβ is called bending angle δ which its positive and negative values correspond tobending of the swimmer towards and away from the wall, respectively. When theswimmer is not bent, δ = 0, we have p = eα = −eβ. The force and torque F , aswell as the bending angle δ, are now unknown in this equation. Extra equations areproduced by the force balance on the assembly. Figure 5.2 shows the free diagramof the spheres and the links. The balance of forces and moments acting on spheresis enforced to keep the swimmer in equilibrium, such thatFα + Fβ = 0,Lα⊥ + Lβ⊥ +L2eα × Fα + L2 eβ × Fβ = 0,Lα∥ = 0,Lβ∥ = 0. (5.1)or compactlyΣF · F = 0, Ωα∥ = −Ωαs∥ , Ωβ∥ = −Ωβs∥ , (5.2)1015.2. Dynamics of a elastic swimmer<latexit sha1_base64="KOS5wXOhDnQEDMU/CuwJuA+6RfM=">AAAB7XicbVBNS8NAEJ3U [binary] </latexit><latexit sha1_base64="KOS5wXOhDnQEDMU/CuwJuA+6RfM=">AAAB7XicbVBNS8NAEJ3U [binary] </latexit>kesp<latexit sha1_base64="yYahNpJg/uXYfFTGHp by8LEw3WE=">[binary] </latexit>kesp<latexit sha1_base64 ="YqGW2/20UTU2o/nPOsOSSViQGZo=">AAACAHic [binary] 3rpiVDNH6A+Mzx+JZJZT</latexit>F<latexit sha1_base64="TywPxzRwfM6MeW3mdG 0whZcfN8g=">[binary]</latexit>F<latexit sha1_base64="O72Iq0+JiaU9YnM2U3 3/vt5h/Ak=">[binary]</latexi t>F↵<latexit sha1_base64 ="KHqbZhTtgHDYIZtAHGA2oJp8HZY=">AAAB+Xic [binary] wx9Ynz9PXJNj</latexit>F↵<latexit sha1_base64 ="1+ADP6p/rDwZpWV/t0BRXHTxQ/w=">AAAB+Hic [binary] f2B9/gDjV5Ms</latexit>Lk<latexit sha1_base64="SEtzmwYpNrcETJa+EV fIcviH8lQ=">[binary] +QOr0ZlC</latexit>L↵k<latexit sha1_base64 ="zec8/7bjy6ksGgKag1+8IA0brYE=">AAACBnic [binary] nqwX6936mLUuWeXMAfgD6/MHfUWZtg==</latex it>L↵?<latexit sha1_base64 ="PFHj7tIikXsBM5fqHME3xB0DxOI=">AAACAnic [binary] rHfrY9a6ZJUzB+APrM8fVY2X+A==</latexit>L?<latexit sha1_base64="LGaVesgwrBCXxAgaLj Da+F7H78E=">[binary] yZeE</latexit>e<latexit sha1_base64="O2EKx5yBZO4xYtIHE1 /nNVeiZnw=">[binary]</latexit>e↵<latexit sha1_base64="6p+RnP/y+QYi6hiadq 0nj4Dj9tA=">[binary]</latexi t>esp<latexit sha1_base64="XN8GYbpeYZ0vkKeCma 38ByqU1HM=">[binary]</latexit>eO<latexit sha1_base64="vOF+dej01gUZkNPIWp IauRGMDPk=">[binary]</latexit>pFigure 5.2: Free diagram of spheres and rods in an elastic swimmer model.e?<latexit sha1_base64="nOMRF5dmKC3ywuwZQsmqv9ZRgzA=">[binary]</latexit>ek<latexit sha1_base64="8GtlkUMM/wbldbLGwfEhUYlNY80=">[binary]</latexit>e<latexit sha1_base64="UkKZolDPAIEgZ6Zg8yanpw+zeVg=">[binary]</latexit>x↵<latexit sha1_base64="1QIiPrf8D8wjZHhgApJpLWAD5Os=">[binary]</latexit>x<latexit sha1_base64="EXwVbBIT6epH0TtZ5HEwtnUf95A=">[binary]</latexit>xO<latexit sha1_base64="GaSRNWd/ZL7BAtE1BVwgRJo7ngk=">[binary]</latexit>⌦↵?<latexit sha1_base64="vSofrI/NqQmczxXaNMJg9FHih9E=">[binary]</latexit>⌦↵k<latexit sha1_base64="cgEldKizOuTe3k9QGG7751cxePw=">[binary]</latexit>U↵<latexit sha1_base64="UCNQcOe2TrhC4s9yE9cQemsCgVk=">[binary]</latexit>⌦?<latexit sha1_base64="bzst57jxYrCF4LnQMIXhvMUqgIU=">[binary]</latexit>⌦k<latexit sha1_base64="RKXVp0DSnCQ/0kfwnaboGMmINZ0=">[binary]</latexit>U<latexit sha1_base64="JDZ2LsvnQnvAHH7ChIDWrI1nN/M=">[binary]</latexit>eO<latexit sha1_base64="ASBW3enL3ZwjmExu+q2SFXnzvDA=">[binary]</latexit>e↵<latexit sha1_base64="h50z48J+LMZ1LFRk+RWCCnDhJvs=">[binary]</latexit>e<latexit sha1_base64="f6k4gPEQS5YW7O580H5TScq5weU=">[binary]</latexit>⌦O<latexit sha1_base64="20L2AQZZhe8AlrZq1dVbkXohufE=">[binary]</latexit>UO<latexit sha1_base64="TEjeP96FDxL33+5FtC3LotuBIqM=">[binary]</latexit>Xr<latexit sha1_base64="Tk9H66fXNNP73AKz47Lrl5bcEDo=">[binary]</latexit>p<latexit sha1_base64="Q7JGk32PEYYk6FH/urQWFg5x15o=">[binary]</latexit><latexit sha1_base64="V2dNNplsElqpnkQACSrwHmYv9x0=">[binary]</latexit><latexit sha1_base64="V2dNNplsElqpnkQACSrwHmYv9x0=">[binary]</latexit>✓<latexit sha1_base64="J9ZO1LuihcGJXJ29819AjQ7PvlE=">[binary]</latexit>✓<latexit sha1_base64="J9ZO1LuihcGJXJ29819AjQ7PvlE=">[binary]</latexit>✓<latexit sha1_base64="J9ZO1LuihcGJXJ29819AjQ7PvlE=">[binary]</latexit>Figure 5.3: The rigid body motion of the swimmer.whereΣF =[I I 0 0− L2 eα · ϵ − L2 eβ · ϵ I I](5.3)Figure 5.3 depicts changes in spatial configuration of the swimmer after a time-sequence dt to help comprehend the swimmer’s rigid body motion. The swimmer’sdynamics are provided by1025.2. Dynamics of a elastic swimmere˙O = ΩO × eO = θ˙esp × eO,e˙r = ΩO × p = θ˙esp × p,e˙α = Ωα⊥ × eα = (θ˙ + δ˙)esp × eα,e˙β = Ωβ⊥ × eβ = (θ˙ − δ˙)esp × eβ,Uα = UO +L2e˙α,Uβ = UO +L2e˙β, (5.4)where esp is the unit vector perpendicular to eα, and eβ, while eO is the bisectorunit vector of them. Also, UO, andΩO are translational and rotational velocities ofthe midpoint of the swimmer. The dynamics of each sphere isUαUβΩαΩβ = −0 0 0 00 0 0 00 0 eαeα 00 0 0 eβeβ ·UαsUβsΩαsΩβs +I L2 eα · ϵI L2 eβ · ϵ0 I0 I ·[UOΩO]+L2 eα · ϵ− L2 eβ · ϵI−I · δ˙esp(5.5)or compactlyU = −ΣUU s · U s + ΣUUO · UO + ΣU δ˙ · δ˙esp, (5.6)From above, the dynamics of the assembly isUO = −(ΣF ·RFU · ΣUUO)−1 · ΣF ·[RFU · U s +RFE : E s −RFU · ΣUU s · U s +RFU · ΣU δ˙ · δ˙esp]. (5.7)To complete the system of equations, we still need another relation for the bendingangle δ. The restoring moment of the spring balances the torque dipole and themoment of the rods. For finite stiffness, the relative motion caused by a change inthe angle of the torsional spring iskδesp =12ΣδF · F = 12[L2 eα·ϵ − L2 eβ·ϵ −I+eαeα I−eβeβ ] · FαFβLαLβ. (5.8)We compute R at each time step and use eq.5.7 to obtain the dynamics of theelastic swimmer for initial zero values of δ = 0 and δ˙ = 0. Following that, wecalculate δ using eq.5.8, and then δ˙ using first-order temporal discretization. Fol-lowing that, we update the geometry and the solver and continue the procedureuntil we get the desired convergence accuracy. The time derivative of the swim-mer’s position and orientation is then computed using an RK4 approach.1035.3. Discussion of results5.3 Discussion of resultsThis chapter adds to a comprehensive examination of the impact of surface activ-ity on a swimmer’s movement near a wall. The assembly’s sphere α mimics thetail and produces swim force via surface activity, uαs, which results in squirmingmodes of B1 and B2. The sphere β represents the head and is passive, with nosurface activity uβs = 0. We’ll look at each squirming mode’s contribution individ-ually.5.3.1 First squirming mode Bα1The fact that pullers are unaffected by elasticity is a significant finding in our un-derstanding of the role of elasticity in swimming motion. When a swimmer ispulled, it seeks to maintain its connections tight and stiff in order to transfer moreof the driving force into the body. The body being dragged will eventually tend tokeep in line with the power leading it. Comparable to a train with an engine at thefront, the pullers are more stable than pushers. When the swimmer is pushed, onthe other hand, the links are squeezed, and even a tiny deviation in the directionof force bends the swimmer and makes the motion less stable. In other words, apusher swimmer easily buckles. The torque dipole on the pusher bends the swim-mer, as seen in fig.5.4. The swimmer bends more as the stiffness of the torsionalspring decreases. Order unity k values result in a steady circular trajectory, butsmaller k values bring the spheres together and make them contact. The methodis hampered by the lack of a contact model. The motion calculation is terminatedwhen the spheres collide or when the swimmer collides with the wall. The effect ofλ on pushers is analogous to the stiffness of a spring. The swimmer’s flexibility in-creases as the size ratio λ decreases. Swimmers with smaller tails are more flexiblethan swimmers with larger tails, according to the results shown in fig.5.5. Figure5.6 depicts swimmer trajectories for various initial orientations. The stretched po-sition is unstable regardless of the initial orientation or the height of the pusherwith respect to the wall, and a tiny deviation leads the pusher to bend and fall intoa more stable configuration. The wall asymmetric traction is responsible for thisdeviation. Figure 5.7 displays swimmer trajectories for different lengths. As pre-viously explained, the impetus from the active particle is immediately transferredto the passive sphere via the links, and unlike in the second squirmer mode B2,the interactions do not fade away as the length increases. As L grows, so does thebending angle and the radius of the orbit.1045.3. Discussion of resultsFigure 5.4: Trajectories of a swimmer for different values of stiffness k ={∞, 1.5, 1, 0.5, 0.1} near a wall. Particle α is a neutral squirmer with first squirmingmodes of Bα1 = {−1, 1}. Results are for L = hi = 3d¯, θi = 0, λ = 1 and the pointson the trajectories show increments of τ/5 and the direction of the swimmer’s linkseα, eβ. Trajectories are plotted for a total time, τ, required for the swimmer to reach|x∥| = 20d¯ unless it falls into an orbit or the spheres collide. Solid and dashed-linescorrespond to δ ≤ 0, and δ > 0, respectively.1055.3. Discussion of resultsFigure 5.5: Trajectories of a swimmer for different values of λ = {0.1, 0.5, 1, 2, 10}near a wall. Particle α is a neutral squirmer with first squirming modes ofBα1 = {−1, 1}. Results are for k = 1, L = hi = 3d¯, θi = 0 and the points onthe trajectories show increments of τ/5 and the direction of the swimmer’s linkseα, eβ. Trajectories are plotted for a total time, τ, required for the swimmer to reach|x∥| = 20d¯ unless it falls into an orbit or the spheres collide. Solid and dashed-linescorrespond to δ ≤ 0, and δ > 0, respectively. Dotted lines correspond to a rigidswimmer.1065.3. Discussion of results(a) (b)Figure 5.6: Trajectories of a swimmer for different values of initial orientation θinear a wall for a) puller, and b) a pusher. Increments of θi are π/16. Particle α isa squirmer with the first squirming mode of Bα1 = {−1, 1}. Results are for λ = 1,L = 3d¯, hi = 20d¯, k = 1 and the points on the trajectories show increments of τ/5and the direction of the swimmer’s links eα, eβ. Trajectories are plotted for a totaltime, τ, required for the swimmer to reach |x∥| = 20d¯, unless it falls into an orbit orthe swimmer collides with wall. Solid and dashed-lines correspond to δ ≤ 0, andδ > 0, respectively. Dotted lines correspond to a rigid swimmer.1075.3. Discussion of resultsFigure 5.7: Trajectories of a swimmer for different values of length L/d¯ ={1.1, 2, 3, 5, 10} near a wall. Particle α is a neutral squirmer with first squirmingmodes of Bα1 = {−1, 1}. Results are for λ = 1, k = 1, hi = 5d¯, θi = 0 and the pointson the trajectories show increments of τ/5 and the direction of the swimmer’s linkseα, eβ. Trajectories are plotted for a total time, τ, required for the swimmer to reach|x∥| = 20d¯ unless it falls into an orbit or the spheres collide. Solid and dashed-linescorrespond to δ ≤ 0, and δ > 0, respectively. Dotted lines correspond to a rigidswimmer.1085.3. Discussion of results5.3.2 Second squirming mode Bα2As shown in figs.5.8,5.9, a pusher squirmer in the assembly propels the swimmerin the same way as a pusher moves away from the wall, nearly independently ofthe stiffness value k. A puller squirmer, on the other hand, pulls the other spheretowards it and bends owing to the torque dipole. Our data show that the spheres areunlikely to collide. A puller squirmer repels from its sides, as stated in the preced-ing chapters. When the swimmer bends, the passive sphere enters the squirmer’srepulsion area. The swimmer bends more to counteract the torque dipole on theswimmer by reducing the stiffness of the torsional spring. This twisting completelyaffects the swimmer’s trajectory. Flexible pullers with smaller tails than their headsswim towards the wall, but stiffer pullers flee away from it. According to the re-sults for λ = 1, flexible pullers conduct a bicycle kick near a wall! Fluffy pullers, orswimmers with large flagella, escape the wall, except for extremely flexible swim-mers, which collide with the wall.Unlike pushers, which may avoid the wall even if the swimmer is initially ori-entated towards it, pullers are doomed to clash with it (see fig.5.10). As the lengthincreases, the interaction between the spheres decreases, and the swimmer becomesweaker. The torque dipole of a flexible swimmer reduces and the swimmer bendsless. A flexible puller with a shorter length bends more and its trajectory is morechanged than a rigid puller. For L = 2d¯, the results in fig.5.11 indicate a spiraltrajectory that comes closer to the wall until the swimmer collides.1095.3. Discussion of results(a)(b)(c)Figure 5.8: Trajectories of a swimmer for different values of stiffness k near a wallfor a) λ = 0.5, b) λ = 1 and c) λ = 2. Particle α is a squirmer with secondsquirming mode of Bα2 = {−10, 10}. Results are for Bα1 = 0, L = hi = 3d¯, θi = 0,and the points on the trajectories show increments of τ/5 and the direction of theswimmer’s links eα, eβ. Trajectories are plotted for a total time, τ, required for theswimmer to reach |x∥| = 20d¯ or |x⊥| = 20d¯, or collapse to the wall, whicheverhappens first. Solid and dashed-lines correspond to δ ≤ 0, and δ > 0, respectively.1105.3. Discussion of results(a) (b)Figure 5.9: Trajectories of a swimmer for different values of λ near a wall for a)k = 1, and b) k = 0.3. Particle α is a squirmer with second squirming mode ofBα2 = {−10, 10}. Results are for Bα1 = 0, L = hi = 3d¯, θi = 0 and the points onthe trajectories show increments of τ/5 and the direction of the swimmer’s linkseα, eβ. Trajectories are plotted for a total time, τ, required for the swimmer to reach|x∥| = 20d¯ or |x⊥| = 20d¯, or collapse to the wall, whichever happens first. Solid anddashed-lines correspond to δ ≤ 0, and δ > 0, respectively. Dotted lines correspondto a rigid swimmer.1115.3. Discussion of results(a) (b)Figure 5.10: Trajectories of a swimmer for different values of initial orientationθi near a wall for a) puller, and b) a pusher. Increments of θi are π/16. Sphereα is a squirmer with the second squirming mode of Bα2 = {−10, 10}. Results arefor Bα1 = 0, L = 3d¯, hi = 20d¯, k = 0.3 and the points on the trajectories showincrements of τ/5 and the direction of the swimmer’s links eα, eβ. Trajectories areplotted for a total time, τ, required for the swimmer to reach |x∥| = 20d¯, unless theswimmer touches the wall. Solid and dashed-lines correspond to δ ≤ 0, and δ > 0,respectively. Dotted lines correspond to a rigid swimmer.1125.3. Discussion of resultsFigure 5.11: Trajectories of a swimmer for different values of length L/d¯ ={1.1, 2, 3, 5, 10} near a wall. Sphere α is a squirmer with the second squirmingmodes of Bα2 = {−10, 10}. Results are for λ = 1, k = 0.3, hi = 5d¯, θi = 0 and thepoints on the trajectories show increments of τ/5 and the direction of the swim-mer’s links eα, eβ. Trajectories are plotted for a total time, τ, required for theswimmer to reach |x∥| = 20d¯ unless the swimmer collide. Solid and dashed-linescorrespond to δ ≤ 0, and δ > 0, respectively. Dotted lines correspond to a rigidswimmer.1135.4. Conclusions5.4 ConclusionsThis chapter examined the effect of flexibility on a swimmer’s movement near awall. One of the assembly’s spheres is a squirmer with modes B1 and B2 thatsimulates the tail. The results for B1 reveal that when a swimmer is pulled, theswimmer seeks to maintain its linkages tight and under tension in order to transmitmore of the driving energy into the body. Pushers, on the other hand, are easilybent for smaller ratios λ and stiffness. They then enter a circular orbit with a fixedbending angle. The results for B2 reveal that a puller pulls the other sphere towardsit and bends because of the torque dipole. Flexible pullers with smaller tails swimtoward the wall, whereas stiffer pullers flee. Pushers are less affected by flexibilityand are able to avoid the wall.Future research on the role of other elements in the swimmer’s motion, suchas the correct contact model, may broaden the explanations of dynamics and theunderlying physics. Buckling instability should also be quantified in the case ofpushers.In the following chapter, we will look at another technique to deal with thedynamics of an elastic swimmer. We simply employ far-field singular multipolesto construct the solution and grasp the theory behind an elastic swimmer interactingwith a surface and what causes the swimmer to bend.114Chapter 6An elastic swimmer near a wall(far-field approach)6.1 IntroductionTheoretically, studies on the swimming trajectories of a microswimmer near a wallcan be divided into two groups: those where the swimmer is very close to the walland lubrication equations govern the motion [9], and those where the swimmer’sdistance from the wall is large compared to its body size and far-field approxima-tions are used [19, 70, 154]. However, far-field hydrodynamics has been demon-strated to yield correct findings even at distances as small as a fraction of the cellbody [190]. Therefore, far-field hydrodynamics are frequently utilized to explaininteractions between a swimmer with nearby boundaries [22, 190]. Using thismethod a number of studies have quantified the dynamics of the swimmers closeto walls and have showed that swimmers tend to accumulate walls [19, 141]. Moststudies keep the geometry of swimmers fixed despite the fact that many swimmerstend to be quite flexible [189] and flexibility can alter trajectories [151]. Whilewe used active Stokesian Dynamics method in chapter 5, in this chapter we in-vestigate the trajectories of flexible swimmers near boundaries using a far-fieldapproximation. Thus, we can comprehend the theory underlying the numerical re-sults obtained in Chapter 5. The toy model is still made up of two spheres linkedby two rods and a torsional spring. One of the spheres in the toy model createsthe swim force FT , which drives the swimmer. We construct the velocity field dueto the presence of spheres using singular Stokeslet multipoles, Stokeslet dipoles(doublets), and Stokeslet quadrupoles. Then, we employ Blake’s image solutioncreated by higher-order terms at the place of the sphere image at the wall to meetthe no-slip boundary condition on the wall [22].The following is how this chapter is organised. The problem formulation, ge-ometry and the background disturbance velocity field are demonstrated in section6.2. The results of the swim force FT are reported in section 6.3, followed by swim-ming in an unbounded fluid as a reference point in section 6.4. In an alternativetechnique, the velocity field is constructed using the velocity of the spheres rather1156.2. Problem formulationthan the link force (see section 6.5). Finally, section 6.6 concludes the findings inthis chapter.6.2 Problem formulationFigure 6.1 shows toy models for a pusher and a puller swimmer to quantify theeffect of flexibility on the hydrodynamic interactions of a swimmer near a wall.More specifically, we investigate the hydrodynamics of a pusher in details as thesetwo toy models are only different in the direction of the imposed thrust force.6.2.1 GeometryWe consider a two-sphere model to capture the physics of the head (experiencingmost of the hydrodynamic drag) and the tail (generating the hydrodynamic thrust)with radii a∗β and a∗α, respectively, and the ratio of λ = a∗α/a∗β (asteroid symbol∗ denotes dimensional quantity). Sphere α propels forward by generating thrustF∗T = γF∗T eα and is called a pusher for γ = −1 or a puller for γ = 1. The centres ofthese two spheres are connected by two hydrodynamically insignificant thin rods(each length L∗/2) and a linear torsional spring (stiffness k∗) in the middle of them.Sphere α is at distance h∗α from the wall and we refer to the angle between thesymmetry axis p and unit vector parallel to the wall e∥ as the swimming angle θwhere θ > 0, θ = 0 and θ < 0 are orientations away from, parallel to, and towardsthe wall, respectively. The angle between p and the thrust direction eα is calledbending angle δ which its positive and negative values correspond to bending ofthe swimmer towards and away from the wall, respectively. When the swimmer isstretched, δ = 0, the thrust is aligned with the swimmer axis of symmetry eα = p.Orientations and unit vectors in Cartesian coordinates are defined asp = − cos(θ) e∥ − sin(θ) e⊥,eα = Rz · p,eβ = eα − 2 cos(δ)p, (6.1)where Rz(δ) is the rotational matrix of the formRz(δ) =cos(δ) − sin(δ) 0sin(δ) cos(δ) 00 0 1 . (6.2)1166.2. Problem formulation ✓L/2L/2eˆ?eˆkPusher<latexit sha1_base64="Yi2gaTSrvlzY5I38o2aoFJk9j4g=">[binary]</latexit>✓Puller<latexit sha1_base64="JBcRJJRxXwCA+K0ijizr4enKZCM=">AAACBXicbVDNS8Mw [binary]</latexit>a↵<latexit sha1_base64="pb83BWG8IEqs0MrOZ1bKtObPueg=">[binary]</la texit>a<latexit sha1_base64="6NnlR2cUxsIFEzKXydD7jAQ+Ytw=">AAAB7nicbVDLSgNBEJz1GeMr6tHLYBA8hd0o [binary]</latexit>e↵<latexit sha1_base64="lb9i0ddo/POGiouGOUSvSLlrzVs=">AAAB+HicdVDLS [binary]</latexit>e<latexit sha1_base64="Y/5EOLXHf6ne3aZjPCF6lzjLv7o=">AAAB9XicdVDLS [binary]</latexit>p<latexit sha1_base64="9nGZyi523xGJtCdzu5t0BknYkNY=">AAAB73icdVDLS [binary]</latexit>FT e↵<latexit sha1_base64="WBSOX86iYjTsNoKVPvIM++dQKQ4=">AAAB/XicdVDLS [binary]</latexit>e↵<latexit sha1_base64="lb9i0ddo/POGiouGOUSvSLlrzVs=">A [binary]</latexit>e<latexit sha1_base64="Y/5EOLXHf6ne3aZjPCF6lzjLv7o=">A [binary]</latexit>p<latexit sha1_base64="9nGZyi523xGJtCdzu5t0 BknYkNY=">[binary]</latexit>FT e↵<latexit sha1_base64="fU7C3B52TBi CWqIwO2Ofaih8nl8=">[binary]</lat exit>↵<latexit sha1_base64="xhbvynZRyri Heb9F86uFx8M3yvI=">[binary] eDkM8fwB87nD4pnjxQ=</latexit><latexit sha1_base64="jSfhZpPPFzV WvjVMnfIHAendRGo=">[binary] WZO4A+czx/CvI6g</latexit>h<latexit sha1_base64="z2xU1ORh51LGZcg8XXl9Md13WpA=">AAAB7nicbVBNS [binary]</latexit>h↵<latexit sha1_base64="6pVte1MQdOwIP2aKnvSkbE/WxAk=">AAAB73icbVBNS [binary]</latexit>Figure 6.1: Schematic of the elastic swimmer.Initial positions of spheres arexα∗0 = h∗αe⊥,xβ∗0 = xα∗0 − L∗ cos(δ)p,h∗β = h∗α + L∗ cos(δ) sin(θ). (6.3)6.2.2 Background velocityA background flow field u∗∞(x∗) occurs due to the existence of a wall. We assumethat all geometries (two spheres and a wall) are relatively far apart so that we maysuperpose the generated flow field due to each sphere and expand it around a pointin the centre of each sphere xα0 and xβ0 using far-field approximation. We constructthe velocity field using singular multipoles of Stokeslet, Stokeslet dipole (doublet),1176.2. Problem formulationand Stokeslet quadrupole as of the formStokeslet G∗(x∗, e) =e|x∗| +e · x∗x∗|x∗|3 ,Doublet GD∗(x∗,d, e) =(d · x∗)e − (e · x∗)d − (e · d)x∗|x∗|3+ 3(e · x∗)(d · x∗)x∗|x∗|5 ,Quadrupole GQ∗(x∗, c,d, e) =(d · e)c + (c · e)d − (c · d)e|x∗|3 −3g∗(x∗, c,d, e)|x∗|5+ 15(c · x∗)(d · x∗)(e · x∗)x∗|x∗|7 , (6.4)whereg∗(x∗, c,d, e) =((d · e)(c · x∗) + (c · e)(d · x∗) + (c · d)(e · x∗))x∗+ (d · x∗)(e · x∗)c + (c · x∗)(e · x∗)d − (c · x∗)(d · x∗)e. (6.5)The source doublet and source quadrupole that are later employed have the formofSource Doublet SD∗(x∗, e) = − e|x∗|3 + 3e · x∗x∗|x∗|5 ,Source quadrupole SQ∗(x∗,d, e) = −3(d · x∗)e + (e · x∗)d + (d · e)x∗|x∗|5+ 15(e · x∗)(d · x∗)x∗|x∗|7 . (6.6)To meet the no-slip boundary condition on the wall, we employ Blake’s imagesolution generated by higher-order terms at the point of the sphere image at thewall to capture the impact of neighbouring surface using far-field approximations.The following are image solutions for a Stokeslet and a source doublet:G∗w(h∗,x∗, e, e∥, e⊥) = eˆ · eˆ∥(−G∗(x∗, e∥) + 2h∗GD∗(x∗, e∥, e⊥) − 2h∗2SD∗(x∗, e∥))+ eˆ · eˆ⊥(−G∗(x∗, e⊥) − 2h∗GD∗(x∗, e⊥, e⊥) + 2h∗2SD∗(x∗, e⊥)),SDw∗(h∗,x∗, e, e∥, e⊥) = eˆ · eˆ∥(SD∗(x∗, e∥) + 2GQ∗(x∗, e∥, e⊥, e⊥) − 2h∗SQ∗(x∗, e∥, e⊥))+ eˆ · eˆ⊥(− 3SD∗(x∗, e⊥) − 2GQ∗(x∗, e⊥, e⊥, e⊥) + 2h∗SQ∗(x∗, e⊥, e⊥)),(6.7)1186.2. Problem formulationat the image positions ofxα∗0im = xα∗0im − 2h∗αe⊥,xβ∗0im = xβ∗0im − 2h∗βe⊥. (6.8)Thus, we can describe the velocity field u∗(x∗) around a translating sphere withradius r∗ in terms of its moments at x0 due to the force f ∗e using a Stokeslet(fundamental solution for a point force) and a potential dipole (this must be addedto account for the finite sphere size) of the formu∗( f ∗, x∗, e, r∗) =f ∗8πµ∗(G∗(x∗, e) − r∗23SD∗(x∗, e)), (6.9)whereu∗( f ∗, x∗, e, a) =f e6πµaon the sphere. (6.10)Also its image velocity isu∗im( f∗, h∗, x∗, e, e∥, e⊥, r∗) =f ∗8πµ∗(G∗w(h∗, x∗, e, e∥, e⊥)− r∗23SDw∗(h∗, x∗, e, e∥, e⊥)). (6.11)Individual imposed velocity fiels for each sphere and its image areuα∗(x∗) = u∗(FαR∗, x∗ − xα∗0 , eαR, a∗α),uβ∗(x∗) = u∗(FβR∗, x∗ − xβ∗0 , eβR, a∗β),uα∗im(x∗) = u∗im(FαR∗, h∗α, x∗ − xα∗0im, eαR, e∥, e⊥, a∗α),uβ∗im(x∗) = u∗im(FβR∗, h∗β, x∗ − xβ∗0im, eβR, e∥, e⊥, a∗β), (6.12)where FR is the link force on the spheres, and eαR is the unit vector parallel toFαR. Each sphere interacts with the nearby sphere and the nearby wall as if in abackground flow field. The background velocity field for each sphere isBackground velocity field at sphere α uα∗∞ (x∗) = uβ∗(x∗) + uα∗im(x∗) + uβ∗im(x∗),Background velocity field at sphere β uβ∗∞ (x∗) = uα∗(x∗) + uα∗im(x∗) + uβ∗im(x∗).(6.13)1196.2. Problem formulationk<latexit sha1_base64="62nv0bb5EuPjZt4kBzEq4ouYGOg=">[binary]</ latexit> > 0<latexit sha1_base64="pO8aqF+w2EybpemzWyT3nZ60lPM=">[binary]</latexit> < 0<latexit sha1_base64="JmhP8eXfexlEaho/MhTkfDUkPDU=">[binary]</latexit>↵<latexit sha1_base64="xhbvynZRyriHe b9F86uFx8M3yvI=">[binary]</latexit><latexit sha1_base64="jSfhZpPPFzVWvjVMnfIHAendRGo=">AAAB7HicbVBNS8NAE [binary] nXNKWZO4A+czx/CvI6g</latexit>↵<latexit sha1_base64="xhbvynZRyriHeb9F86uFx8M3yvI=">AAAB7XicbVBNS8NAE [binary] seiteDkM8fwB87nD4pnjxQ=</latexit><latexit sha1_base64="jSfhZpPPFzVWvjVMnfIHAendRGo=">AAAB7HicbVBNS8NAE [binary] nXNKWZO4A+czx/CvI6g</latexit>↵<latexit sha1_base64="xhbvynZRyriHeb9F86uFx8M3yvI=">AAAB7XicbVBNS8NAE [binary] seiteDkM8fwB87nD4pnjxQ=</latexit><latexit sha1_base64="jSfhZpPPFzVWvjVMnfIHAendRGo=">AAAB7HicbVBNS8NAE [binary] nXNKWZO4A+czx/CvI6g</latexit>FT<latexit sha1_base64="v5r0SI1EQb/hU N8Fmg+6xkzp/Xg=">[binary]</latex it>T↵R<latexit sha1_base64="WQfWe6CsgaPAR UirRnaOpbRD+BY=">[binary] RlAc=</latexit>T↵R<latexit sha1_base64="qmXWvjVi+VviD C5FAE18cTqzJCw=">[binary] KJZQ+</latexit>T↵D<latexit sha1_base64="Blwzm8e8icGVq w5E0+klloBfOTs=">[binary] 3k/k=</latexit>TR<latexit sha1_base64="5MEepF1fCyuBW0+r1lcCDIm4PkI=">AAAB+nicbVDLTsMwE [binary] B7BM3gFb9aT9WK9Wx/z1oKVzxyAP7A+fwD9vpPK</latexit>TR<latexit sha1_base64="2yIj/Goarrt5f2I1ldTVtBtM4WA=">AAAB+XicbVBNS8NAE [binary] /SK3qzcerHerY9F65pVzJygP7A+fwCRmZOT</latexit>TD<latexit sha1_base64="2Y47kX5d0vgQJazcXMfgD0XhiBs=">AAAB+XicbVBNS8NAE [binary] Ayv8Gbl1ov1bn0sWtesYuYE/sD6/AF8DZOF</latexit>FR<latexit sha1_base64="OIGlrp0x4ikUXN/LbKZmgCwdv2E=">AAAB+XicbVBNS8NAE [binary] Ixe0ZuVWy/Wu/Uxb12xipkj9AfW5w9745OF</latexit>FR<latexit sha1_base64="OuyYXYV3WWd0X qI32eNIxbvrBQg=">[binary] oCJO8</latexit>F↵R<latexit sha1_base64="HS38/UoVRAPzt T82gzb/CewKAuk=">[binary] 0YZQw</latexit>F↵R<latexit sha1_base64="nsVLYj0+mTk9q rOu8Ew8AJ+AusM=">[binary] Nk/k=</latexit>F↵D<latexit sha1_base64="T+kJT6A7jpSiE rSDDyYX2BOP5DY=">[binary] zk+s=</latexit>FD<latexit sha1_base64="NGYt2tB8PQTDiqIv8dS2VvzjGuc=">AAAB+XicbVBNS8NAE [binary] md4hTcrt16sd+tj3rpiFTNH8AfW5w9mV5N3</latexit>Figure 6.2: Free diagram of spheres and rods.Faxe´n’s laws of the kind shown below capture the interaction between the spheresFαD∗= 6πµ∗a∗α(uα∗∞ (xα∗0 ) +a∗2α6∇∗2uα∗∞ (xα∗0 ) − U∗α),FβD∗= 6πµ∗a∗β(uβ∗∞ (xβ∗0 ) +a∗2β6∇∗2uβ∗∞ (xβ∗0 ) − U∗β),TαD∗= 8πµ∗a∗3α(ωα∗∞ (xα∗0 ) −Ω∗),TβD∗= 8πµ∗a∗3β(ωβ∗∞ (xβ∗0 ) −Ω∗), (6.14)where the background vorticity field is ω∗∞(x∗) = 12∇∗ × u∗∞(x∗) and the angularvelocity Ω∗ is constant for the assembly andU∗β = U∗α − L∗ cos(δ)Ω∗ × p. (6.15)The free diagram of the spheres and rods is shown in Figure 6.2 . Balance of forcesand moments acting on spheres is enforced to keep the swimmer in equilibrium.Sphere α F∗T + FαR∗+ FαD∗= 0,TαR∗+ TαD∗= 0, (6.16)Links FαR∗+ FβR∗= 0,TβR∗+ TαR∗ − L∗ cos(δ)p × FαR∗ = 0,TβR∗ − TαR∗ = k∗δe∥ × e⊥, (6.17)sphere β FβR∗+ FβD∗= 0,TβD∗+ TβR∗= 0. (6.18)Figure 6.2 additionally shows the bending caused by a torque dipole acting on theswimmer as a result of the background flow field. This torque dipole is in balance1206.2. Problem formulationwith the torque imparted by the spring and adds an equation to the systems ofequations Eq.6.17. To get analytical solutions, we disregard the rate of bendingand assume δ˙ = 0, whereas in chapter 5, we considered the rate of bending. Thereare now enough equations to solve the unknowns U∗α, U∗β, Ω∗, FαR∗, FβR∗, FαD∗, FβD∗,TαR∗, TβR∗, TαD∗, TβD∗, and δ. For simplicity, the sphere sizes are considered to besignificantly smaller than the scale of the body, and the swimmer is supposed to befar away from the wall in order to capture the impact of neighbouring surface usingfar-field approximations a∗β << L∗ << H∗α. Now we specify the small perturbationparameters of the formϵ =L∗H∗α,ξ =a∗βL∗, (6.19)where H∗α is the initial height of the swimmer. Velocity and forces are relatedlinearly and hydrodynamic interactions are obtained through the double asymptoticexpansions of solutions in ϵ and ξ of the formA =9∑m=−1m∑i=−1ϵiξm−iAim, (6.20)where A = {U∗,Ω∗, FαR∗}. In other words, unknowns are expanded over all poten-tial combinations of ϵ, and ξ.For non-dimensionalizing the quantities, we have used the initial distance ofthe sphere α from the wall H∗α for the length, µ∗H∗αU∗ for the forces, µ∗H∗αU∗ forthe torques and stiffness k∗, and U∗/H∗α for the angular velocity, where µ∗ is thedynamical viscosity and U∗ is the swimming velocity in an unbounded medium.Non-dimensional lengths are thenx =x∗H∗α,hα =h∗αH∗α,L =L∗H∗α= ϵ,aβ =a∗βH∗α= ϵξ,aα =a∗αH∗α= λϵξ. (6.21)1216.2. Problem formulationNon-dimensional velocity of the sphere β isUβ = Uα − ϵ cos(δ)Ω × p. (6.22)Non-dimensional forces and torques get the form ofFαD = 6πλϵξ(uα∞(xα0 ) +λ2ϵ2ξ26∇2uα∞(xα0 ) − Uα),FβD = 6πϵξ(uβ∞(xβ0) +ϵ2ξ26∇2uβ∞(xβ0) − Uβ),TαD = 8πλ3ϵ3ξ3(ωα∞(xα0 ) −Ω),TβD = 8πϵ3ξ3(ωβ∞(xβ0) −Ω). (6.23)As shown above, the link torque is orders of magnitude smaller than the link force,which justifies the removal of Rotlet from the flow field, resulting in lower orderflows in the far-field. The set of equations is solved until order 9, which is whenthe total of the powers of ϵ and ξ equals 9. The leading order solution of the systemof equations for a rigid swimmer isUα · e∥ = − γFT cos(θ)ϵξ6π(1 + λ)(1 − ξ3(1 − λ)2(1 + λ)− ξ2 9λ(1 − λ)2(1 + λ)2+ ξ31 + λ − 27λ2 + 27λ3 − λ4 − λ52(1 + λ)3− ϵ2ξγ9 sin(θ)16h2α)+ O(ϵ2, ϵξ, ϵ−1ξ3),Uα · e⊥ = FT sin(θ)ϵξ6π(1 + λ)(1 − ξ3(1 − λ)2(1 + λ)− ξ2 9λ(1 − λ)2(1 + λ)2+ ξ31 + λ − 27λ2 + 27λ3 − λ4 − λ52(1 + λ)3+ ϵ2ξγ9(3 cos(2θ) − 1)64h2α sin(θ))+ O(ϵ2, ϵξ, ϵ−1ξ3),Ω · (e∥ × e⊥) = 3FT (7 − cos(2θ)) sin(2θ)512π(1 + λ)(− ϵ 1h3α− ϵ2γ3 sin(θ)2h4α− ϵξ 3λh3α(1 + λ))+ O(ϵ3, ϵ2ξ, ϵξ2). (6.24)As we can see, when the particle is aligned with the boundary, θ = 0, the angularvelocity disappears, and we must dig deeper to uncover the leading order non-zeroterm.Ωθ=0 · (e∥ × e⊥) = ϵ2ξ2γFT (2 + λ − λ2 − 2λ3)64πh4αλ+ O(ϵ2ξ3, ϵ4ξ). (6.25)1226.2. Problem formulationThe leading order terms due to the bending k, areUkα · e∥ =ϵ3ξ2kF2T (1 − λ3)(3 − cos(2θ))cos(θ) sin2(θ)64h3απλ(1 + λ)+ O(ϵ3ξ3k,ϵ4ξ2k),Ukα · e⊥ =ϵ3ξ2kγF2T (1 − λ3)(3 − cos(2θ))sin(θ)cos2(θ)64h3απλ(1 + λ)+ O(ϵ3ξ3k,ϵ4ξ2k),Ωk · (e∥ × e⊥) =F2T cos(2θ)256π(1 + λ)(− ϵ2ξ2kγ4(1 − λ3)(3 − cos(2θ))sin(θ)h3αλ− ϵ2ξ3kγ12(1 − λ3)(3 − cos(2θ))sin(θ)h3α(1 + λ)− ϵ3ξ2k5 − 37λ3 − 3(1 − λ3)(8 cos(2θ) − cos(4θ))2h4αλ)+ O(ϵ2ξ4k,ϵ3ξ3k,ϵ4ξ2k),δk = −3FT cos(θ)128(1 + λ)(ϵ4ξ3k4(1 − λ3)(3 − cos(2θ))sin(θ)h3α+ϵ4ξ4k12λ(1 − λ3)(3 − cos(2θ))sin(θ)h3α(1 + λ)+ϵ5ξ3kγ5 − 37λ3 − 3(1 − λ3)(8 cos(2θ) − cos(4θ))2h4α)+ O(ϵ4ξ5k,ϵ5ξ4k,ϵ6ξ3k). (6.26)The swimming parallel to the wall where θ = 0 is of particular relevance in this1236.2. Problem formulationstudy. The leading order non-zero terms areU0α · e∥ = −γFTϵξ6π(1 + λ)(1 − ξ9λ(1 − λ)2(1 + λ)2)+ O(ϵ−1ξ, ϵmξnkfor any m + n = 10), (6.27)U0α · e⊥ = ϵ3γFT64πh2α(1 + λ)− ϵ4ξ2kF2T (1 − λ + λ2)32πλh4α+ O(ϵξ, ϵ4ξ3k), (6.28)Ω0 · (e∥ × e⊥) = ϵ2ξ2γFT (1 − λ)(2 + 3λ + 2λ2)64πλh4α+ϵ3ξ2kF2T (1 − λ + λ2)32πλh4α+ O(ϵ2ξ3, ϵmξnkfor any m + n = 10), (6.29)δ0 =ϵ5ξ3k3γFT (1 − λ + λ2)16h4α+ O(ϵ5ξ4k). (6.30)The findings up to this order of calculation reveal that the parallel velocity is unaf-fected by elasticity and that its direction is determined by the type of swimmer γ.However, vertical velocity is affected by both the type of swimmer and elasticity.We observe attraction for pushers and repulsion for pullers. Elasticity would pullthe swimmer (regardless of type or λ) to the wall. The angular velocity reorientsa puller or a pusher swimmer towards the wall for λ < 1 and away from the wallfor λ > 1. For identical spheres λ = 1, there is no rotation. Elasticity reorients apusher away from the wall and a puller towards the wall. Finally, elasticity bendsa puller towards the wall while bending a pusher away from the wall.The velocity at the boundaries can be used to calculate the flow field’s error.The flow field velocity is u(x) = uα(x) + uβ(x) + uαim(x) + uβim(x). The error of thetotal flow field velocity on the wall Erwall, the flow field produced by the sphere βon its skin Erβskin, and the total flow field on the sphere β’s skin Erskin are definedasErwall = u(x ∈ wall) = 0,Erβskin = |uβ(x ∈ sphere β) − Uβ| =1ϵFT4π(1 + λ)+ξϵ3FTλ4π(1 + λ)2+ O(ϵ),Erskin = |u(x ∈ sphere β) − Uβ| = ξϵFT16π(1 + λ)(3 sin(2θ) cos(Φ)+ (1 − 3 cos(2θ)) sin(Φ) cos(Θ))+ O(ϵ, ξ), (6.31)where Φ, and Θ are azimuthal and revolution angles on the sphere β. As demon-strated above, Erwall is zero as this constraint has been already enforced by Blake’s1246.3. Resultsimage solution, while Erβskin is in order of 1/ϵ due to the difference between the en-forced boundary condition of FβR/(6πµa) at Eq.6.10 and the sphere’s velocity Uβ.Finally, Erskin is the error of the total velocity field (which is the superposition ofthe flow field induced by each sphere and the wall) on the skin of the sphere β.6.3 ResultsTo determine the swimmer’s trajectory, the swimmer’s instantaneous velocity ismeasured, and the swimmer’s new location and orientation are accurately esti-mated. Because the flow field is independent of time and solely depends on theswimmer’s current configuration, the system instantaneously adjusts to the newconfiguration. We define the swimmer’s initial orientation as θi, where the thrustpushes (pulls) the swimmer towards the wall for θi < 0 and away from the wall forθi > 0. The hydrodynamic torque, on the other hand, reorients the swimmer andchanges its course.Figure 6.3a depicts the trajectories of a flexible swimmer at various initial ori-entations θi for pushers and pullers, whereas fig.6.3b depicts their y phase portraits.To save space, we have shown pullers moving in negative x direction. Swimmers’trajectories are highly dependent on their starting configuration. For small λ, theresults show that an originally parallel oriented pusher moves away from the wallwhereas a puller draws to it (look at eq. 6.28). A pusher that is slightly orientatedtowards the wall swims near the wall for a long time with an initial attraction untilit ultimately escapes almost linearly as the hydrodynamic effect of the wall decaysquadratically by hα. The sharp turn overs in the figure are caused by the log-scaledisplay of the trajectories. Pullers, on the other hand, prefer to go towards the wall.Furthermore, when swimming towards the wall, the swimmer prefers to lean to-wards it. The results in fig 6.3b for a pusher swimming slightly towards the wallindicate that the vertical velocity slows until it switches direction and then acceler-ates until the swimmer escapes the wall at a constant velocity. Pullers, on the otherhand, accelerate towards the wall (Note that the accuracy drops as the swimmergets close to the wall). Also, there is little difference between pushers and pullersescaping the wall if θi > 0. These findings are comparable to those obtained foran elastic swimmer propelled by mode B2 in chapter 5, as illustrated in fig.5.8a.Elastic pushers with small λ escape the wall, whereas pullers are drawn to it.Figure 6.4 illustrates how the trajectories alter as a function of λ, bending stiff-ness k, the swimmer’s initial position relative to the wall α, and the distance be-tween the tail and head β for initial orientation of θi = 0. Theoretical findings infig. 6.4a for a rigid swimmer (k → ∞) with δ = 0 indicate that the balance betweenhydrodynamic torque and force causes the swimmer to maintain a nearly parallel1256.3. Results-20 -10 0 10 20x10-1100101y0.30.20.10-0.10-0.3-0.30.30.20.1i=-0.2-0.1(a) (b)Figure 6.3: a) Trajectories and b) y phase portrait for different initial orientations θi.Solid and dashed lines show δ < 0, δ > 0, respectively. In a) blue lines correspondto the position of the swimmer after T = 1, 2, 3, 4, and 5 (scaled with H∗α/U∗).In b) black and blue lines correspond to pushers and pullers, respectively, andred dots show the initial position while blue dots across the curves demonstratethe position of the swimmer after increments of T = 1. Results are obtained forHα = 1, FT = 1, k = 10−4, λ = 0.1, ϵ = ξ = 0.11266.3. Resultstrajectory with the wall for a long time. In contrast, the results for elastic swimmersdemonstrate repulsion and attraction for pushers and pullers swimming parallel tothe wall (θi = 0), respectively. The effect of λ qualitatively is similar to the effect ofelasticity k. As λ approaches zero or k decreases, as seen in figures 6.4a, 6.4b, theswimmer travels less time before escaping from or attracting to the wall. However,for bigger λ or rigid swimmers, the swimmer remains parallel to the wall for anextended period of time. Figures .6.4c,6.4d shows that raising ϵ and/or ξ causesthe hydrodynamic torque to reorient the swimmer quicker.The results in fig.6.5 suggest that the bending angle δ decreases as the ratio getscloser to λ = 1, where the bending angle reduces to δ = 0+O(ϵ5ξ3/k) (see eq.6.26).The bending angle of a pusher pointed towards the wall is negative, indicating thatthe swimmer is bent towards the wall. When the ratio exceeds λ = 1, the bendingangle reverses sign and bends away from the wall. Also, an elastic swimmer withλ > 1 (where the thrust is produced by the larger sphere and the flagella feel greaterdrag than the head) bends more than a swimmer with λ < 1. The bending anglereduces to eq.6.30 at θi = 0 and is always positive for a pusher and always negativefor a puller.Figure 6.6 show the contour plots of normal velocity U · e⊥ = 0, rotation di-rection Ω · (e∥ × e⊥) = 0, and bending angle δ = 0, as a function of the initialorientation θi, and λ for pullers and pushers. It should be noted that the resultsare for the swimmer’s immediate response, and its trajectory may alter as it re-orients and the distance to the wall changes. A closer examination at the dottedand dashed-line curves shows that the bending is mostly in favor with angular ve-locity to direct the swimmer either towards the wall or away from it e.g., when apusher bends towards the wall δ > 0, the angular velocity re-orients it downwardΩ < 0. The normal velocity is another essential aspect in defining the swimmer’strajectory. The swimmer’s attraction or repulsion is determined by a mixture ofthese factors. The grey area in figure 6.6 depicts the terminus attraction zone aftersome time, starting at θi. The comparison of grey areas for pushers and pullerssuggests that a puller swimming almost parallel to the wall is likely to swim to-wards it, whereas for a pusher, the angle of initial orientation θi should be directedextensively to the wall in order for the pusher to swim towards it (θi < −0.25 forλ = 0.1).Figure 6.7 depicts the visual form of the swimmers and their trajectories forvarious initial orientations. Sharp turn overs previously seen in fig. 6.3, are nolonger apparent in this linear presentation of the trajectories. As the swimmer ap-proaches the wall, the accuracy decreases by 1/h2α, and the lubrication equations forsqueezing flow should be used to estimate the precise behaviour of the swimmer.For the time being, the findings indicate that the swimmer orients to the verticalposition θ = −π/2 while it is relatively far away from the wall, and that owing1276.3. Results(a) (b)(c) (d)Figure 6.4: Trajectories for different a) spring stiffness k, b) ratio λ, c) α, and d)β. Blue lines correspond to the position of the swimmer after T = 1, 2, 3, 4, and 5.Results are obtained for θi = 0,Hα = 1, FT = 1, k = 10−4, λ = 0.1, ϵ = ξ = 0.1unless otherwise mentioned.1286.3. Results10-3 10-2 10-1 100 1010.0020.00150.0010.00050-0.0005-0.001-0.0015-0.0020.20.150.10.050-0.05-0.1-0.15-0.2| i |=0| i |=0.1| i |= /8| i |= /4| i |=3 /8| i |= /210-3 10-1 1010.0020.0010-0.001-0.0020.20.1-0-0.1-0.210110-3 10-1- /2- /40/4/2/43/8/80.1- /2,0, /2-0.1-/8-3/8i =-/4i =010-3-10-3-10 -410 -4-10-3-10-210-210-300Figure 6.5: Variation of bending angle δ of a pusher as a function of λ for differentorientations θ. The right axis magnifies the results for higher λ > 1. The Contourplot within the figure shows negative bending angles with solid lines and positivevalues with dashed lines. Same results are shown for a puller within the figure.Results are obtained for FT = 1, k = 10−4, ϵ = ξ = 0.1.1296.3. Results10-3 10-2 10-1 100 101-0.3-0.2-0.100.10.20.3i10-3 10-2 10-1 100 101- /2- /40/4/2iU < 0> 0 > 0U > 0 < 0U > 0> 0 > 0U < 0 < 0Puller< 0 < 0 (a)10-3 10-2 10-1 100 101-0.3-0.2-0.100.10.20.3i10-3 10-2 10-1 100 101- /2- /40/4/2iU > 0 < 0 > 0U > 0> 0U < 0 < 0 > 0U < 0> 0Pusher < 0 < 0(b)Figure 6.6: Contour plots of normal velocity U · e⊥ = 0 (solid lines), angularvelocity Ω · (e∥ × e⊥) = 0 (dashed lines), and bending angle δ = 0 (dotted lines) asa function of λ and initial orientation θi. Blue lines are for a rigid swimmer. Grey(white) area is the terminus attraction (escape) zone for a flexible swimmer, whilethe hatched area is the terminus attraction zone for a rigid swimmer. Results areobtained for FT = 1, k = 10−4, ϵ = ξ = 0.1.1306.3. ResultsFigure 6.7: Swimmers alongside their trajectories for different initial orientationsθi. Blue and red links show δ < 0, δ > 0, respectively. Thin solid and dotted linesare the trajectories of the spheres A and B, respectively. Results are obtained forHα = 1, FT = 1, k = 4 ∗ 10−5, λ = 0.1, ϵ = 0.2, ξ = 0.2to symmetry and the nature of the Stokes equations, the swimmer maintains thistrajectory with a nearly zero bending angle (δ = 0).In the following section, we will examine the hydrodynamics of a swimmer inan unbounded fluid.1316.4. Swimming in an unbounded fluidL<latexit sha1_base64="GTYqfQBoIxfQYV3RRqT1x0FkHLQ=">AAAB6HicbVDLSsNA [binary] x7q0n69V6m7cWrHxmH/2C9f4Fux2NAw==</latexit>FT<latexit sha1_base64="v839nAyUYTZeVs+5k 58a3jeXKhU=">[binary] Vq5w5An9gff4A8jmWHQ==</latexit>a↵<latexit sha1_base64="pb83BWG8IEqs0MrOZ1b KtObPueg=">[binary]</latexit>a<latexit sha1_base64="6NnlR2cUxsIFEzKXydD7jAQ+Ytw=">AAAB7nicbVDLSgNBEJz1 [binary] =</latexit>Figure 6.8: Schematic of the swimmer in an unbounded fluid.6.4 Swimming in an unbounded fluidIn an unbounded medium with no wall, by considering L∗ as the length scale (ratherthan H∗α), the solution of the system of equations simplifies toU =FT6πξ(1 + λ)(1 − ξ3(1 − λ)2(1 + λ)− ξ2 9λ (1 − λ)2(1 + λ)2+ ξ31 + λ − 27λ2 + 27λ3 − λ4 − λ52(1 + λ)3)+ O(ξ3). (6.32)The hydrodynamic torque disappears, and the swimmer does not bend or re-orient. In fact, the swimmer swims in the direction of hydrodynamic thrust like arigid swimmer. The velocity of an assembly with λ = 1 will simplify to the well-known Stokes velocity of U = 12FT6πξ . As λ approaches zero, the velocity reduces toU = FT6πξ (1 − 32ξ + 12ξ3) + O(ξ3)6.5 An alternative formulationIn this section, we begin by examining the flow field generated by a sphere movingin a fluid at velocity U (rather than using the external link force FR in the previoussection) asu(U, x, r) =3r4(G(x,U) − r23SD(x,U)). (6.33)The motivation for investigating this formulation is because it allows us to satisfythe boundary condition on the skin of each sphere (Erβskin = 0 in eq.6.31). The1326.5. An alternative formulationimage solution of the velocity field takes the form ofuim(h, x,U, e∥, e⊥, r) =3r4(G⋆(h, x,U, e∥, e⊥) − r23SD⋆(h, x,U, e∥, e⊥)), (6.34)where on the sphere the velocity simplifies to u(U, x ∈ on the sphere, r) = U.Individual imposed velocity fields for each sphere and its image solution areuα(x) = u(x − xα0 ,Uα, λϵξ),uβ(x) = u(x − xβ0,Uβ, ϵξ),uαim(x) = uim(hα, x − xα0im,Uα, e∥, e⊥, λϵξ),uβim(x) = uim(hB, x − xβ0im,Uβ, e∥, e⊥, ϵξ), (6.35)whereUβ = Uα − ϵ cos(δ)Ω × p. (6.36)By eliminating the link force (which was not possible in the previous formula-tion since the velocity field of each sphere was constructed by FR), the system ofequations simplifies toFT + FβD + FαD = 0,TβD + TαD + ϵ cos(δ)p × FβD = 0,TβD − TαD + kδe∥ × e⊥ = 0, (6.37)and eventually more simplification leads toUα =11 + λ(FT6πϵξ+ uβ∞(xβ0) + λuα∞(xα0 )+ϵ2ξ26∇2(uβ∞(xβ0) + λ2uα∞(xα0 ))+ ϵ cos(δ)Ω × p),Ω =11 + λ3 − 34ξ2 cos2(δ)(ωβ∞(xβ0) + λ3ωα∞(xα0 )+34ϵξ2cos(δ)p ×(uβ∞(xβ0) +ϵ2ξ26∇2uβ∞(xβ0) − Uα)),δ =8πϵ3ξ3k(− ωβ∞(xβ0) + λ3ωα∞(xα0 ) +Ω(1 − λ3))· (e∥ × e⊥). (6.38)The leading order solution of the system of equations for a rigid swimmer isUα · e∥ = − γFT cos(θ)ϵξ6π(1 + λ)(1 + ξ3λ1 + λ− ϵξ9(1 + λ)16hα+ ξ29λ2(1 + λ)21336.5. An alternative formulation− ϵ2ξγ9(7 − cos(2θ))64h2αsin(θ) + ϵξ227(1 − 19λ − (1 − 3λ) cos(2θ))128hα− ξ3λ(2 + 5λ − 49λ2 + 3λ3 + λ4)2(1 + λ)3)+ O(ϵ2, ϵξ, ξ2, ϵ−1ξ3),Uα · e⊥ = FT sin(θ)ϵξ6π(1 + λ)(1 + ξ3λ1 + λ− ϵξ9(1 + λ)8hα+ ξ29λ2(1 + λ)2− ϵξ2 27(1 + 29λ + (1 − 3λ) cos(2θ))128hα− ξ3λ(2 + 5λ − 49λ2 + 3λ3 + λ4)2(1 + λ)3)− ϵ 3γFT (11 − 20 cos(2θ) + cos(4θ))512πh2α(1 + λ)+ O(ϵ2, ϵξ, ξ2, ϵ−1ξ3),Ω · (e∥ × e⊥) = 3FT cos(θ)ϵ128πhα(ϵ3 − cos(2θ)hα− ξγ3(1 − λ) sin(θ)+ ϵ2γ19 + 5λ − (5 + 3λ) cos(2θ)4h2α(1 + λ)sin(θ) − ϵξ3(1 − 4λ − (1 − 2λ) cos(2θ))2hα(1 + λ)− ξ2γ9λ(1 − λ)1 + λsin(θ))+ O(ϵ2, ϵξ, ξ2, ϵ−1ξ3). (6.39)The leading order terms due to the bending effect k, areUkα · e∥ = −F2T sin(2θ)64πλhα(ϵξ3k3(1 − 2λ − 2λ3 + λ4) sin(θ)− ϵ2ξ2kγ(1 − λ3)(3 − cos(2θ))hα)+ O(ϵ3ξ2k,ϵ2ξ3k,ϵξ4k),Ukα · e⊥ =F2T cos2(θ)32πλhα(− ϵξ3kγ3(1 − 2λ − 2λ3 + λ4) sin(θ)+ϵ2ξ2k(1 − λ3)(3 − cos(2θ))hα)+ O(ϵ3ξ2k,ϵ2ξ3k,ϵξ4k),Ωk · (e∥ × e⊥) = −F2T cos(θ)32πλhα(ϵξ2k(1 − λ3)(3 − cos(2θ))hα− ξ3kγ3(1 − 2λ − 2λ3 + λ4) sin(θ))+ O(ϵ2ξ2k,ϵξ3k,ξ4k),δk =3FT cos(θ)16hα(− ϵ3ξ3kγ(1 − λ3)(3 − cos(2θ))hα+ϵ2ξ4k3(1 − 2λ − 2λ3 + λ4) sin(θ))+ O(ϵ4ξ3k,ϵ3ξ4k,ϵ2ξ5k).(6.40)1346.5. An alternative formulationMost terms vanish when the particle is aligned with the wall, θ = 0. The leadingorder non-zero terms areU0α · e∥ = −γFTϵξ6π(1 + λ)(1 + ξ3λ1 + λ)+ O(1, ϵ−1ξ, ϵ4ξ3k),U0α · e⊥ = ϵ3γFT64πh2α(1 + λ)+ϵ2ξ2kF2T (1 − λ3)16πλh2α+ O(ϵξ, ϵ2ξ3k),Ω0 · (e∥ × e⊥) = − 3γFT64πh2α− ϵξ2kF2T (1 − λ3)16πλh2α+ O(ξ, ϵξ3k),δ0 = −ϵ3ξ3k3γFT (1 − λ3)8h2α+ O(ϵ3ξ4k), (6.41)which differ from previous estimations. The elasticity in the previous equationwould constantly drive the swimmers vertically towards the wall, but now it isdependent on λ. Swimmers with lower ratios λ are able to escape the wall. Also,the ordering differ, and the elasticity effect is more significant in this calculation.The rotation also helps to reorient the swimmers away from the wall.Furthermore, the previously defined flow-field errors areErwall = u(x ∈ wall) = 0,Erβskin = |uβ(x ∈ sphere β) − Uβ| = 0,Erskin = |u(x ∈ sphere β) − Uβ| = 1ϵFTλ4π(λ + 1)− 3FT32πhα(1 − 2 cos2(θ))+ξϵFTλ4π(λ + 1)2(3λ − (1 + λ)(cos(θ) cos(Φ) + sin(θ) sin(Φ) cos(Θ)))+ O(ϵ, ξ), (6.42)where Φ, and Θ are azimuthal and revolution angles on sphere β. The first twoerror definitions are zero, as they have been forced as constraints.Figure 6.9 shows the trajectories of a swimmer almost parallel to the wall fordifferent ratios λ = 0.1, 1, 3, and different swimmer stiffness k. While the outcomesfor pushers are qualitatively the same, pullers have a better probability of escapingthe wall as compared to the prior formulation. Flexible pushers escape the wallquickly while pullers escape gradually for smaller λ and vice versa for larger λ.Furthermore, results for pushers and pullers with λ = 1 are identical. These rigidswimmer results in figure 6.9 are qualitatively consistent with the findings for arigid swimmer propelled by mode B2 in chapter 4 fig. 4.5a.1356.5. An alternative formulation-30 -20 -10 0 10 20 30x01234567y0.2-0.2-0.2i=0.200(a)-30 -20 -10 0 10 20 30x012345678y0.20-0.2-0.20i=0.2(b)-25 -20 -15 -10 -5 0 5 10 15 20 25x0510152025y(c)-20 -15 -10 -5 0 5 10 15 20x00.511.522.533.544.55y0.2-0.20-0.20i=0.2(d)-20 -15 -10 -5 0 5 10 15 20x00.511.522.533.544.55y0.2-0.20 0-0.2i=0.2(e)-20 -15 -10 -5 0 5 10 15 20x0123456789yi=0.20.20 0-0.2 -0.2(f)-10 -8 -6 -4 -2 0 2 4 6 8 10x00.511.522.533.5yi=0.20-0.2-0.200.2(g)-10 -8 -6 -4 -2 0 2 4 6 8 10x00.511.522.533.544.55y-0.200.2i=0.20-0.2(h)-8 -6 -4 -2 0 2 4 6 8x012345678910y(i)Figure 6.9: Trajectories for different initial orientations −0.2 ⩽ θi ⩽ 0.2 and fora,d,g) a rigid swimmer with k = ∞, b,e,h) a moderately flexible swimmer and c,f,i)a very flexible swimmer. Also, the rows are for λ = 0.1, 1, 3, respectively. Solidand dashed lines show δ ⩽ 0, δ > 0, respectively. Blue lines correspond to theposition of the swimmer after T = 1, 2, 3, 4, and 5 (scaled with H∗α/U∗). Resultsare obtained for Hα = 1, FT = 1, ϵ = ξ = 0.11366.6. Conclusion6.6 ConclusionAnalysis of an elastic swimmer near a flat wall using a far-field approximation hasbeen carried out. The toy model is formed by two spheres linked by two rods and atorsional spring in the middle. The spheres capture the physics of a head and a tailwhile the torsional spring allows the swimmer to bend. The motion of the elasticswimmer is force and torque free and the thrust generated by the tail is completelybalanced by the hydrodynamic drag. This analysis is limited to Stokes regimewhere no Brownian motion takes place in the absence of inertia. Eventually, thewall would lead pushers and pullers to bend and this elastic bending can alter theirdynamics and trajectories. We have shown different scenarios of motion dependingon the initial orientation, and the geometry of the swimmer. We demonstrated thatwall-trapping dynamics of microorganisms can be significantly altered by shapechange due to flexibility. Qualitatively, a pusher swims near the wall for a long timewith an initial attraction, eventually escaping almost linearly as the hydrodynamicinfluence of the wall decays quadratically by hα, whereas a puller draws to the wall.Furthermore, λ has a qualitatively similar impact to stiffness. As the tail shrinksor elasticity increases, the swimmer swims less distance before escaping or beingdrawn to the wall.We can isolate elasticity’s contribution and analyse how it affects the dynamicsof a rigid swimmer. Consider a swimmer which is swimming parallel to a wall.Elasticity has no effect on the parallel velocity up to this degree of calculation. Arigid pusher is drawn to the wall, but a rigid puller escapes it. A rigid puller orpusher swimmer is reoriented towards the wall when λ < 1 and away from the wallwhen λ > 1. No rotation is observed for λ = 1. While the attraction and repulsionare consistent with the findings in chapter 4 for a rigid swimmer propelled by B1,the effect of λ on the direction of rotation is the opposite (see 4.2).Elasticity, on the other hand, would draw the swimmer vertically to the wall(regardless of type or λ), while reorienting a pusher away from the wall and apuller towards the wall. Elasticity also causes a puller to bend towards the wall anda pusher to bend away from the wall. All in all, elasticity allows a pusher to escapewhile pulling a puller towards the wall. These results are close to those obtained inchapter 5 for an elastic swimmer propelled by mode B2, as shown in fig.5.8: Elasticpullers with small λ are drawn to the wall, whereas pushers are pushed away fromit.It is also crucial to note the distinction between the elastic swimmer studiedhere and the one in chapter 5. Figure 6.10 depicts the far-field results for an elasticswimmer parallel to the wall and the ”equivalent” elastic swimmer analysed usingthe active SD method. Note that these elastic swimmers have distinct propellingmodes. Squirming modes are responsible for the previous swimmer’s propulsion,1376.6. Conclusion-1 -0.5 0 0.5 1x00.20.40.60.811.21.41.61.82yPusherPullerFigure 6.10: The comparison of a far-field study of an elastic swimmer (solid lines)with the elastic swimmer investigated in chapter 5 (dashed lines). Results are ob-tained for θi = 0,Hα = 1, FT = 1, k = 4 ∗ 10−5, λ = 0.1, ϵ = 0.2, ξ = 0.2.however in this case, the tail generates the swim force. We can observe that the ve-locity in the active SD method is linear with the propelling magnitude, but only theleading order solutions are linear with FT in the far-field approach. Furthermore,the swimmer studied in this chapter, is static in the sense that the relative rotationof the head and tail is ignored, and the swimmer rotates as a rigid body (δ˙ = 0). Wealso assumed that the bending angle is small and that it could be expanded asymp-totically. We previously demonstrated that an elastic pusher is unstable, and anydeviation causes the swimmer to bend until a more stable configuration is achieved.Figure 6.10 depicts how a pusher with no bending angle constraint falls into an or-bit while the latter swimmer escapes the wall. Finally, in this model, near-fieldcontributions are ignored.138Chapter 7Jeffery orbits in shear-thinningfluids7.1 IntroductionPastes [52], paints [80], pulps [43], and ceramics [207], among many other chemi-cal substances [7], contain suspensions of (anisotropic) particles. Biological fluidsalso carry nutrition [96], Bacillus species [55], and other mobile micro-organismsof mostly rod shape [57, 194]. These are but a few examples that demonstrate theimportance of understanding the underlying dynamics and rheology of particlessuspended in fluid flows. As we will highlight below, this area has a long literaturewhen the suspending fluid is Newtonian, but our understanding is much more basicwhen the fluid is non-Newtonian.The presence of particles in a fluid can dramatically change its rheology. Ein-stein [78, 79] calculated an effective viscosity for a dilute suspension of non-colloidal hard spheres and showed that the effective viscosity of the suspension in-creases linearly with the volume fraction of spheres. For higher volume fractions,hydrodynamic (and contact) interactions become more significant. Suspensionsgenerally shear thin with increasing the shear [210] while beyond a threshold ex-hibit shear thickening or even jamming [212]. If the particles are anisotropic, theneven small deviations from spherical geometry can significantly impact the rheol-ogy even for dilute suspensions [133]. One reason for this is because elongatedor rod-like particles tend to align with the flow thus changing the properties of thefluid under shear. Therefore, understanding the dynamics of even a single parti-cle in shear is important to determine the rheology of a suspension of anisotropicparticles.The dynamics of elongated particles, for example axisymmetric ellipsoids andspheroids, display significantly more complicated behaviour than spherical parti-cles under shear [38, 73, 113, 159]. At zero Reynolds number, spheroids and longslender bodies in shear flow undergo a periodic motion. A century ago, Jeffery[113] solved the motion of a neutrally buoyant ellipsoid of revolution in a simpleuniform shear flow in the absence of inertial and Brownian forces. He found that1397.1. Introductionthe particle’s axis of revolution rotates on infinitely many degenerate periodic or-bits called “Jeffery orbits”. Jeffery’s solution is degenerate in the sense that theorientation of the body at long times depends on its initial orientation. Jefferysuggested that this degeneracy would be lifted by inertia and speculated that theparticle would evolve to an orbit corresponding to the minimum mean energy dissi-pation. A year after Jeffery’s calculations, Taylor [202] experimentally showed thatan ellipsoid of revolution in a simple shear flow drifts through the continuous fam-ily of Jeffery orbits until the ellipsoid is rotating in a final preferred orbit. A prolatespheroid, after approximately 180 complete revolutions, would settle into a log-rolling final orbit, rotating perpendicular to the shear plane so that its long axis isparallel to the vortex direction. In contrast, an oblate spheroid, after approximately40 revolutions, would assume a tumbling orbit in which its axis of revolution isin the shear plane and rotates with variable angular velocity. In other words, Tay-lor confirmed Jeffery’s minimum energy hypothesis for spheroidal particles withinthe range of his experiments. Harper and Chang [97] later showed theoreticallyand experimentally that the preferred constant orbit for a dumb-bell shaped bodycorresponds to maximum dissipation (tumbling in the flow-shear plane). With therecent use of direct numerical simulations (DNS) and heavy computations, Dingand Aidun [62] showed that the period of a prolate particle diverges for higherReynolds numbers till the particle remains motionless when it is nearly alignedwith the flow direction. Recently, Einarsson et al. [35, 36, 74–76, 177] have showntheoretically that in the limit of weak flow and particle inertia, the degeneracy ofJeffery orbits is indeed lifted. The first effects due to weak inertia cause a pro-late spheroid in simple shear to drift to a stable tumbling limit cycle, whatever theinitial condition [74].Bretherton [32] investigated the motion of a particle of more general shape inshear flow and in the presence of rigid boundaries. He showed that axisymmetricparticles follow Jeffery’s equation of motion if the aspect ratio is replaced with anapproximate effective aspect ratio. The theoretical work of Hinch and Leal [104]on non-axisymmetric ellipsoids in shear flow shows that even a small deviationfrom axisymmetric geometry results in profound changes in the nature of the orbit.Consequently, theoretical models quantifying real solutions of particles, on theassumption of axisymmetric particles, can be inaccurate. The motion of a non-axisymmetric ellipsoid has two periodic parts called “doubly periodic” tumbling:a rapid rotation similar to Jeffery orbits and a slower drift in the orbits. Recently,Masoud et al. [144] showed numerically that porous ellipsoids follow Jeffery orbits(of impermeable ellipsoids) to very good approximation.As noted above, most prior studies have focused on Newtonian fluids, but forsuspensions in non-Newtonian fluids the fundamental building blocks governingthe rheology are still being developed. Non-Newtonian carrier fluids can produce1407.1. Introductionqualitative changes in the rheological behaviour of suspensions in comparison withNewtonian fluids, as an example, recent studies have calculated modifications ofthe Einstein viscosity for dilute suspensions of spheres in weakly nonlinear vis-coelastic fluids [77], and shear-thinning fluids [58]. The dynamics of individualparticles in flows can also be substantially modified by complex fluid rheology.Early work focused on the dynamics of spherical particles in weakly nonlinear vis-coelastic fluids, and much of that work is summarized in the wonderful review byLeal [132]. The dynamics of anisotropic particles in shear flows of viscoelasticfluids has also received attention. In experimental work, Saffman [180] reportedthat spheroidal particles deviate from Jeffery orbits in viscoelastic fluids. Exper-iments conducted by Bartram et al. [11] in viscoelastic fluids showed that at lowshear rates (in comparison to the relaxation time of the fluid) slender bodies tendto see an increase in period of rotation and a drift towards the log-rolling position;however, at higher shear rates, they tend to face the flow direction and stop rotating.Leal [131] found similar results, calculated theoretically, for the motion of rod-likeparticles in second-order fluids. Brunn [33] found theoretically that the effect ofa second-order fluid on ellipsoidal particles in shear flow is as follows: a prolatespheroid drifts to log-rolling, whereas an oblate tumbles around the vorticity axisin direct contrastdistinction to the effects of weak inertia [74]. Gunes et al. [93]carried out experiments for prolate spheroids of moderate aspect ratio in severalsuspending fluids. They found that elastic effects tend to increase the period ofrotation and that the orbits start to drift towards log-rolling. In recent numericalsimulations D’Avino et al. [61] have shown that a prolate particle in a viscoelas-tic fluid, achieves a log rolling orbit at low shear rates while the particle tends toalign in the flow direction at high shear rates. Sobhani et al. [188] using the lattice-Boltzmann method studied the dynamics of an elliptic particle in a yield-stress fluidin a finite, close-ended channel. The particle releases from the centreline and ini-tially sediments toward one of the side-walls before moving toward the centerline.The particle eventually keeps falling at a distance from the midplane independentof the initial inclination angle.Much of the work on the dynamics of particles in complex fluids have focusedon the effects of viscoelasticity, see for example the recent review by Shaqfeh [184]that summarizes prior research and recent advances on the rheology of particlesuspensions in viscoelastic fluids. However, many realistic complex fluids tendto exhibit both viscoelasticity and shear-dependent rheology [21, 34]. RecentlyDatt and Elfring [58] explored different dynamics of spherical particles in shear-thinning fluids, and observed for example that the rotation rate of a sphere in shearflow is unaffected by shear-thinning rheology, but the impacts of shear rheologyon the dynamics of other, (anisotropic) particles have received almost no attention.Recently, Fe´rec et al. [87] investigated the dynamics of a two-dimensional ellip-1417.2. Mathematical modelsoidal particle in shear flow of a power-law fluid using a finite element simulation.In this proceeding, the authors show a slight reduction in angular velocity that di-minishes with particle aspect ratio. In this chapter we consider the dynamics of aneutrally buoyant three-dimensional prolate spheroid in a shear flow of a weaklyshear-thinning fluid. The particle is small enough that inertial forces are negligibleand but big enough that Brownian forces do not play a role. To capture the leading-order effects of shear-thinning rheology on the dynamics, we solve for the motionof the particle for asymptotically weak shear thinning using the Carreau model.The question we ask is whether shear-thinning rheology affects the orientationaldynamics and to what extent, and in particular, is shear-thinning rheology suffi-cient to lift the degeneracy of Jeffery orbits in Newtonian fluids. We will show thatthe orbits are indeed modified but the degeneracy must remain due to the symme-try of the constitutive equations. This paper is organized as follows: we start withthe rheology of shear-thinning fluids, then move to the problem of a single prolatespheroid in the linear flow of Newtonian fluid. Finally, we study the deviation ofthe dynamics due to weak shear thinning.7.2 Mathematical model7.2.1 A spheroid in shearWe consider here a prolate spheroid B with surface ∂B, whose axis of symmetry isdefined by the unit vector p as shown in Fig. 7.1. The polar angle of the particle,with respect to a fixed lab frame whose origin is at the particle center, is θ whileϕ is the azimuthal angle. The major (minor) axis length is denoted by a (b) andthe aspect ratio is λ = a/b = 1/√1 − e2 > 1. The spheroid is immersed in anotherwise linear velocity field defined by u∞ = A∞ ·x, whose origin is at the centerof the particle to eliminate any constant translation. The velocity gradient tensor∇u∞ = A∞ is constant and may be decomposed in the usual way into symmetricand antisymmetric parts A∞ = E∞ + Ω∞ × I where Ω∞ is the angular velocity ofthe background flow (I is the identity). The flow-shear plane is defined by the basisex, and ey.Given a fluid velocity field u, we define a disturbance velocity field relative tothe background asu′ = u − u∞. (7.1)We assume no slip on the surface of the spheroid and that the disturbance flowis zero far from the spheroid, written in terms of the disturbance quantities the1427.2. Mathematical modelp⌦12a2b✓Figure 7.1: A prolate spheroid in simple shear flow.boundary conditions areu′ = (Ω −Ω∞) × x − E∞ · x x ∈ ∂B, (7.2)u′ = 0 |x| → ∞, (7.3)where Ω is the angular velocity of the spheroid (the translation velocity U = 0 byconstruction). The difference in the angular velocity of the spheroid and the undis-turbed background flow Ω′ = Ω −Ω∞ is sometimes referred to as the slip angularvelocity [75] (although the no-slip condition is indeed satisfied). The evolution ofthe particle director isp˙ = Ω × p. (7.4)7.2.2 Shear-thinning fluidsWe consider here fluids that shear thin, meaning the viscosity of the fluids, η, de-creases with increasing strain rates, γ˙. To capture this behavior and its effect onthe dynamics of a prolate ellipsoid in shear we use the Carreau model [21] for ageneralized Newtonian fluid with deviatoric stressτ = η(γ˙)γ˙, (7.5)where the functional dependence of the viscosity on the strain-rate,η(γ˙) = η∞ + (η0 − η∞)[1 + λ2t |γ˙|2] (n−1)2 , (7.6)is characterized by a zero-shear viscosity η0, an infinite-shear viscosity η∞, a timeconstant λt, and a power-law index n < 1. The magnitude of strain-rate is defined|γ˙| = √γ˙ : γ˙. In this study, we explore only the weakly shear-thinning effects onthe dynamics of the particle. In this regard, we always assume that λt ≪ 1/γ˙c,where we define γ˙c ≡√2E∞ : E∞ as the characteristic strain rate of the flowdefined by the external imposed flow. The deviatoric stress may be conveniently1437.2. Mathematical modeldecomposed into Newtonian and non-Newtonian parts, τ = η0γ˙ + τNN where thenon-Newtonian part,τNN = (η(γ˙) − η0)γ˙, (7.7)is defined as the amount of shear thinning relative to the zero-shear viscosity η0.7.2.3 Motion of a particle in a complex fluidIn the absence of inertia, the velocity of a particle in a background flow of a fluidof arbitrary rheology may be written asU = Rˆ−1FU · [Fext + FT + FNN] , (7.8)where U = [U Ω]⊤ is a 6-dimensional vectors containing translational and rota-tional velocities, likewise F = [F L]⊤ represents both force and torque [81]. Weconsider here a particle that is neutrally buoyant and that no other external forceacts on the particle and thus Fext = 0.The termFT = −∫∂Bu∞ · (n · TˆU) dS , (7.9)is the drag force and torque on a particle held fixed in a background flow u∞ ofa Newtonian fluid of constant viscosity η0. In a Newtonian fluid this force, alongwith the rigid-body resistance tensor,RˆFU =[RˆFU RˆFΩRˆLU RˆLΩ], (7.10)for the particle in the same Newtonian fluid, would entirely determine the dynamicsof a freely moving particle U0 = Rˆ−1FU · FT .In a non-Newtonian fluid, there is an extra force/torque on the particle dueto the extra deviatoric stress τNN in the fluid volume V in which the particle isimmersed [132]. This force given byFNN = −∫VτNN : EˆU dV. (7.11)We may write for simplicity that velocity is composed of a Newtonian part and anon-Newtonian correction U = U0 + Rˆ−1FU · FNN . The tensors EˆU and TˆU , in ad-dition to RˆFU , are linear operators that are calculated from the resistance/mobilityproblem (for the same particle) in a Newtonian fluid with viscosity η0. The tensors1447.2. Mathematical modelEˆU and TˆU are functions of position in space that map the rigid-body motion Uˆ of aparticle to the fluid strain-rate ˆ˙γ = 2EˆU ·Uˆ and stress fields σˆ = TˆU ·Uˆ, respectively.In our study the translational velocity of the particle is zero, U = 0, and bysymmetry the only relevant component of the hydrodynamic resistance is RˆLΩ.Because of this the above general expressions simplify considerably so that theangular velocity of the particle in a non-Newtonian fluid isΩ = Ω0 − Rˆ−1LΩ ·∫VτNN : EˆΩ dV, (7.12)where the angular velocity in a Newtonian fluidΩ0 = −Rˆ−1LΩ ·∫∂Bu∞ · (n · TˆΩ) dS . (7.13)Clearly, the Newtonian dynamics are well known, and to determine the correctionwe must (only) resolve the integral on the right-hand side of (7.12). The tensorsRˆLΩ, EˆΩ, and TˆΩ for prolate spheroids are also well known (see details in appendixC.3), but to calculate the integral one must also know the non-Newtonian stressτNN in the entire fluid domain and thus requires resolution of the non-Newtonianflow field. To bypass this difficulty we employ a perturbative approach whereinthe leading-order contributions to the non-Newtonian stress are determined by theflow-field of the Newtonian solution τNN ∼ τNN(u0).7.2.4 Asymptotic solutionFirst we non-dimensionlize our equations (denoted by *), lengths are scaled by themajor axis length a and stresses by η0γ˙c. The dimensionless non-Newtonian stressis thusτ∗NN = −(1 − β)(1 − [1 +Cu2|γ˙∗|2](n−1)/2)γ˙∗. (7.14)The Carreau number Cu = γ˙cλt is the ratio of the characteristic strain rate γ˙cto the crossover strain-rate 1/λt, while the viscosity ratio is β = η∞/η0. Theshear-thinning correction in constitutive equation appears with the factor of Cu2.Thus, we assume that the viscosity is not substantially deviant from η0 and theflow quantities are expanded in regular perturbation series in powers of Cu2, u∗ =u∗0 + Cu2u∗1 + O(Cu4), and τ∗ = τ∗0 + Cu2τ∗1 + O(Cu4) where u∗, and τ∗ are thedimensionless velocity field, and deviatoric stress fields respectively. In this way,the non-Newtonian deviatoric stressτ∗NN = Cu2τ∗NN,1 + O(Cu4)= −12Cu2(1 − β)(1 − n)|γ˙∗0|2γ˙∗0 + O(Cu4). (7.15)1457.3. ResultsWriting similarly for the orientational dynamics of the spheroid Ω∗ = Ω∗0 +Cu2Ω∗1 + O(Cu4) we find, by way of equations (7.12) and (7.15), thatΩ∗1 = −Rˆ∗−1LΩ ·∫Vτ∗NN,1 : Eˆ∗Ω dV∗=12(1 − β)(1 − n)Rˆ∗−1LΩ ·∫V|γ˙∗0|2γ˙∗0 : Eˆ∗Ω dV∗. (7.16)We see that the change in the orientational dynamics depends only on the Newto-nian velocity field u∗0 to leading order in Cu. The main mathematical task of thiswork is to calculate and integrate the tensor |γ˙∗0|2γ˙∗0 : Eˆ∗Ω over the entire fluid do-mainV. This is a rather involved integral and therefore we want to use as amenablea representation of the Newtonian solution as possible. To this end, we use a solu-tion based on an expansion in spheroidal multipoles taken directly from Einarssonet al. [75]. For completeness we repeat details of that solution in the appendix ofthis work.For simplicity we now drop the *’s and use only dimensionless variables fromthis point on unless explicitly stated otherwise.7.3 Results7.3.1 Jeffery OrbitsThe solution to the zeroth order (Newtonian) problem yields the classical Jefferyorbits of an ellipsoidal particle in shear flow. The solution obtained by Jeffery [113]involved solving for the disturbance flow field, but it can also be found by directlyintegrating (7.13). The angular velocity of a prolate ellipsoid in a Newtonian fluidisΩ0 = Ω∞ + Λp × E∞ · p, (7.17)where Λ = λ2−1λ2+1 , while the evolution of the director isp˙ = Ω0 × p = Ω∞ × p + Λ (I − pp) · E∞ · p. (7.18)Jeffery’s results show that, unlike a point particle or a sphere, a prolate spheroidrotates not only with the (constant) local angular velocity of the flow but, giventhat object is elongated and unevenly samples the velocity field about its center,has a rotational component that depends on the orientation and aspect ratio of thespheroid.1467.3. ResultsFigure 7.2: Trajectories of a prolate spheroid’s pole with aspect ratio 5 in a linearshear flow of Newtonian fluid for different initial positions.Assuming a background flow field (as we do throughout this paper)u∞ = yex (7.19)and ϕ = 0 at t = 0, leads to Jeffery orbits of the formtan ϕ = λ tan(λt1 + λ2),tan θ =Cλ√sin2ϕ + λ2cos2ϕ, (7.20)where C is a constant of integration and the axis of revolution rotates in one ofinfinitely many possible periodic orbits depending on the value of C. Fig. 7.2shows Jeffery orbits for different values of C on a unit sphere. The Jeffery orbit onthe equator of the sphere, C → ∞, is called the tumbling orbit because the vectorp tumbles in the flow-shear plane. The orbit at the pole of the sphere, C = 0,where p is aligned with the vorticity direction, is called log-rolling. The periodT0 = 2π(λ2+1)/λ is constant for particles of same aspect ratio and does not dependon the initial orientation of the particle (in dimensional terms the period scales with1/γ˙c). Note that the angular velocity of the particle is not constant in time but thatthe particle slows down when p tends to the flow direction [60].1477.3. Results7.3.2 Dynamics of a prolate spheroid in shear flow of ashear-thinning fluidThe zeroth order (Newtonian) solution of the flow field is linear, and thus we mustbe able to write γ˙0 = M : E∞ where M is a fourth order tensor that depends onthe orientation of the particle alone. Substituting into (7.16) we obtain for eachcomponent of Ω1Ω1 j =(1 − β)(1 − n)2Rˆ−1LΩ, jtE∞ls E∞pqE∞g f∫VMimslMmipqMuv f gEˆΩ,vut dV, (7.21)where repeated indices are summed. Although M is constructed from a knownNewtonian solution the details are quite complicated (as shown in appendix). Thistensor product contains hundreds of terms and analytical evaluation of the integralproves more or less intractable and so this integral is performed numerically. Aftercalculation of all the tensors, a trapezoidal rule in spheroidal coordinates is usedfor the integration with singular terms evaluated analytically.Upon resolution of Ω1 we calculate the periodic orbits of p in a shear-thinningfluid to leading order in Cu, namely we integratep˙ = (Ω0 +Cu2Ω1) × p (7.22)forward in time. An RK4 method is used for the time derivative of the particle’sorientation, and the orientation is expressed by angles θ and ϕ rather than vector pto ensure unity of its magnitude.The dynamics of the spheroid can be divided into a rotation around the vorticityaxis, ϕ (spinning), and rotation about the velocity axis, θ (oscillating). While theperiod in a Newtonian fluid, T0, is determined only by the particle shape and shear-rate, the period in a non-Newtonian fluid, T , is highly dependent on initial positionof the particle and Cu number in shear-thinning fluids.In Fig. 7.3 we show Jeffery orbits modified by shear thinning for differentsituations: a) different initial positions θi (for ϕi = 0), b) different initial positionson a specific Newtonian Jeffery orbit passing (θi, ϕi) = (5π/12, 0) c) different valuesof Cu, and finally d) different aspect ratios λ. In these results and all that followwe take values of β = 0.5, and n = 0.5. We also take Cu = 0.1, and λ = 5unless otherwise stated. The first thing to notice in the figures is that shear-thinningrheology does not lift the degeneracy of the Jeffery orbits observed in Newtonianfluids, unlike the effects of fluid elasticity or inertia. There are still infinitely manymodified ‘Jeffery’ orbits, selected by the initial condition, that repeat periodicallyfor all time. Indeed this is somewhat expected given that the generalized Newtonianfluid constitutive equation (7.5), maintains the symmetries of the Stokes equationsand so one should not expect a symmetry-breaking drift of the orbits in time. In1487.3. Results(a) (b)(c) (d)Figure 7.3: Modified orbits (red lines) in the presence of shear thinningand Newtonian orbits (blue lines) for a) different initial conditions of θi =0, π/12, π/4, π/3, 5π/12, 11π/24, π/2 and ϕi = 0, b) different initial conditions ofϕi = π/2, π/4, 0 on a specific Jeffery orbit passing (θi, ϕi) = (5π/12, 0), c) differentCarreau numbers of Cu = 0, 0.1, 0.12, 0.14, 0.16 for two different initial conditionsof (θi, ϕi) = (π/3, 0), (0.33, π/2) belonging to a Jeffery orbit, and d) different valuesof aspect ratios λ = 1, 2, 3, 4, 5 for (θi, ϕi) = (π/3, 0). Calculations are carried outfor Cu = 0.1, and λ = 5 unless otherwise stated.1497.3. Results(a) (b) (c)Figure 7.4: Phase portraits of a) θ and b) ϕ for different Cu numbers. Dashed-lineshows the Newtonian case. Dotted-lines in a is for different initial condition of(θi, ϕi) = (0.33, π/2) following Fig.7.3c. Results are for (θi, ϕi) = (π/3, 0), andλ = 5.general we observe that shear thinning tends to narrow the orbits in much the sameway that an elongation of the particle aspect ratio does (compare Fig. 7.3a), c)with d). The change in a particular Newtonian Jeffery orbit due to shear thinningdepends on the initial position as shown in Fig. 7.3 b), because of course the orbitsare not continuously overlapping.As the aspect ratio of the particle increases, the particle spends a larger amountof time aligned with the flow as the torque due to the applied background flow isdiminished in comparison to the hydrodynamic resistance to rotation, as dictated by(7.13), and thus the period of rotation increases. Much of the same thing happenswith shear thinning , where the torque is reduced due to changes in the viscosity.These changes in the dynamics are illustrated in Fig. 7.4a) and 7.4b). In particular,note in Fig. 7.4b) that the effects of shear thinning on the particle spin (ϕ˙) are onlyapparent when the particle is aligned with the flow, ϕ ≈ nπ, where shear dominatesand hence the particle is slowed further due to a reduction of the viscosity. Whenthe (slender) particle is aligned with the velocity gradient, changes in viscosity areless relevant, as the particle is essentially pushed around its orbit . Similar resultswere given for two-dimensional particles in Fe´rec et al. [87]. We note that whilechanges in the absolute value of the the rotation rate might be small, the changesin the period of rotation can be dramatic if the angular velocity is close to zero asshown in Fig. 7.4; however, while the instantaneous velocity of the particle may beaccurate in either the Newtonian case, or the correction we calculate here, neglectedeffects such as inertia, particle eccentricity and idealizations used in Carreau model,will cumulatively affect the orbit of particle integrated over time.1507.4. Conclusion(a) (b) (c)Figure 7.5: Phase portraits of a) θ and b) ϕ for different initial conditions of θi =π/12, π/4, π/3, 5π/12, 11π/24. Dashed-lines show the Newtonian cases. Thick,and thin lines correspond to Cu = 0.15, and Cu = 0.1, respectively. Results arecarried out for λ = 5.As shown in Fig. 7.5, the effect of shear thinning varies quantitatively depend-ing on the particular orbit but the qualitative picture is similar. In particular, theperiod of each orbit depends on the initial position, unlike in a Newtonian fluid, ina shear-thinning fluid each orbit has a different period.Different orbits for different aspect ratios, λ, are shown in Fig. 7.6. In caseof a sphere, λ = 1, the period doesn’t change at all. In other words, the sphericalparticle is unaffected by shear thinning, as discussed in Datt and Elfring [58]. Forhigher aspect ratios, the orbit slows when the particle is aligned with the flow asshown in Fig. 7.6b), and the period T thereby increases, as shown in Fig. 7.6c).We find that shear-thinning rheology exacerbates this effect, more substantiallyincreasing the period for larger λ as the shear stresses, which are needed to rotatea long slender particle aligned with the flow, are reduced by shear thinning.7.4 ConclusionIn this work, we investigated the orientational dynamics of a prolate spheroid im-mersed in a background shear flow of a shear-thinning Carreau fluid. An equationof motion for the rotation of the prolate particle was derived for asymptoticallyweak shear thinning using a regular perturbation expansion in the Carreau num-ber and then integrated numerically. We found that shear-thinning rheology doesnot lift the degeneracy of the Jeffery orbits observed in Newtonian fluid. In shear-thinning fluids there are still infinitely many orbits that repeat periodically for alltime, each selected by the initial condition. However, the instantaneous rotation1517.4. Conclusion(a) (b) (c)Figure 7.6: Phase portraits of a) θ and b) ϕ for different aspect ratios of λ =1, 2, 3, 4, 5. Dashed-lines show the Newtonian cases. Thick, and thin lines cor-respond to Cu = 0.15, and Cu = 0.1, respectively. Results are carried out for(θi, ϕi) = (π/3, 0).rate and trajectories of the orbits are modified. Qualitatively, shear thinning has asimilar effect as elongating a particle in a Newtonian fluid: shear thinning tends toincrease the period of rotation as the particle slows down more when aligned withthe flow due to a reduction of shear stress. Unlike for Jeffery orbits in Newtonianfluids, in shear-thinning fluids the period of the orbits does depend on the specifictrajectory (or initial orientation of the particle) because the effect of shear thinningvaries depending on the orientation of the particle. The results presented in thiswork can serve as a base for further investigation of dilute particles in shear flowof biological fluids and other shear-thinning substances e.g., blood and pastes.152Chapter 8Machine learning for theinteractions of two squirmersWe have made our two squirmer implementations available as part of a trainedneural network model using synthetic data generated by our computations. It al-lows anyone to simulate the interaction of two identically sized squirmers. Whentwo squirmers are at close distance, it utilises the findings from chapter 2, andwhen they are far apart, it uses the data from chapter 3. To train the model, wemust produce a large number of synthetic simulations, and the model may thenlearn to forecast unknown data. There are two inputs for the type of squirmer andthree for their relative positions and orientations. We have six outputs for the two-dimensional translational and angular velocity of the squirmers. Figure 8.1a showsthe input and output data for two interacting squirmers. The network consisted ofeleven hidden layers in the following order 8, 16, 32, 64, 128, 256, 128, 64, 32,16, and 8 nodes. The network schematic is shown in fig.8.1b. This network de-codes low-dimensional input into a high-dimensional latent space and then encodesback into the low-dimensional space. Each layer has the ReLU activation formula,whereas the output layer has a linear activation. We employed batch normalisa-tion after each hidden layer to minimise overfitting, provide regularisation, andincrease the learning performance of the model. The 18M training data sets arepassed through hidden layers, then backpropagated using the Adam optimizer toreweight the feed forward neural networks for 10 epochs and an adaptive learningrate. Following the training of the network, a validation set is used to demonstratethe model’s accuracy. The mean absolute error is used to calculate the accuracyand loss. The trained model was utilised to create a software capable of predictingthe trajectory of two squirmers. The algorithm functions as follows. The neuralnetwork generates a vector of velocity field of squirmers based on their current po-sition in each time step, which is then used by the RK4 technique to forecast thenext position. Figure 8.2 shows a snapshot of the software. It is worth noting thatthe addition of a wall results in two more input data points to account for the wall’sposition.153Chapter 8. Machine learning for the interactions of two squirmers(a)B↵2<latexit sha1_base64="+54g28S/h2/Lvd9Xz+DOOL+RTcY=">[binary] zsplD+APn8wf2nZBx</latexit>B2<latexit sha1_base64="OmR7giLRESyCI+1N7O4/Yvss7hY=">[binary] kM4fkD5zPHy3Ij/0=</latexit>r<latexit sha1_base64="AcQuwJUYmZzDq8q7VmxDoa5ELdw=">[binary] e4RXenEfnxXl3PhatBSefOYY/cD5/AN07jPY=</latexit><latexit sha1_base64="Ts9hwIzQ4mWsJxfPSkZncKyen7I=">[binary] KWZO4Q+czx8Too4/</latexit>✓<latexit sha1_base64="a9g4R2QJN0NpFcnDn3suEQSQSYM=">[binary]</latexit>Uk<latexit sha1_base64="aeFYF7Ykcrt1KhA5xLYYlQ5E/WA=">AAAB+nicbVBNS8NAEN34WetXqkcv [binary]</latexit>U↵k<latexit sha1_base64="HRTV4LEofXYYFSryPQazkRSzD4Q=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16 [binary]</latexit>U↵?<latexit sha1_base64="Ij/d1nK06PND+nl5CW7+gqlGjZ4=">AAAB9XicbVBNTwIxEJ3FL8Qv1KOX [binary]</latexit>U?<latexit sha1_base64="GOrXh3/OlJdTRyZE/JnmEXObzAk=">AAAB9HicbVBNTwIxEO3iF+IX6tFL [binary]</latexit>⌦↵<latexit sha1_base64="q1r3ftGcVpp6KokiedoqiJpbDPY=">AAAB9HicbVBNSwMxEM36WetX1aOX [binary]</latexit>⌦<latexit sha1_base64="PJjVaVlZHZcz0WAdFUJudbs6qjo=">AAAB83icbVBNS8NAEJ34WetX1aOX [binary]</latexit>Input layerHiddenlayersOutputlayer(b)Figure 8.1: a) Geometry inputs (Blue) and velocity outputs (red) of two interact-ing squirmers. b) The schematic diagram of a multilayer neural network for twointeracting squirmers.(a) (b)Figure 8.2: Interface of the software for two interacting squirmers.154Chapter 9Concluding remarksIn this dissertation, we used theoretical and computational methods to examinethe hydrodynamic interactions of microswimmers near a boundary. To model mi-croswimmers, we use spheres with a fixed tangential slip velocity to mimic cilia,flagella, or phoretic slip.We began with the lubrication problem for two unequal squirmers interactingclose together, as well as a squirmer and a wall. The final findings are providedas an active swim force, which may be summed across a suspension of squirmersalong a wall at close ranges. The results reveal that when approaching anotherswimmer or a surface, a swimmer slows down and reorients away. As a result,the swimmers escape or move in a cyclical pattern. We also showed that a strongpusher can swim backwards over a surface which had previously been observedexperimentally.Then, the hydrodynamics of a suspension of N active particles near a surfacewere examined. We used the Stokesian Dynamics approach created for passiveparticles in a viscous fluid to model the behaviour of a poly disperse suspension ofactive particles near a rigid boundary. The presence of the wall is integrated intothe SD by using image representation of singular multipoles in the Boundary Inte-gral equations. Also, surface activity of particles adds up to the system’s velocitymoments or force moments, but the resistance tensor remains unchanged and sim-ilar to that of passive particles. Furthermore, three contact models are presented toavoid swimmers overlapping. We showed that pusher-like swimmers attract otherswimmers or objects from their sides while repelling them from their front or back.Puller swimmers, on the other hand, reject from the sides and drag other objectsalong their head axis. In addition, we demonstrated that interactions along thehead axis are twice as powerful as interactions along the sides. As a result, pullerstend to cluster in a puller suspension, but pushers scatter in a pusher suspensionas shown in fig.9.1. Saintillan and Shelley [181] have reviewed different patternformations due to hydrodynamic interactions in active suspensions. The interac-tion of two squirmers near a surface produced trajectories that included converging,diverging, oscillatory, and orbital motions. Orbital motions occur when two swim-mers are in dynamic equilibrium and revolve indefinitely around one other. Wealso showcased a chase-and-run manoeuvre between two pushers near a wall. This155Chapter 9. Concluding remarks(a) (b)Figure 9.1: a) Puller and b) pusher suspensions with random initial orientationsafter T=10.behaviour was seen in an experimental study of neural crest (NC) cells, and placo-dal cells. NC cells chase placodal cells, but placodal cells run when contacted byNC cells. The chase-and-run motion in cells can explain the coordinated migrationof cells, from embryo development to cancer metastasis [204]. It is worth notingthat the first squirming mode B1 does not contribute to hydrodynamic interactionsin the active SD approach. Neutral swimmers, for example, are ignorant of eachother and the nearby surfaces up to this order of calculations. This mode, however,adds to the swim force calculated in chapter 2, which may be employed in futureactive SD study. We used the swim force from near-field calculations in the activeSD approach and calculated the dynamics of two squirmers for 18M potential con-figurations to train a neural network capable of predicting the trajectories of twointeracting squirmers.Then we presented a toy model to differentiate the hydrodynamics of the headand the tail. We utilized a simple toy model consisting of two spheres and a rod toinvestigate how a wall affects a swimmer’s mobility. The head of the swimmer isrepresented by a passive sphere, while the tail is represented by a squirmer, whichis responsible for the propulsion. The swimmer imitates a pusher when B1 < 0and a puller when B1 > 0. A pusher has a lower chance of escaping the wallthan a puller. Only pushers with low tail/head ratios and lengths are less likelyto be attracted to walls. Also, the tail’s push is promptly sent to the head via thelink between them. However, in the second squirming mode B2, the swimmer156Chapter 9. Concluding remarkspushes forward due to spheres’ interactions. As a result, as the length increases,the interactions fade away. Finally, the azimuthal squirming mode C1 consider-ably changes the swimmer’s orientation while making minimal contribution to theattraction/repulsion process. Separating the head and tail is also the process thatexplains E. coli’s circular motion over a surface as the C1 mode does not cause asquirmer to move in a circular route on its own.We then adjusted our toy model to account for swimmer flexibility. To do this,we put a torsional spring into the centre of the toy model, allowing the swimmerto bend due to torque dipole and higher order interactions. Again, one of the as-sembly’s spheres is a squirmer with modes B1 and B2 that replicates the tail. TheB1 results show that when a swimmer is pulled, the swimmer tries to keep its con-nections tight and under strain in order to send more driving force into the body.Pushers, on the other hand, may be easily bent for lower tail/head ratios and stiff-ness. The B2 results show that a puller swimmer pulls the other sphere towardsit and bends because of the torque dipole. Stiffer pullers escape, whereas flexiblepullers with smaller tail/head ratio swim towards the wall.We also looked into another method for dealing with an elastic swimmer’sdynamics. The velocity field was constructed using just far-field approximationsutilising singular multipoles, and the link forces were calculated using Faxe´n law.In this model, the tail provides thrust force rather than slip velocity as it does in thesquirming model. The swimmer functions as a pusher or a puller depending on thedirection of the thrust force. The wall would eventually cause pushers and pullersto bend, and this elastic bending would change their dynamics and trajectories.We showed that a pusher swims near the wall for a long time with an initial attrac-tion, later escaping almost linearly as the wall’s hydrodynamic impact decays, but apuller attracts to the wall. Furthermore, the tail/head ratio has a qualitatively equiv-alent effect as stiffness. The swimmer swims a shorter distance before escaping orbeing dragged to the wall when the tail/head ratio decreases or elasticity increases.We segregated elasticity’s contribution in our far-field study with regard to a rigidswimmer in order to adequately analyse the consequences of elasticity. We con-sidered a swimmer swimming parallel to a wall. Up to this order of calculation,elasticity has no effect on parallel velocity. A rigid pusher is attracted to the wall,whereas a rigid puller manages to avoid it. When λ < 1, a rigid puller or pusherswimmer is reoriented towards the wall and away from the wall when λ > 1. Forλ = 1, no rotation is seen. While the attraction and repulsion are consistent withour previous numerical results for a rigid swimmer propelled by squirming modeB1, the influence of λ on the rotational direction is the opposite. Elasticity alonewould a) pull the swimmer vertically to the wall (regardless of type or λ), b) reori-ent a pusher (puller) away from (towards) the wall, and c) cause a pusher (puller)to bend away from (towards) the wall. Overall, elasticity causes a pusher to escape157Chapter 9. Concluding remarkswhile attracting a puller against the wall. These results are similar to those reportedearlier numerically for an elastic swimmer driven by mode B2. Elastic pushers withsmall size ratio are pushed away from the wall, while pullers are dragged againstthe wall.Based on these findings, we may conclude that flexibility plays an importantrole in the dynamical behaviour of swimmers and has the potential to drasticallymodify hydrodynamic interactions. Our findings show the need of accountingfor elasticity when modelling hydrodynamic interactions of swimmers near walls.However, there are always limitations to the techniques used, which are worth ex-amining.Our toy model’s linear torsional spring may be oversimplifying the complicatedgeometry of flexible swimmers. We demonstrated the linear effects of elasticity onswimmer dynamics in chapters 5 and 6. This concept, however, cannot explainmassive deformations of cells, such as those found in Euglena gracilis specimens[178]. This model is also incapable of predicting the dynamics of elastic filamentsconnected to microorganisms [51]. One option is to increase the number of tor-sional springs in the toy model. Furthermore, the two-bead toy model may also failto explain complex ciliary motion. The impetus in chapters 4 and 5 is generatedby slip velocity on one of the spheres in the toy model, but the impetus in chapter6 is a swim force that arises from an intrinsic mechanism within the body. Cili-ated dynamics, on the other hand, can be far more complicated than a two-beadmodel can capture [160, 161]. More microorganism features, such as flagellumor cilium motility mechanisms, should be incorporated to fully understand the hy-drodynamic interaction between the swimmer and the surrounding fluid medium[120]. The vast majority of true biological microorganisms do not squirm andthe slip velocity is not prescribed. Furthermore, our assembly results show thatswimmer shape has a significant role in the dynamics of swimmers near surfaces.Microswimmer models that are overly simplified, such as the squirmer swimmer,may be inadequate. Likewise, no shape change is possible in the squirming model.Another major ongoing problem in active soft matter has been to understand theimpact of surrounding fluid hydrodynamic interactions on the behaviour of activematter systems. Lately, inspired by the complex dynamics of living organisms,learning algorithms may give new insights into the functioning of active mattersystems [41]. Both experimental and physics-driven data may be utilized to buildmachine learning models capable of predicting swimmer dynamics. Finally, whilewe concentrated on two active particles near a wall, including the toy models, thedeveloped active Stokesian Dynamics method in chapter 3 and the analytic expres-sions for the near-field swim force in chapter 2 can handle a wide range of activematter suspensions in confinement that can be the subject of future research.As previously stated, the simplest squirmers are spherical colloids. As a mat-158Chapter 9. Concluding remarkster of fact, biological microorganisms such as E. coli are not spherical and do notsquirm (with the exception of Volvox) [169]. Volvox is the only true sphericalorganism for which the fluid dynamical description of its locomotion may be im-proved by using slender-body theory for each individual flagellum [91]. Undershear, the dynamics of elongated particles, such as axisymmetric ellipsoids andspheroids, are significantly more complex than those of spherical particles [128].Furthermore, most past studies has focused on Newtonian fluids, but the major-ity of biological fluids are non-Newtonian. In fact, microswimmers in Newtonianfluids are more understood than those in complex fluids. When compared to New-tonian fluids, non-Newtonian carrier fluids clearly cause qualitative changes in therheological behaviour of particles. While the ultimate objective is to investigatethe hydrodynamic interactions of non-spherical active matter in complex fluids,we begin with a simpler scenario in order to build the mathematics needed forsuch calculations. We explored the orientational dynamics of a prolate spheroidsubmerged in a background shear flow of a shear-thinning Carreau fluid. We dis-covered that shear-thinning rheology did not lift the Jeffery orbit degeneracy seenin Newtonian fluid. There are still an endless number of stable orbits each chosenby the initial condition. However, the instantaneous rotation rate and trajectories ofthe orbits are altered. Shear thinning has a qualitative impact comparable to elon-gating a particle in a Newtonian fluid. It tends to lengthen the period of rotationas the particle slows down more when aligned with the flow. Unlike Jeffery orbitsin Newtonian fluids, the period of the orbits in shear-thinning fluids depends onthe initial orientation of the particle. The developed mathematical models may beused to modify the active SD method for non-spherical active particles in complexfluids such as biological fluids.Finally, in this dissertation, we developed the machinery to investigate the ef-fect of elasticity on the dynamics of microorganisms in fluids, but there are stillmore questions to be answered. We believe that experimental data and numeri-cal simulations should be used in conjunction with theoretical research to betterunderstand this phenomena.159Bibliography[1] S. A. Abtahi and G. J. Elfring. Jeffery orbits in shear-thinning fluids. Phys.Fluids, 31(10):103106, 2019.[2] S. A. Abtahi and G. J. Elfring. Active particles near a wall. to be submitted,2022.[3] S. A. Abtahi and G. J. Elfring. Confined flexible swimmers. to be submitted,2022.[4] P. Ahana and S. P. Thampi. Confinement induced trajectory of a squirmer ina two dimensional channel. Fluid Dyn. Res., 51(6):065504, 2019.[5] L. Angelani, C. Maggi, M. L. Bernardini, A. Rizzo, and R. 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E, 41(5), 2018.177Appendix ANear-field hydrodynamics ofswimmersA.1 Slip velocity break downConsider two spheres in fig.2.1b with head directions pα and pβ. In a sphericalcoordinate system (R, Θ,Φ) where Φ and Θ are the axis of rotation and the polarangle with respect to pα, sphere α has slip velocity ofus = (B1 sinΘ + B2 sinΘ cosΘ) eΘ +C1 sinΘ eΦ, (A.1)In a cartesian coordinate system (X,Y,Z) attached to the centre of sphere α, whereZ is aligned with the head pα, the slip velocity takes the form ofus = (XZ(B1 + B2Z) −C1Y) eX + (YZ(B1 + B2Z) +C1X) eY+((Z2 − 1)(B1 + B2Z))eZ . (A.2)In a fixed cartesian coordinate system (x, y, z), where z is aligned with the centresof the two spheres, the head direction ispα = sin χ ex + cos χ ez (A.3)Furthermore, the rotation matrix from (X,Y,Z) to (x, y, z) isR(χ) = cos χ 0 sin χ0 1 0− sin χ 0 cos χ , (A.4)andX = x cos χ − z sin χ, Y = y, Z = x sin χ + z cos χ. (A.5)178A.1. Slip velocity break downNow we can write the slip velocity in (x, y, z) asus =12(cos χ(2B1xz − 2C1y) + 2B1(x2 − 1)sin χ − B2x cos 2χ(2x2 + y2 − 2)+B2(2x2 − 1)z sin 2χ − B2xy2)ex +12(2 cosχ(B1yz + 2B2xyz sin χ +C1x)+2 sinχ(B1xy −C1z) − B2y cos(2χ)(x2 − z2)+ B2(y − y3) )ey+12(2 sinχ(B1xz +C1y) − 2B1 cos χ(x2 + y2)+ B2x sin(2χ)(2z2 − 1)−B2z cos(2χ)(2x2 + y2)− B2y2z)ez (A.6)Finally, in a fixed spherical coordinate system (r, θ, ϕ) at the centre of sphereα, where ϕ and θ are the axis of rotation and the polar angle with respect to ez, theslip velocity takes the form ofus(B1,C1, B2, χ) =(B1 (sin θ cos χ − cos θ cos ϕ sin χ) −C1 sin ϕ sin χ+B2(12sin 2θ − 14(3 + cos 2ϕ) sin 2θ sin2 χ − 12cos 2θ cos ϕ sin 2χ))eθ+(B1 sin ϕ sin χ +C1(sin θ cos χ − cos θ cos ϕ sin χ)+B2(12sin θ sin 2ϕ sin2 χ +12cos θ sin ϕ sin 2χ))eϕ (A.7)For two special cases, where the head is parallel with and perpendicular to z, wehavepα|| ez → us(B1,C1, B2, 0) =(B1 sin θ +12B2 sin 2θ)eθ +C1 sin θeϕ,pα|| ex → us(B1,C1, B2, π/2) =(− B1 cos θ cos ϕ −C1 sin ϕ + 12 B2 sin 2θ− 14B2(3 + cos 2ϕ) sin 2θ)eθ+(B1 sin ϕ −C1 cos θ cos ϕ + 12 B2 sin θ sin 2ϕ)eϕ(A.8)179A.2. Axisymmetric problemsNow we can construct the general form of us via these special cases asus(B1,C1, B2, χ) = us(B1, 0) cosχ + us(B1, π/2) sinχ+ us(C1, 0) cosχ + us(C1, π/2) sinχ+ us(B2, 0) cos2 χ + us(B2, π/2) sin2 χ+ usresidual sin χ cos χ, (A.9)where usresidual is the residual slip velocity of the formusresidual = − B2 cos 2θ cos ϕeθ + B2 cos θ sin ϕeϕ. (A.10)which is equivalent to a sphere with pα|| ex + ez and slip velocity ofusresidual =14B2(3 + cos 2Φ) sin 2Θ eΘ − 12 B2 sinΘ sin 2Φ eΦ. (A.11)Note that the slip velocities in table 2.1 are reported in a spherical coordinate sys-tem where the axis of rotation is aligned with the head direction.A.2 Axisymmetric problemsThe governing equations for axisymmetric problems is E2E2ψ = 0 where E2 in thecylindrical coordinate system takes the form ofE2 =∂2∂z2+∂2∂r2− 1r∂∂r. (A.12)After the expansion of stream function and stretching the coordinate system, theleading, and the first order systems take the form ofψ(0)ZZZZ = 0, ψ(1)ZZZZ =2ψ(0)RZZR− 2ψ(0)RRZZ . (A.13)Boundary conditions for problem 1 are4ψ(0)|Zα = −ψ(0)Z |Zα = Bα1 R4, ψ(0)|Zβ = ψ(0)Z |Zβ = 0,ψ(1)|Zα = −18R4ψ(0)Z |Zα , ψ(1)Z |Zα =Bα1 R42− 18R4ψ(0)ZZ |Zα ,ψ(1)|Zβ =R4ψ(0)Z |Zβ8β3, ψ(1)Z |Zβ =R4ψ(0)ZZ |Zβ8β3. (A.14)180A.2. Axisymmetric problemsLikewise boundary conditions for problem 3 can be calculated in the same way.The leading order solution for problem 1 isψ(0) =(R3 + 2βRZ)2 ((5β + 3)R4 + (14β − 4(2β + 1)Z + 4)R2 − 8β(Z − 1))4(2β + (β + 1)R2)3 Bα1 .(A.15)The rest of solution for ψ(1) is lengthy and not shown here. The force acting on theswimmer isF =∫Sn · σ dS . (A.16)We define a system of coordinates intrinsic to the surface of the swimmer, with unitvectors (n, s, t) where n is the unit outer normal and s is a unit tangent vector onthe surface. The gradient and velocity fields takes the form of∇ = n ∂∂n+ s∂∂s+ t1r∂∂t, (A.17)un = −1r∂ψ∂s, us =1r∂ψ∂n. (A.18)Then, the stress tensor σ across the outer normal n isn · σ = −np − 2µ∇(1r∂ψ∂s) + sµ1rE2ψ, (A.19)and the element of surface area is dS = 2πr ds. The total impact of the stressesoperating across the whole surface of the body results in a force only acting parallelto the axis of revolution due to the symmetry of flow F = Fzez. The force on thebody is thenFz = π∫r2∂p∂sds − 2πµ∫∂r∂nE2ψ ds. (A.20)It is possible to establish a link between the pressure in a continuous creeping flowand the stream function as∂p∂s=µr∂∂n(E2ψ), (A.21)which finally results in a force of the formFz = πµ∫r3∂∂n(E2ψr2) ds. (A.22)181A.3. Asymmetric problemsIn order to calculate the differentiation of unit vectors, the method of chain-rule isused,∂∂n=∂∂z∂z∂n+∂∂r∂r∂n, (A.23)along with identities of the form∂z∂n= − drds= −1/√1 + (dzdr)2, (A.24)∂r∂n=dzds= 1/√1 + (drdz)2, (A.25)ds =√dr2 + dz2, (A.26)where z = −√1 − r2,z = √1 − r2 for the lower half, and upper half of the swimmerrespectively. Finally in the stretched coordinate system we haveFzπµ=∫ R00R3∂∂n(E2ψR2)√1 + ϵ(dZdR)2 dR, (A.27)Fαzπµ= −∫ R00(Rψ(0)ZZZ +(Rψ(1)ZZZ + 2Rψ(0)ZZ − ψ(0)RZ − R2ψ(0)RZZ + Rψ(0)RRZ)ϵ)∣∣∣∣∣∣Z=1+ R22dR,(A.28)where R0 denotes the lubrication region’s limit. As explained by Ishikawa et al.[105], O’Neill and Stewartson [163], in the process of matching the inner lubrica-tion region and the outer solution, we choose R0 = Cϵ−1/2, where C is a constantdetermined from the outer solution. The leading terms of the force is related to thelubrication force while the outer solution contributes to the O(1) terms.A.3 Asymmetric problemsFor asymmetric problems the leading order governing equations areu(0)R + R∂u(0)R∂R+∂u(0)θ∂θ+ R∂u(0)Z∂Z= 0, (A.29)µ∂2u(0)R∂Z2− ∂p(0)∂R= 0, µ∂2u(0)θ∂Z2− ∂p(0)∂θ= 0,∂p(0)∂Z= 0. (A.30)with boundary conditions of the formu(0)R |Zα = Bα2 cos θ, u(0)θ |Zα = −Bα2 sin θ, u(0)Z |Zα = Bα2 R cos θ,u(0)R |Zβ = u(0)θ |Zβ = u(0)Z |Zβ = 0, (A.31)182A.3. Asymmetric problemsand constraints ofp(0)(0, θ, 0) = 0, p(0)(∞, θ,Z) = 0, (A.32)for problem 7. The rest of boundary conditions can be obtained in a similarway. From the momentum equation A.30 in Z direction, we conclude that p(0) =p(0)(R, θ). The rest of momentum equations giveu(0)R = −(R2 + 2βZ) (R2 − 2Z + 28µβp(0)R −Bα2 cos θ2β + (β + 1)R2),u(0)θ = −(R2 + 2βZ) (R2 − 2Z + 28µβp(0)θ +Bα2 R sin θ2β + (β + 1)R2). (A.33)In order to satisfy boundary conditions on u(0)Z , we integrate continuity equationalong two surfaces of the form∫ ZαZβu(0)RR + ∂u(0)R∂R+∂u(0)θR∂θ dZ = u(0)Z |Zβ − u(0)Z |Zα = −Bα2 R cos θ, (A.34)which results inR2 p(0)RR +(R + 3H−1R3(1 + β−1))p(0)R + p(0)θθ = 6µBα2 H−3R3(1 + β−1) cos θ (A.35)Using the pressure constraints, we getp(0) = −6/5µBα2 H−2R cos θ. (A.36)Finally, we substitute pressure back into continuity to getu(0)Z =Bα2 R cos θ5(2β + (β + 1)R2)4 [5(β + 1)2R8 + 28β(β + 1)R6 + 4β(β + 4)R4 − 64β2R2+ 4βZ(3(β + 1)2R6 + (β + 1)R4(28β +(β2 + 5β + 4)Z − 8)+ 4βR2(7β + 2(β + 1)2Z2 +(7β2 − β − 8)Z + 10)− 4β2(8(β + 1)Z2 − (13β + 16)Z + 8) )]. (A.37)183A.4. Far-field swim force near a surfaceNext, the leading order force and torque in Cartesian coordinate system areFα =∫ 2π0∫ R00R[(− R cos θp(0) − µ cos θ∂u(0)R∂Z+ µ sin θ∂u(0)θ∂Z)ex+ =+(− R sin θp(0) − µ sin θ∂u(0)R∂Z− µ cos θ∂uθ∂Z)ey +( p(0)√ϵ)ez]Zαdθ dR,Fβ =∫ 2π0∫ R00R[(− Rβcos θp(0) + µ cos θ∂u(0)R∂Z− µ sin θ∂u(0)θ∂Z)ex+(− Rβsin θp(0) + µ sin θ∂u(0)R∂Z+ µ cos θ∂uθ∂Z)ey −( p(0)√ϵ)ez]Zβdθ dR,Tα =∫ 2π0∫ R00Rµ[(− sin θ∂u(0)R∂Z− cos θ∂u(0)θ∂Z)ex+(cos θ∂u(0)R∂Z− sin θ∂uθ∂Z)ey]Zαdθ dR,Tβ =∫ 2π0∫ R00Rβµ[(− sin θ∂u(0)R∂Z− cos θ∂u(0)θ∂Z)ex+(cos θ∂u(0)R∂Z− sin θ∂uθ∂Z)ey]Zβdθ dR. (A.38)As mentioned before, we choose R0 = Cϵ−1/2 and getFα = −Fβ = 45πµBα2β2 + 4β(β + 1)2log ϵ ex + O(1),Tα = −165πµBα2ββ + 1log ϵ ey + O(1),Tβ = −45πµBα2β2β + 1log ϵ ey + O(1). (A.39)A.4 Far-field swim force near a surfaceAlternatively, we can derive forces on a swimmer near a wall usingFs = −RwFU · Us − RwF E : Es (A.40)184A.4. Far-field swim force near a surfacewhere RwFU , and RwFE are second-rank, and third-rank resistance tensors for a rigidparticle-wall interactions and containsRwFU =RαwFU RαwFΩRαwLU RαwLΩ , RwF E =RαwFERαwLE . (A.41)Using lubrication theory presented in Cooley and O’Neill [48], Jeffrey [114], wegetRαwFU33 = ϵ−1 +15ln ϵ−1 + O(1),RαwFUii =815ln ϵ−1 + O(1),RαwLUi j =215ϵi j3 ln ϵ−1 + O(1),RαwLΩ33 = O(1),RαwLΩii =815ln ϵ−1 + O(1),RαwFE333 = −23ϵ−1 +815ln ϵ−1 + O(1),RαwFEii3 =13ϵ−1 +415ln ϵ−1 + O(1),RαwFE3ii = RαwFEi3i = −715ln ϵ−1 + O(1),RαwLE3i j = RαwLEi3 j = −215ϵi j3 ln ϵ−1 + O(1), (A.42)where {i, j} = {1, 2} and i , j. As seen before, the slip velocity is decomposed intosquirming modes. Then the zeroth and the first moments take the form ofUs =1S∫SusdS = −23B1p,Ωs =12V∫Sn × usdS = C1p,Es =12V∫S(nus + usn)dS = − 35aB2(pp − 13I), (A.43)185A.4. Far-field swim force near a surfaceWhere S and V are surface area and volume of the swimmer. Eventually, usingeq.A.40 we can obtain swim force/torque up to leading order asF6πµ=[23B1(e⊥ · p)e⊥ + 15 B2(e⊥ · p)p −15B2(e⊥ · p)2e⊥ + 415 B2e⊥]ϵ−1+ O(log ϵ),T8πµ=[115B1e⊥ × p + 25C1(p − (e⊥ · p)e⊥) +350B2(e⊥ · p)e⊥ × p)]log ϵ+ O(1, ϵ log ε) (A.44)one may compare these equation by the ones presented in eq.2.15-2.16.186Appendix BActive particles near a wallB.1 Details of far-field hydrodynamicsContribution of a wall in terms of force singularities areGwi j(R) = − 3δi j4R+6y⊥(y⊥ − R⊥)δi j + 6y⊥e⊥ jRi + 6y⊥e⊥i(R j − 2y⊥e⊥ j)− 3RiR j4R3−9y⊥(y⊥ − R⊥)Ri(R j − 2e⊥ jR⊥)2R5,GwPi (R) =6y⊥e⊥i − 3Ri2R3+9y⊥e⊥ jR j (Ri − 2R⊥e⊥i)R5,GwSi jk(R) =92R5(2y2⊥(2R⊥e⊥k − Rk)δi j + 2y⊥(R⊥ − y⊥)(Riδ jk + R jδik)+ 4y⊥ (y⊥ − R⊥) e⊥k(e⊥ jRi + e⊥iR j) − 2y⊥e⊥kRiR j + RiR jRk)− 45y⊥RiR j (y⊥ − R⊥) (2R⊥e⊥k − Rk)R7. (B.1)187B.1. Details of far-field hydrodynamicsA special care for the derivatives of above with respect to y must be taken. Someof the usefull derivatives are∇yiGw jk(R) =34R3[δi j (Rk − 2y⊥e⊥k) + δik(R j − 2y⊥e⊥ j)+ δ jk (2y⊥e⊥i − Ri)]+184R5[y⊥δi j(Rk (y⊥ − e⊥lRl) − 2e⊥ke⊥lRl (y⊥ − e⊥mRm))+ 6y⊥δ jk(Ri (4y⊥ − 3e⊥lRl) − 6e⊥ie⊥lRl (y⊥ − e⊥mRm))+ 6y⊥R jδik (4y⊥ − 3e⊥lRl)− 9R j(e⊥k (4e⊥ie⊥le⊥mRlRm − 2y⊥Ri)+ Rk (2e⊥i (y⊥ − 2e⊥lRl) + Ri))+ 18y⊥e⊥ j(Rk (Ri − 2y⊥e⊥i)− 2e⊥k (2e⊥ie⊥lRl (e⊥mRm − 2y⊥) + y⊥Ri))]− 452R7[y⊥R j(2e⊥ie⊥lRl(2e⊥ke⊥mRm (y⊥ − e⊥nRn) + Rk (e⊥mRm − y⊥))+ Ri(Rk (y⊥ − e⊥lRl) − 2e⊥ke⊥lRl (y⊥ − e⊥mRm)))],∇y2Gwi j(R) = − 32R3 δi j+92R5[(2e⊥ke⊥lRkRlδi j − 2y⊥e⊥kRkδi j+ 2e⊥ j(2e⊥ie⊥kRk (2y⊥ − e⊥lRl) + Ri (y⊥ − 2e⊥kRk))− 2y⊥e⊥iR j + RiR j)]+45R7[e⊥kRiRk(R j (y⊥ − e⊥lRl) − 2e⊥ je⊥lRl (y⊥ − e⊥mRm)) ].(B.2)188B.1. Details of far-field hydrodynamicsWe have used 6πµanα and λ = aα/aβ to non-dimensionalize the mobilities. Self-interaction mobilities areMααUFi j = λ−1MββUFi j = δi j,MααΩLi j = λ−3MββΩLi j =3δi j4,MααESi jkl = λ−3MββESi jkl =920(δilδ jk + δikδ jl − 23δi jδkl), (B.3)while sphere-sphere mobilities in an unbounded flow field areMαβUFi j =3(δi j + erier j)4r−(λ2 + 1) (3erier j − δi j)4λ2r3,MαβULi j =3erkϵi jk4r2,MαβUSi jk =3eri(3er jerk − δ jk)4r2+3(5λ2 + 3) (erkδi j + er jδik + eriδ jk − 5erier jerk)20λ2r4,MαβΩFi j =3erkϵi jk4r2,MαβΩLi j =3(3erier j − δi j)8r3,MαβΩSi jk =9erl(erkϵi jl + er jϵikl)8r3,MαβEFi jk =3erk(δi j − 3erier j)4r2−3(3λ2 + 5) (erkδi j + er jδik + eriδ jk − 5erier jerk)20λ2r4,MαβELi jk = −9erl(er jϵikl + eriϵ jkl)8r3,MαβESi jkl =18r3[9(er jerlδik + 2erkerlδi j + erker jδil + erierlδ jk + erierkδ jl+ 2erier jδkl − 10erier jerkerl)− 6δi jδkl]+9(λ2 + 1)20λ2r5[− 5(er jerlδik + erkerlδi j + erker jδil + erierlδ jk + erierkδ jl+ erier jδkl − 7erier jerkerl)+ δikδ jl + δilδ jk + δi jδkl],(B.4)189B.1. Details of far-field hydrodynamicswhere r = rer, and r⊥ = e⊥ · r. The effect of a solid flat surface is captured bymobilities below:Mαα′UF i j = −δi j + e⊥ie⊥ j16h5α+δi j + 3e⊥ie⊥ j8h3α− 9δi j + e⊥ie⊥ j16hα,Mαα′UL i j =3e⊥kϵki j32h4α,Mαα′US i jk =3(e⊥ jδik + e⊥kδi j + e⊥i(e⊥ je⊥k − δ jk))32h6α−9(4e⊥ jδik + 4e⊥kδi j + e⊥i(7e⊥ je⊥k − 5δ jk))160h4α+9(e⊥ jδik + e⊥kδi j + e⊥i(e⊥ je⊥k − δ jk))32h2α,Mαα′ΩF i j = −3e⊥kϵki j32h4α,Mαα′ΩL i j =3(3e⊥ie⊥ j − 5δi j)64h3α,Mαα′ΩS i jk =9(e⊥ke⊥lϵli j + e⊥ je⊥lϵlik)80h5α−9(e⊥ke⊥lϵli j + e⊥ je⊥lϵlik)64h3α,Mαα′EF i jk =3(e⊥ jδik − e⊥kδi j + e⊥i(δ jk + e⊥ je⊥k))32h6α−9(4e⊥ jδik − 5e⊥kδi j + e⊥i(4δ jk + 7e⊥ je⊥k))160h4α+9(e⊥ jδik − e⊥kδi j + e⊥i(δ jk + e⊥ je⊥k))32h2α,Mαα′EL i jk = −9(e⊥ je⊥lϵlik + e⊥ie⊥lϵl jk)80h5α+9(e⊥ je⊥lϵlik + e⊥ie⊥lϵl jk)64h3α,Mαα′ES i jkl = −27640h7α[3e⊥ie⊥k(δ jl + e⊥ je⊥l)+ δi j (δkl − 5e⊥ke⊥l) + δil(δ jk + 3e⊥ je⊥k)− 5e⊥ie⊥ jδkl + 3e⊥ie⊥lδ jk + δik(δ jl + 3e⊥ je⊥l) ]+980h5α[3e⊥ie⊥k(δ jl + e⊥ je⊥l)+ δi j (δkl − 5e⊥ke⊥l) + δil(δ jk + 3e⊥ je⊥k)190B.1. Details of far-field hydrodynamics− 5e⊥ie⊥ jδkl + 3e⊥ie⊥lδ jk + δik(δ jl + 3e⊥ je⊥l) ]− 364h3α[3e⊥ie⊥k(2δ jl + e⊥ je⊥l)+ δi j (δkl − 9e⊥ke⊥l)+ 3δil(δ jk + 2e⊥ je⊥k)− 9e⊥ie⊥ jδkl + 6e⊥ie⊥lδ jk+ 3δik(δ jl + 2e⊥ je⊥l) ], (B.5)Mαβ′UF i j = −34R[δi j + eRieR j]+3hβ2R2[eR je⊥i − eR⊥δi j + eRi((1 − 6eR2⊥)e⊥ j + 3eR⊥eR j) ]+14λ2R3[ (6eR2⊥ + 6λ2h2β − λ2 − 1)δi j + 3eRi(4eR⊥(5eR2⊥+ 3λ2h2β − 1)e⊥ j + eR j(−10eR2⊥ − 6λ2h2β + λ2 + 1) )− 12(eR2⊥ + λ2h2β)e⊥ie⊥ j]+3hβ(λ2 − 1)2λ2R4[eR je⊥i + eRi((10eR2⊥ − 1)e⊥ j − 5eR⊥eR j)+ eR⊥(δi j − 4e⊥ie⊥ j) ]+12λ2R5[− 10eR⊥eR je⊥i +(1 − 5eR2⊥)δi j + 10eR⊥(2 − 7eR2⊥)eRie⊥ j+ 2(15eR2⊥ − 2)e⊥ie⊥ j + 5(7eR2⊥ − 1)eRieR j],Mαβ′UL i j = −3eRk4R2[6eR⊥eRie⊥lϵl jk + ϵi jk]+3hβ2R3[3eRieRke⊥lϵl jk + e⊥lϵli j]+32R4[eRk (5eR⊥eRi − e⊥i) e⊥lϵl jk + eR⊥e⊥lϵli j],Mαβ′US i jk = −3eRi4R2[12e⊥ je⊥keR2⊥ − 6eRke⊥ jeR⊥ − δ jk + 3eR j (eRk − 2eR⊥e⊥k)]+3hβ2R3[6δike⊥ jeR2⊥ + 6δi je⊥keR2⊥ − 12e⊥ie⊥ je⊥keR2⊥ − 3eRkδi jeR⊥191B.1. Details of far-field hydrodynamics+ 3eRi(20e⊥ je⊥keR2⊥ − 10eRke⊥ jeR⊥ − δ jk+ 5eR j (eRk − 2eR⊥e⊥k))eR⊥ + 3eR j (eRke⊥i − eR⊥δik) − e⊥iδ jk]+320λ2R4[− 840eRie⊥ je⊥keR4⊥ − 60eRke⊥ie⊥ jeR2⊥ − 60eR je⊥ie⊥keR2⊥− 600λ2h2βeRie⊥ je⊥keR2⊥ + 100λ2eRie⊥ je⊥keR2⊥+ 180eRie⊥ je⊥keR2⊥ + 300λ2h2βeRieRke⊥ jeR⊥+ 10(−5λ2 + 42eR2⊥ − 6)eRieRke⊥ jeR⊥ − 60λ2h2βδike⊥ jeR⊥+ 2(5λ2 − 30eR2⊥ + 6)δike⊥ jeR⊥ + 300λ2h2βeRieR je⊥keR⊥+ 10(−5λ2 + 42eR2⊥ − 6)eRieR je⊥keR⊥ − 60λ2h2βδi je⊥keR⊥+ 2(5λ2 − 30eR2⊥ + 6)δi je⊥keR⊥ + 240λ2h2βe⊥ie⊥ je⊥keR⊥+ 8(−5λ2 + 30eR2⊥ − 3)e⊥ie⊥ je⊥keR⊥− 5(30λ2h2β − 5λ2 + 42eR2⊥ − 3)eRieR jeRk+(30λ2h2β − 5λ2 + 30eR2⊥ − 3)eRkδi j+(30λ2h2β − 5λ2 + 30eR2⊥ − 3)eR jδik − 60λ2h2βeRke⊥ie⊥ j+ 10λ2eRke⊥ie⊥ j +(30λ2h2β − 5λ2 + 30eR2⊥ − 3)eRiδ jk− 60λ2h2βeR je⊥ie⊥k + 10λ2eR je⊥ie⊥k]−3hβ(5λ2 − 3)10λ2R5[10δike⊥ jeR2⊥ + 10δi je⊥keR2⊥ − 60e⊥ie⊥ je⊥keR2⊥− 5eRkδi jeR⊥ + 20eRke⊥ie⊥ jeR⊥ − δike⊥ j+ 5eRi( (1 − 14eR2⊥)eRke⊥ j + eR j(7eR⊥eRk +(1 − 14eR2⊥)e⊥k)+ eR⊥(4(7eR2⊥ − 1)e⊥ je⊥k − δ jk) )+ e⊥iδ jk − δi je⊥k + 4e⊥ie⊥ je⊥k− 5eR j (eRke⊥i + eR⊥ (δik − 4e⊥ie⊥k))]+32λ2R6[252eRie⊥ je⊥keR4⊥ + 42eRke⊥ie⊥ jeR2⊥ + 42eR je⊥ie⊥keR2⊥− 84eRie⊥ je⊥keR2⊥ − 14eR jeRke⊥ieR⊥ + 14(2 − 9eR2⊥)eRieRke⊥ jeR⊥+ 2(7eR2⊥ − 2)δike⊥ jeR⊥ + 2e⊥iδ jkeR⊥ + 14(2 − 9eR2⊥)eRieR je⊥keR⊥192B.1. Details of far-field hydrodynamics+ 2(7eR2⊥ − 2)δi je⊥keR⊥ + 8(3 − 14eR2⊥)e⊥ie⊥ je⊥keR⊥+ 7(9eR2⊥ − 1)eRieR jeRk +(1 − 7eR2⊥)eRkδi j+(1 − 7eR2⊥)eR jδik− 4eRke⊥ie⊥ j +(1 − 7eR2⊥)eRiδ jk − 4eR je⊥ie⊥k + 2eRie⊥ je⊥k],Mαβ′ΩF i j = −14R2[3eRkϵi jk]+3hβ2R3[e⊥lϵli j − 3eRk(eR j − 2eR⊥e⊥ j)e⊥lϵlik]− 32λ2R4[eRk((10eR2⊥ − 1)e⊥ j − 5eR⊥eR j)e⊥lϵlik + eR⊥e⊥lϵli j],Mαβ′ΩLi j = −38R3[6eR2⊥δi j − 6eR⊥eR je⊥i − 6eRkeRle⊥lϵlike⊥lϵl jl − δi j + 3eRieR j],Mαβ′ΩS i jk = −9eRl8R3[eRk(2e⊥ je⊥lϵlil + ϵi jl)+ eR j (2e⊥ke⊥lϵlil + ϵikl)− 2eR⊥(e⊥kϵi jl + e⊥ j (4e⊥ke⊥lϵlil + ϵikl)) ]+9hβ2R4[− 2eR⊥e⊥ je⊥lϵlik + eRle⊥lϵlilδ jk+ eRk(e⊥lϵli j − 5eRl(eR j − 2eR⊥e⊥ j)e⊥lϵlil)− 2eR⊥e⊥k(e⊥lϵli j − 5eRl(eR j − 2eR⊥e⊥ j)e⊥lϵlil)+ eR je⊥lϵlik]+910λ2R5[10eR2⊥e⊥ je⊥lϵlik − 5eR⊥eRle⊥lϵlilδ jk+ 5e⊥keRl(4eR⊥(7eR2⊥ − 1)e⊥ j +(1 − 14eR2⊥)eR j)e⊥lϵlil+(10eR2⊥ − 1)e⊥ke⊥lϵli j− 5eRk(eRl((14eR2⊥ − 1)e⊥ j − 7eR⊥eR j)e⊥lϵlil + eR⊥e⊥lϵli j)− 5eR⊥eR je⊥lϵlik − e⊥ je⊥lϵlik],Mαβ′EF i jk =14R2[3eRk(3eRieR j − δi j) ]193B.1. Details of far-field hydrodynamics+3hβ2R3[− 6δi je⊥keR2⊥ + 3eRkδi jeR⊥ + 3eR jδikeR⊥ − 6eR je⊥ie⊥keR⊥ + δi je⊥k+ 3eRi(eR j((10eR2⊥ − 1)e⊥k − 5eR⊥eRk)+ eR⊥(δ jk − 2e⊥ je⊥k)) ]+320λ2R4[− 300eR⊥eRieR je⊥kλ2h2β + 60eR⊥δi je⊥kλ2h2β+ 5(30λ2h2β − 3λ2 + 70eR2⊥ − 5)eRieR jeRk − 50eR⊥eR jeRke⊥i−(30λ2h2β − 3λ2 + 50eR2⊥ − 5)eRkδi j+(−30λ2h2β + 3λ2 − 50eR2⊥ + 5)eR jδik − 50eR⊥eRieRke⊥ j+ 10eR⊥δike⊥ j +(−30λ2h2β + 3λ2 − 50eR2⊥ + 5)eRiδ jk+ 10eR⊥e⊥iδ jk − 100eR⊥(7eR2⊥ − 1)eRieR je⊥k+ 10(6λ2h2β + 20eR2⊥ − 1)eR je⊥ie⊥k + 20eR⊥(5eR2⊥ − 1)δi je⊥k+ 10(6λ2h2β + 20eR2⊥ − 1)eRie⊥ je⊥k − 40eR⊥e⊥ie⊥ je⊥k]−3hβ(3λ2 − 5)10λ2R5[− 10δi je⊥keR2⊥ + 5eRkδi jeR⊥ − δike⊥ j − e⊥iδ jk + δi je⊥k+ 4e⊥ie⊥ je⊥k + 5eR j (eRke⊥i + eR⊥ (δik − 4e⊥ie⊥k))+ 5eRi(eRke⊥ j + eR j((14eR2⊥ − 1)e⊥k − 7eR⊥eRk)+ eR⊥(δ jk − 4e⊥ je⊥k) )],+32λ2R6[7(1 − 9eR2⊥)eRieR jeRk + 14eR⊥eR je⊥ieRk +(7eR2⊥ − 1)δi jeRk+ 14eR⊥eRie⊥ jeRk − 2e⊥ie⊥ jeRk +(7eR2⊥ − 1)eR jδik − 2eR⊥δike⊥ j+(7eR2⊥ − 1)eRiδ jk − 2eR⊥e⊥iδ jk + 14eR⊥(9eR2⊥ − 2)eRieR je⊥k+ 2(2 − 21eR2⊥)eR je⊥ie⊥k + 2eR⊥(2 − 7eR2⊥)δi je⊥k+ 2(2 − 21eR2⊥)eRie⊥ je⊥k + 12eR⊥e⊥ie⊥ je⊥k],Mαβ′EL i jk =98R3[− 4eR⊥eRlδi je⊥mϵmkl + eR j(eRl(20eR⊥eRie⊥mϵmkl− 2e⊥ie⊥mϵmkl + ϵikl)+ 2eR⊥e⊥mϵmik)194B.1. Details of far-field hydrodynamics+ eRi(eRl(ϵ jkl − 2e⊥ je⊥mϵmkl)+ 2eR⊥e⊥mϵm jk) ],− 9hβ2R4[− eRlδi je⊥mϵmkl + eR j (5eRieRle⊥mϵmkl + e⊥mϵmik) + eRie⊥mϵm jk]− 910λ2R5[− 5eR⊥eRlδi je⊥mϵmkl − e⊥ j (5eRieRle⊥mϵmkl + e⊥mϵmik)+ 5eR j (eRl (7eR⊥eRi − e⊥i) e⊥mϵmkl + eR⊥e⊥mϵmik)+ 5eR⊥eRie⊥mϵm jk − e⊥ie⊥mϵm jk],Mαβ′ES i jkl =38R3[− 24δi je⊥ke⊥leR2⊥ + 12eRlδi je⊥keR⊥ + 6eR jδile⊥keR⊥ + 6eR jδike⊥leR⊥− 24eR je⊥ie⊥ke⊥leR⊥ − 3eR jeRlδik + 6eR jeRle⊥ie⊥k + 2δi jδkl− 3eRk(2eRlδi j − 4eR⊥e⊥lδi j + eR j (δil − 2e⊥ie⊥l))+ 3eRi(− eRkδ jl + 2eR⊥e⊥kδ jl − eRl(δ jk − 2e⊥ je⊥k)+ 2eRke⊥ je⊥l + 2eR⊥δ jke⊥l − 8eR⊥e⊥ je⊥ke⊥l+ 2eR j(20e⊥ke⊥leR2⊥ − 10eRle⊥keR⊥ − δkl + 5eRk (eRl − 2eR⊥e⊥l)) )]− 9hβeR⊥2R4[− 20δi je⊥ke⊥leR2⊥ + 10eRlδi je⊥keR⊥ + 10eR jδile⊥keR⊥+ 10eR jδike⊥leR⊥ − 40eR je⊥ie⊥ke⊥leR⊥ − 5eR jeRlδik + δilδ jk+ δikδ jl + 10eR jeRle⊥ie⊥k − 2δile⊥ je⊥k − 2e⊥iδ jle⊥k + δi jδkl − 2δike⊥ je⊥l− 2e⊥iδ jke⊥l + 8e⊥ie⊥ je⊥ke⊥l− 5eRk(eRlδi j − 2eR⊥e⊥lδi j + eR j (δil − 2e⊥ie⊥l))+ 5eRi(− eRkδ jl + 2eR⊥e⊥kδ jl − eRl(δ jk − 2e⊥ je⊥k)+ 2eRke⊥ je⊥l + 2eR⊥δ jke⊥l − 8eR⊥e⊥ je⊥ke⊥l+ eR j(28e⊥ke⊥leR2⊥ − 14eRle⊥keR⊥ − δkl + 7eRk (eRl − 2eR⊥e⊥l)) )]+320λ2R5[7560eRieR je⊥ke⊥leR4⊥ − 840δi je⊥ke⊥leR4⊥+ 4200λ2h2βeRieR je⊥ke⊥leR2⊥ − 420λ2eRieR je⊥ke⊥leR2⊥− 1260eRieR je⊥ke⊥leR2⊥ − 600λ2h2βδi je⊥ke⊥leR2⊥ + 60λ2δi je⊥ke⊥leR2⊥195B.1. Details of far-field hydrodynamics+ 180δi je⊥ke⊥leR2⊥ − 210eR jeRkeRle⊥ieR⊥ − 210eRieRkeRle⊥ jeR⊥+ 30eRlδike⊥ jeR⊥ + 30eRkδile⊥ jeR⊥ + 30eRle⊥iδ jkeR⊥+ 30eRke⊥iδ jleR⊥ − 2100λ2h2βeRieR jeRle⊥keR⊥+ 210(λ2 − 18eR2⊥ + 2)eRieR jeRle⊥keR⊥ + 300λ2h2βeRlδi je⊥keR⊥− 30(λ2 − 14eR2⊥ + 2)eRlδi je⊥keR⊥ + 300λ2h2βeR jδile⊥keR⊥− 30(λ2 − 14eR2⊥ + 2)eR jδile⊥keR⊥ − 120eRle⊥ie⊥ je⊥keR⊥+ 300λ2h2βeRiδ jle⊥keR⊥ − 30(λ2 − 14eR2⊥ + 2)eRiδ jle⊥keR⊥+ 30eR je⊥iδkleR⊥ + 30eRie⊥ jδkleR⊥ − 2100λ2h2βeRieR jeRke⊥leR⊥+ 210(λ2 − 18eR2⊥ + 2)eRieR jeRke⊥leR⊥ + 300λ2h2βeRkδi je⊥leR⊥− 30(λ2 − 14eR2⊥ + 2)eRkδi je⊥leR⊥ + 300λ2h2βeR jδike⊥leR⊥− 30(λ2 − 14eR2⊥ + 2)eR jδike⊥leR⊥ − 120eRke⊥ie⊥ je⊥leR⊥+ 300λ2h2βeRiδ jke⊥leR⊥ − 30(λ2 − 14eR2⊥ + 2)eRiδ jke⊥leR⊥− 1200λ2h2βeR je⊥ie⊥ke⊥leR⊥+ 120(λ2 − 21eR2⊥ + 2)eR je⊥ie⊥ke⊥leR⊥− 1200λ2h2βeRie⊥ je⊥ke⊥leR⊥+ 120(λ2 − 21eR2⊥ + 2)eRie⊥ je⊥ke⊥leR⊥+ 105(10λ2h2β − λ2 + 18eR2⊥ − 1)eRieR jeRkeRl− 3(50λ2h2β − 5λ2 + 70eR2⊥ − 5)eRkeRlδi j− 15(10λ2h2β − λ2 + 14eR2⊥ − 1)eR jeRlδik− 15(10λ2h2β − λ2 + 14eR2⊥ − 1)eR jeRkδil− 15(10λ2h2β − λ2 + 14eR2⊥ − 1)eRieRlδ jk− 3(−10λ2h2β + λ2 − 10eR2⊥ + 1)δilδ jk− 15(10λ2h2β − λ2 + 14eR2⊥ − 1)eRieRkδ jl− 3(−10λ2h2β + λ2 − 10eR2⊥ + 1)δikδ jl+ 30(10λ2h2β − λ2 + 28eR2⊥ − 1)eR jeRle⊥ie⊥k+ 30(10λ2h2β − λ2 + 28eR2⊥ − 1)eRieRle⊥ je⊥k+ 6(−10λ2h2β + λ2 − 20eR2⊥ + 1)δile⊥ je⊥k+ 6(−10λ2h2β + λ2 − 20eR2⊥ + 1)e⊥iδ jle⊥k196B.1. Details of far-field hydrodynamics− 3(50λ2h2β − 5λ2 + 70eR2⊥ − 5)eRieR jδkl+(30λ2h2β − 3λ2 + 30eR2⊥ − 3)δi jδkl+ 30(10λ2h2β − λ2 + 28eR2⊥ − 1)eR jeRke⊥ie⊥l+ 30(10λ2h2β − λ2 + 28eR2⊥ − 1)eRieRke⊥ je⊥l− 6(10λ2h2β − λ2 + 20eR2⊥ − 1)δike⊥ je⊥l+ 6(−10λ2h2β + λ2 − 20eR2⊥ + 1)e⊥iδ jke⊥l− 24(−10λ2h2β + λ2 − 30eR2⊥ + 1)e⊥ie⊥ je⊥ke⊥l]+9hβ(λ2 − 1)2λ2R6[− 28δi je⊥ke⊥leR3⊥ + 14eRlδi je⊥keR2⊥ + 14eRkδi je⊥leR2⊥− 7eRkeRlδi jeR⊥ + δilδ jkeR⊥ + δikδ jleR⊥ − 4δile⊥ je⊥keR⊥− 4e⊥iδ jle⊥keR⊥ + δi jδkleR⊥ − 4δike⊥ je⊥leR⊥ − 4e⊥iδ jke⊥leR⊥+ 4δi je⊥ke⊥leR⊥ + 24e⊥ie⊥ je⊥ke⊥leR⊥ + eRlδike⊥ j + eRkδile⊥ j+ eRle⊥iδ jk + eRke⊥iδ jl − eRlδi je⊥k − 4eRle⊥ie⊥ je⊥k − eRkδi je⊥l− 4eRke⊥ie⊥ je⊥l + eRi(14δ jle⊥keR2⊥ + 14δ jke⊥leR2⊥− 84e⊥ je⊥ke⊥leR2⊥ − 7eRlδ jkeR⊥ − 7eRkδ jleR⊥ + 28eRle⊥ je⊥keR⊥+ 28eRke⊥ je⊥leR⊥ − 7eRkeRle⊥ j − δ jle⊥k + e⊥ jδkl − δ jke⊥l+ 4e⊥ je⊥ke⊥l + 7eR j( (1 − 18eR2⊥)eRle⊥k+ eRk(9eR⊥eRl +(1 − 18eR2⊥)e⊥l)+ eR⊥(4(9eR2⊥ − 1)e⊥ke⊥l − δkl) ))+ eR j(14δile⊥keR2⊥ + 14δike⊥leR2⊥ − 84e⊥ie⊥ke⊥leR2⊥− 7eRl (δik − 4e⊥ie⊥k) eR⊥ − δile⊥k + e⊥iδkl− δike⊥l + 4e⊥ie⊥ke⊥l − 7eRk (eRle⊥i + eR⊥ (δil − 4e⊥ie⊥l)))]910λ2R7[− 2772eRieR je⊥ke⊥leR4⊥ + 252δi je⊥ke⊥leR4⊥ + 756eRieR je⊥ke⊥leR2⊥− 84δi je⊥ke⊥leR2⊥ + 126eR jeRkeRle⊥ieR⊥ + 126eRieRkeRle⊥ jeR⊥− 14eRlδike⊥ jeR⊥ − 14eRkδile⊥ jeR⊥ − 14eRle⊥iδ jkeR⊥197B.1. Details of far-field hydrodynamics− 14eRke⊥iδ jleR⊥ + 126(11eR2⊥ − 2)eRieR jeRle⊥keR⊥+ 14(2 − 9eR2⊥)eRlδi je⊥keR⊥ + 14(2 − 9eR2⊥)eR jδile⊥keR⊥+ 84eRle⊥ie⊥ je⊥keR⊥ + 14(2 − 9eR2⊥)eRiδ jle⊥keR⊥− 14eR je⊥iδkleR⊥ − 14eRie⊥ jδkleR⊥+ 126(11eR2⊥ − 2)eRieR jeRke⊥leR⊥ + 14(2 − 9eR2⊥)eRkδi je⊥leR⊥+ 14(2 − 9eR2⊥)eR jδike⊥leR⊥ + 84eRke⊥ie⊥ je⊥leR⊥+ 14(2 − 9eR2⊥)eRiδ jke⊥leR⊥ + 168(6eR2⊥ − 1)eR je⊥ie⊥ke⊥leR⊥+ 168(6eR2⊥ − 1)eRie⊥ je⊥ke⊥leR⊥ + 63(1 − 11eR2⊥)eRieR jeRkeRl+ 7(9eR2⊥ − 1)eRkeRlδi j + 7(9eR2⊥ − 1)eR jeRlδik+ 7(9eR2⊥ − 1)eR jeRkδil − 14eRkeRle⊥ie⊥ j + 7(9eR2⊥ − 1)eRieRlδ jk−(7eR2⊥ − 1)δilδ jk + 7(9eR2⊥ − 1)eRieRkδ jl −(7eR2⊥ − 1)δikδ jl+ 14(2 − 27eR2⊥)eR jeRle⊥ie⊥k + 14(2 − 27eR2⊥)eRieRle⊥ je⊥k+ 2(21eR2⊥ − 2)δile⊥ je⊥k + 2(21eR2⊥ − 2)e⊥iδ jle⊥k+ 7(9eR2⊥ − 1)eRieR jδkl +(1 − 7eR2⊥)δi jδkl + 2e⊥ie⊥ jδkl+ 14(2 − 27eR2⊥)eR jeRke⊥ie⊥l + 14(2 − 27eR2⊥)eRieRke⊥ je⊥l+ 2(21eR2⊥ − 2)δike⊥ je⊥l + 2(21eR2⊥ − 2)e⊥iδ jke⊥l− 14eRieR je⊥ke⊥l + 2δi je⊥ke⊥l − 24(14eR2⊥ − 1)e⊥ie⊥ je⊥ke⊥l],(B.6)where R = ReR and eR⊥ = e⊥ · eR.198B.2. Details of near-field hydrodynamicsB.2 Details of near-field hydrodynamicsThe geometrical tensors di in eq. 3.52 ared1i j = erier j,d2i j = δi j − erier j,d3i j = ϵi jkerk,d4i jk = (erier j −13δi j)erk,d5i jk = eriδ jk + er jδik − 2erier jerk,d6i jk = ϵiklerler j + ϵ jklerleri,d7i jkl =32(erier j − 13δi j)(erkerl −13δkl),d8i jkl =12(eriδ jlerk + er jδilerk + eriδ jkerl + er jδikerl − 4erier jerkerl),d9i jkl =12(δikδ jl + δ jkδil − δi jδkl + erier jδkl + erkerlδi j − erierkδ jl − er jerkδil− erierlδ jk + er jerlδik + erier jerkerl). (B.7)Coefficients gXi (λ) in eq.3.55 aregXFU1 = −gXFU2 = −gXFU8 = −2λ/(1 + λ)3,gXFU4 = −gXFU9 = −(1 + 7λ + λ2)/5(1 + λ)3,gXFU6 = −12gXFU12 = −gXFU13 = −(1 + 18λ − 29λ2 + 18λ3 + λ4)/42λ(1 + λ)3,gYFU4 = −gYFU9 = −4(2 + λ + 2λ2)/15(1 + λ)3,gYFU6 = −12gYFU12 = −gYFU13 = −2(1 − λ)2(16 − 13λ + 16λ2)/375λ(1 + λ)3,gYLU4 = −gYLU9 = (1 + 4λ)/5(1 + λ)2,gYLU6 = −12gYLU12 = −gYLU13 = (43 + 83λ − 33λ2 + 32λ3)/250λ(1 + λ)2,gXLΩ3 = gXLΩ4 = −gXLΩ9 = −gXLΩ10 =13gXLΩ12 = 1/2λ(1 + λ),gYLΩ4 = 2/5(1 + λ),gYLΩ6 = (33 + 6λ + 8λ2)/125λ(1 + λ),gYLΩ9 = −λ2/10(1 + λ),12gYLΩ12 = gYLΩ13 = −λ(43 − 24λ + 43λ2)/500(1 + λ),199B.2. Details of near-field hydrodynamicsgXSU1 = (−65 + 566λ − 33λ2 + 181λ3 + 5λ4)/140λ(1 + λ)3,gXSU2 = −gXSU8 = −3λ/(1 + λ)3,gXSU4 = gXSU9 = 3(−4 + 12λ + λ2)/10(1 + λ)3,gXSU5 = gXSU11 = (−65 + 566λ − 453λ2 + 181λ3 + 5λ4)/140λ(1 + λ)3,gXSU7 = −(1 + 18λ − 29λ2 + 18λ3 + λ4)/42λ(1 + λ)3,gYSU1 = gYSU5 = gYSU11 = −(221 − 356λ + 532λ2 − 179λ3 + 32λ4)/500λ(1 + λ)3,gYSU4 = gYSU9 = −(7 − λ + 4λ2)/10(1 + λ)3,gYSU7 = 2(1 − λ)2(16 − 13λ + 16λ2)/375λ(1 + λ)3,gYSΩ1 = gYSΩ5 = −gYSΩ7 = −(2 + 180λ − 61λ2 + 16λ3)/500λ(1 + λ)2,gYSΩ4 = (1 − 2λ)/10(1 + λ)2,gYSΩ9 = (7 + λ)/20λ(1 + λ)2,gYSΩ11 = (221 − 185λ + 147λ2 + 43λ3)/1000λ2(1 + λ)2,gXSE1 = (415 − 1332λ + 411λ2 − 272λ3 − 5λ4)/350λ(1 + λ)3,gXSE2 = λgXSE8 = 6λ/5(1 + λ)3,gXSE4 = −3(−9 + 17λ + λ2)/25(1 + λ)3,gXSE9 = −3(4 − 17λ + 4λ2)/25λ(1 + λ)3,gXSE5 = −gXSE7 = (415 − 1332λ + 831λ2 − 272λ3 − 5λ4)/350λ(1 + λ)3,gXSE11 = (−65 + 832λ − 1041λ2 + 832λ3 − 65λ4)/350λ2(1 + λ)3,gYSE1 = gYSE5 = −gYSE7 = −3(197 − 394λ + 274λ2 − 67λ3 + 8λ4)/625λ(1 + λ)3,gYSE4 = −6(4 − λ + λ2)/25(1 + λ)3,gYSE9 = 3(7 − 10λ + 7λ2)/50λ(1 + λ)3,gYSE11 = 3(221 − 748λ + 1902λ2 − 748λ3 + 221λ4)/2500λ2(1 + λ)3,gZSE1 = gZSE5 = −gZSE7 = −gZSE11 = 3(1 + λ2)/10λ(1 + λ)3, (B.8)200B.2. Details of near-field hydrodynamicsotherwise gXi (λ) = 0. Furthermore, functions fXk (λ) in eq.3.55 aref {XFU,YFU}k =k∑q=0P{XFU,YFU}1,k−q,qλk−q(1 + λ)k,f YLUk = 2k∑q=0QYLU1,k−q,qλk−q(1 + λ)k,f {XLΩ,YLΩ}k =k∑q=0Q{XLΩ,YLΩ}1,k−q,qλk−q(1 + λ)kλ(−1)k−12 ,f {XSU,YSU}k =34k∑q=0P{XSU,YSU}2,k−q,qλk−q(1 + λ)k,f YSΩk = −38k∑q=0PYSΩ2,k−q,qλk−q(1 + λ)kλ(−1)k−12 ,f {XSE,YSE,ZSE}k =k∑q=0P{XSE,YSE,ZSE}2,k−q,qλk−q(1 + λ)kλ(−1)k−12 ,(B.9)and their corresponding recurrence relations V , P, and Q areVXn,p,q = PXn,p,q + nX1q∑s=1nX2 PXs,q−s,p−n−1,PXn,p,q =q∑s=1nX2(nX3 PXs,q−s,p−n+1 + nX4 PXs,q−s,p−n−1 + nX5 VXs,q−s−2,p−n+)+ nX6 QXs,q−s−1,p−n+1),QXn,p,q =q∑s=1nX2(nX7 QXs,q−s−1,p−n + nX8 PXs,q−s,p−n),with boundary conditions ofV {XFU,XSU}n,0,0 = P{XFU,XSU}n,0,0 = δ1n,V {YFU,YLU,YSU,YSE}n,0,0 = P{YFU,YLU,YSU,YSE}n,0,0 = δ1n , Q{YFU,YLU,YSU,YSE}n,0,0 = 0,QXLΩn,0,0 = δ1n,V {YLΩ,YSΩ}n,0,0 = P{YLΩ,YSΩ}n,0,0 = 0 , Q{YLΩ,YSΩ}n,0,0 = δ1n,VXSEn,0,0 = PXSEn,0,0 = δ2n,V {YSE,ZSE}n,0,0 = P{YSE,ZSE}n,0,0 = δ2n , Q{YSE,ZSE}n,0,0 = 0,(B.10)201B.2. Details of near-field hydrodynamicsand coefficients ofn{XFU,XSU,XSE}1 = −2n/(2n + 3),n{YFU,YLU,YLΩ,YSU,YSΩ,YSE,ZSE}1 = 2n/(2n + 3),n{XFU,XSU,XSE,XLΩ}2 =n+sCn/(n + 1),n{YFU,YLU,YLΩ,YSU,YSΩ,YSE}2 =n+sCn+1/(n + 1),nZSE2 =n+sCn+2/(n + 1),n{XFU,XSU,XSE}3 = n(2n + 1)2ns − n − s + 22(2s − 1)(n + s) ,n{YFU,YLU,YLΩ,YSU,YSΩ}3 = (2n + 1)3(n + s) − (ns + 1)(2ns − s − n + 2)2s(2s − 1)(n + s) ,nYSE3 = (2n + 1)(ns + 4)(n + s) − 2(ns + 1)22s(2s − 1)(n + s) ,nZSE3 = (2n + 1)(ns + 16)(n + s) − 2(ns + 4)(ns + 1)2s(2s − 1)(n + s) ,n{XFU,XSU,XSE}4 = −n(2n − 1)/2,n{YFU,YLU,YLΩ,YSU,YSΩ,YSE,ZSE}4 = n(2n − 1)/2,n{XFU,XSU,XSE}5 = n(4n2 − 1)/2(2s + 1),n{YFU,YLU,YLΩ,YSU,YSΩ,YSE,ZSE}5 = −n(4n2 − 1)/2(2s + 1),n{YFU,YLU,YLΩ,YSU,YSΩ}6 = −2(4n2 − 1)/3,nYSE6 = −(4n2 − 1),nZSE6 = −2(4n2 − 1),n{YFU,YLU,XLΩ,YLΩ,YSU,YSΩ,YSE,ZSE}7 = s,n{YFU,YLU,YLΩ,YSU,YSΩZSE}8 = −3/2ns,nYSE8 = −ns,nZSE8 = −2ns,(B.11)otherwise nXi = 0. Also,nCm is n choose m. The solver including the GRM, forcemoments, and velocity moments has the form of202B.2. Details of near-field hydrodynamicsFα1Fα2Fα3Fβ1Fβ2Fβ3Lα1L α2L α3L β1L β2L β3Sα[1]Sα[2]Sα[3]Sα[4]Sα[5]Sβ[1]Sβ[2]Sβ[3]Sβ[4]Sβ[5] =−RααFU11RααFU12RααFU13RαβFU11RαβFU12RαβFU13RααFΩ11RααFΩ12RααFΩ13RαβFΩ11RαβFΩ12RαβFΩ13RααFE1[1]RααFE1[2]RααFE1[3]RααFE1[4]RααFE1[5]RαβFE1[1]RαβFE1[2]RαβFE1[3]RαβFE1[4]RαβFE1[5]RααFU21RααFU22RααFU23RαβFU21RαβFU22RαβFU23RααFΩ21RααFΩ22RααFΩ23RαβFΩ21RαβFΩ22RαβFΩ23RααFE2[1]RααFE2[2]RααFE2[3]RααFE2[4]RααFE2[5]RαβFE2[1]RαβFE2[2]RαβFE2[3]RαβFE2[4]RαβFE2[5]RααFU31RααFU32RααFU33RαβFU31RαβFU32RαβFU33RααFΩ31RααFΩ32RααFΩ33RαβFΩ31RαβFΩ32RαβFΩ33RααFE3[1]RααFE3[2]RααFE3[3]RααFE3[4]RααFE3[5]RαβFE3[1]RαβFE3[2]RαβFE3[3]RαβFE3[4]RαβFE3[5]RβαFU11RβαFU12RβαFU13RββFU11RββFU12RββFU13RβαFΩ11RβαFΩ12RβαFΩ13RββFΩ11RββFΩ12RββFΩ13RβαFE1[1]RβαFE1[2]RβαFE1[3]RβαFE1[4]RβαFE1[5]RββFE1[1]RββFE1[2]RββFE1[3]RββFE1[4]RββFE1[5]RβαFU21RβαFU22RβαFU23RββFU21RββFU22RββFU23RβαFΩ21RβαFΩ22RβαFΩ23RββFΩ21RββFΩ22RββFΩ23RβαFE2[1]RβαFE2[2]RβαFE2[3]RβαFE2[4]RβαFE2[5]RββFE2[1]RββFE2[2]RββFE2[3]RββFE2[4]RββFE2[5]RβαFU31RβαFU32RβαFU33RββFU31RββFU32RββFU33RβαFΩ31RβαFΩ32RβαFΩ33RββFΩ31RββFΩ32RββFΩ33RβαFE3[1]RβαFE3[2]RβαFE3[3]RβαFE3[4]RβαFE3[5]RββFE3[1]RββFE3[2]RββFE3[3]RββE3[4]RββE3[5]RααLU11RααLU12RααLU13RαβLU11RαβLU12RαβLU13RααLΩ11RααLΩ12RααLΩ13RαβLΩ11RαβLΩ12RαβLΩ13RααLE1[1]RααLE1[2]RααLE1[3]RααLE1[4]RααLE1[5]RαβLE1[1]RαβLE1[2]RαβLE1[3]RαβLE1[4]RαβLE1[5]RααLU21RααLU22RααLU23RαβLU21RαβLU22RαβLU23RααLΩ21RααLΩ22RααLΩ23RαβLΩ21RαβLΩ22RαβLΩ23RααLE2[1]RααLE2[2]RααLE2[3]RααLE2[4]RααLE2[5]RαβLE2[1]RαβLE2[2]RαβLE2[3]RαβLE2[4]RαβLE2[5]RααLU31RααLU32RααLU33RαβLU31RαβLU32RαβLU33RααLΩ31RααLΩ32RααLΩ33RαβLΩ31RαβLΩ32RαβLΩ33RααLE3[1]RααLE3[2]RααLE3[3]RααLE3[4]RααLE3[5]RαβLE3[1]RαβLE3[2]RαβLE3[3]RαβLE3[4]RαβLE3[5]RβαLU11RβαLU12RβαLU13RββLU11RββLU12RββLU13RβαLΩ11RβαLΩ12RβαLΩ13RββLΩ11RββLΩ12RββLΩ13RβαLE1[1]RβαLE1[2]RβαLE1[3]RβαLE1[4]RβαLE1[5]RββLE1[1]RββLE1[2]RββLE1[3]RββLE1[4]RββLE1[5]RβαLU21RβαLU22RβαLU23RββLU21RββLU22RββLU23RβαLΩ21RβαLΩ22RβαLΩ23RββLΩ21RββLΩ22RββLΩ23RβαLE2[1]RβαLE2[2]RβαLE2[3]RβαLE2[4]RβαLE2[5]RββLE2[1]RββLE2[2]RββLE2[3]RββLE2[4]RββLE2[5]RβαLU31RβαLU32RβαLU33RββLU31RββLU32RββLU33RβαLΩ31RβαLΩ32RβαLΩ33RββLΩ31RββLΩ32RββLΩ33RβαLE3[1]RβαLE3[2]RβαLE3[3]RβαLE3[4]RβαLE3[5]RββLE3[1]RββLE3[2]RββLE3[3]RββLE3[4]RββLE3[5]RααSU[1]1RααSU[1]2RααSU[1]3RαβSU[1]1RαβSU[1]2RαβSU[1]3RααSΩ[1]1RααSΩ[1]2RααSΩ[1]3RαβSΩ[1]1RαβSΩ[1]2RαβSΩ[1]3RααSE[1][1]RααSE[1][2]RααSE[1][3]RααSE[1][4]RααSE[1][5]RαβSE[1][1]RαβSE[1][2]RαβSE[1][3]RαβSE[1][4]RαβSE[1][5]RααSU[2]1RααSU[2]2RααSU[2]3RαβSU[2]1RαβSU[2]2RαβSU[2]3RααSΩ[2]1RααSΩ[2]2RααSΩ[2]3RαβSΩ[2]1RαβSΩ[2]2RαβSΩ[2]3RααSE[2][1]RααSE[2][2]RααSE[2][3]RααSE[2][4]RααSE[2][5]RαβSE[2][1]RαβSE[2][2]RαβSE[2][3]RαβSE[2][4]RαβSE[2][5]RααSU[3]1RααSU[3]2RααSU[3]3RαβSU[3]1RαβSU[3]2RαβSU[3]3RααSΩ[3]1RααSΩ[3]2RααSΩ[3]3RαβSΩ[3]1RαβSΩ[3]2RαβSΩ[3]3RααSE[3][1]RααSE[3][2]RααSE[3][3]RααSE[3][4]RααSE[3][5]RαβSE[3][1]RαβSE[3][2]RαβSE[3][3]RαβSE[3][4]RαβSE[3][5]RααSU[4]1RααSU[4]2RααSU[4]3RαβSU[4]1RαβSU[4]2RαβSU[4]3RααSΩ[4]1RααSΩ[4]2RααSΩ[4]3RαβSΩ[4]1RαβSΩ[4]2RαβSΩ[4]3RααSE[4][1]RααSE[4][2]RααSE[4][3]RααSE[4][4]RααSE[4][5]RαβSE[4][1]RαβSE[4][2]RαβSE[4][3]RαβSE[4][4]RαβSE[4][5]RααSU[5]1RααSU[5]2RααSU[5]3RαβSU[5]1RαβSU[5]2RαβSU[5]3RααSΩ[5]1RααSΩ[5]2RααSΩ[5]3RαβSΩ[5]1RαβSΩ[5]2RαβSΩ[5]3RααSE[5][1]RααSE[5][2]RααSE[5][3]RααSE[5][4]RααSE[5][5]RαβSE[5][1]RαβSE[5][2]RαβSE[5][3]RαβSE[5][4]RαβSE[5][5]RβαSU[1]1RβαSU[1]2RβαSU[1]3RββSU[1]1RββSU[1]2RββSU[1]3RβαSΩ[1]1RβαSΩ[1]2RβαSΩ[1]3RββSΩ[1]1RββSΩ[1]2RββSΩ[1]3RβαSE[1][1]RβαSE[1][2]RβαSE[1][3]RβαSE[1][4]RβαSE[1][5]RββSE[1][1]RββSE[1][2]RββSE[1][3]RββSE[1][4]RββSE[1][5]RβαSU[2]1RβαSU[2]2RβαSU[2]3RββSU[2]1RββSU[2]2RββSU[2]3RβαSΩ[2]1RβαSΩ[2]2RβαSΩ[2]3RββSΩ[2]1RββSΩ[2]2RββSΩ[2]3RβαSE[2][1]RβαSE[2][2]RβαSE[2][3]RβαSE[2][4]RβαSE[2][5]RββSE[2][1]RββSE[2][2]RββSE[2][3]RββSE[2][4]RββSE[2][5]RβαSU[3]1RβαSU[3]2RβαSU[3]3RββSU[3]1RββSU[3]2RββSU[3]3RβαSΩ[3]1RβαSΩ[3]2RβαSΩ[3]3RββSΩ[3]1RββSΩ[3]2RββSΩ[3]3RβαSE[3][1]RβαSE[3][2]RβαSE[3][3]RβαSE[3][4]RβαSE[3][5]RββSE[3][1]RββSE[3][2]RββSE[3][3]RββSE[3][4]RββSE[3][5]RβαSU[4]1RβαSU[4]2RβαSU[4]3RββSU[4]1RββSU[4]2RββSU[4]3RβαSΩ[4]1RβαSΩ[4]2RβαSΩ[4]3RββSΩ[4]1RββSΩ[4]2RββSΩ[4]3RβαSE[4][1]RβαSE[4][2]RβαSE[4][3]RβαSE[4][4]RβαSE[4][5]RββSE[4][1]RββSE[4][2]RββSE[4][3]RββSE[4][4]RββSE[4][5]RβαSU[5]1RβαSU[5]2RβαSU[5]3RββSU[5]1RββSU[5]2RββSU[5]3RβαSΩ[5]1RβαSΩ[5]2RβαSΩ[5]3RββSΩ[5]1RββSΩ[5]2RββSΩ[5]3RβαSE[5][1]RβαSE[5][2]RβαSE[5][3]RβαSE[5][4]RβαSE[5][5]RββSE[5][1]RββSE[5][2]RββSE[5][3]RββSE[5][4]RββSE[5][5]  Uα1Uα2Uα3Uβ1Uβ2Uβ3Ωα1Ωα2Ωα3Ωβ1Ωβ2Ωβ3Eα[1]Eα[2]Eα[3]Eα[4]Eα[5]Eβ[1]Eβ[2]Eβ[3]Eβ[4]Eβ[5]  .(B.12)203B.3. Tensorial notationB.3 Tensorial notationThe boundary integral equation isu(x) = −∑β∫S[G(x, y) · fβ(y) + uβ(y) ·Gs(x, y) · n(y)]dS (y), (B.13)whereuβ(y) =Uβ + aβΩβ × n + uβs(y),uβs(y) =m∑0Cβm+1 ⊙ nm,fβ(y) =m∑0Bβm+1 ⊙ nm,G(x, y) =m∑0amβm!(1 +a2β4m + 6∇2y)(nm ⊙∇my )G(x, y)|y=xβ ,n ·Gs(x, y) =m∑0amβm!(1 +a2β4m + 6∇2y)[(nm+1 +m2m + 1∆m ⊙ nm−1)⊙ ∇my]·Gs(x, y)|y=xβ ,∫Snm dS =4πa2 δ0m,∫Snm nm′dS =4πa2 m!(2m + 1)!!δmm′∆m, (B.14)thenu(x) = −∑βm∑04πam+2β(2m + 1)!!F βy (m)[(Bβm+1 ⊙ ∆m ⊙ ∇my)·G(x, y)|y=xβ+((m + 12m + 3Cβm+2 ⊙ ∆m+1 + Cβm ⊙ ∆m)⊙ ∇my): Gs(x, y)|y=xβ], (B.15)whereF βy (m) = 1 +a2β4m + 6∇2y . (B.16)204B.3. Tensorial notationSeparating self terms and other particles’ contribution we getUα +Ωα × n(x)aα+m′∑0Cαm′+1 ⊙ n(x)m′=self terms−m∑04πam+2α(2m + 1)!!F αy (m)[(Bαm+1 ⊙ ∆m ⊙ ∇my)·G(aαn(x))+(( m + 12m + 3Cαm+2 ⊙ ∆m+1 + Cαm ⊙ ∆m)⊙ ∇my): Gs(aαn(x))],disturbed field−∑β,αm∑0m′∑04πam+2β(2m + 1)!!am′αm′!F βy (m)F αx (m′) n(x)m′ ⊙∇m′x[(Bβm+1 ⊙ ∆m ⊙ ∇my)·G(x, y)|x=xαy=xβ+(( m + 12m + 3Cβm+2 ⊙ ∆m+1 + Cβm ⊙ ∆m)⊙ ∇my): Gs(x, y)|x=xαy=xβ],(B.17)also8πµG(r) =43Ir+13r2∇∇(1r),4π3Gs(r) =15[13r2∇∇∇ + I∇ + ∇I + ∇IT](1r),∇nr (r2f(r)) = n(n − 1)I∇n−2r f(r) + 2 n r∇n−1r f(r) + r2∇nr f(r),∇n(1r) =(2n − 1)!!(−1)n an+1αnn, (B.18)205B.3. Tensorial notationwhere r = x − y. Finally we getUα +Ωα × n(x)aα +m∑0Cαm+1 ⊙ n(x)m =self terms−m∑0[aα6µ(2m + 1)Bαm+1 ⊙((4I∆m − 2m(m + 1)(2m + 1)2m + 3∆m ⊙ ∆m+1)⊙ n(x)m +2(m − 1)∆m+2 ⊙ n(x)m+2)+m + 15(2m + 3)Cαm+2 ⊙((− 3I∆m+1 + 2m(m + 2)(2m + 3)2m + 5∆m+1 ⊙ ∆m+2− 3∆m+1 ⊙ ∆m+1 − 3(∆m+1 + ∆m+1T ))⊙ n(x)m+1 +10∆m+3 ⊙ n(x)m+3)+15Cαm ⊙(− 6∆m+1 + 2m(m + 2)(2m + 3)2m + 5∆m ⊙ ∆m+2)⊙ n(x)m+1],disturbed field−∑β,αm∑0m′∑04πam′+2β(2m′ + 1)!!amαm!F βy (m′)F αx (m) n(x)m ⊙∇mx[(Bβm′+1 ⊙ ∆m′ ⊙ ∇m′y)·G(rαβ)+(( m′ + 12m′ + 3Cβm′+2 ⊙ ∆m′+1 + Cβm′ ⊙ ∆m′) ⊙ ∇m′y ) : Gs(rαβ)]. (B.19)Matching self terms we get:n(x)0 → Uα + Cα1 =−2aα3µBα1 ,n(x)1 → aαΩα : ϵ + Cα2 = −aα45µ(12Bα2 − 3Bα2 T − 3II : Bα2)+15,n(x)2 → Cα3 = +aα3µBα1 · ∆2 −2 aα15µBα3 ⊙(I∆2 +157∆3 ⊙ ∆2)+625Cα3 ⊙(I∆2 − 107∆3 ⊙ ∆2 + ∆2 · ∆2 + (∆2 · ∆2)T),n(x)m−1 → Cαm = f(Bαm−2,Bαm,Cαm−2,Cαm), (B.20)206B.3. Tensorial notationwhereBα1 =< f >=F4πa2α,Bα2 =< 3 n f >=34πa3α(S˜α +12ϵ · Lα),Cα1 =< us >= Usα,Cα2 =< 3 n us >= aα(Esα + ϵ ·Ωsα). (B.21)After absorbing the double-layer contribution into the stresslet of the form S =S˜ − 2µVEs, the self mobilities upto n(x)1 get the form ofMααUF =I6πµaα,MααΩL =I8πµa3α,MααES =3I20πµa3α. (B.22)Matching disturbed terms: for n(x)0 we getUα + Usα = −∑β,αm∑0m′∑04πam′+2β(2m′ + 1)!!F βy (m′)(1 + a2α6∇2x)[(Bβm′+1 ⊙ ∆m′ ⊙ ∇m′y)·G(rαβ)+(( m′ + 12m′ + 3Cβm′+2 ⊙ ∆m′+1 + Cβm′ ⊙ ∆m′) ⊙ ∇m′y ) : Gs(rαβ)]. (B.23)We expand this series up to m′ = 1Uα + Usα = −∑β,α[F βy (0)F αx (0) Fβ ·G(rαβ)+12F βy (1)F αx (0) (Lβ × ∇y) ·G(rαβ)+F βy (1)F αx (0) S˜β : ∇yG(rαβ)+4πa3β3F βy (0)F αx (0) Es β : Gs(rαβ)+4πa3β3F βy (1)F αx (0) (Cβ3 + Cβ3T) · ∇y : Gs(rαβ) + . . .]. (B.24)207B.3. Tensorial notationThen the mobilities areMαβUF = F βy (0)F αx (0) G(rαβ),MαβUL =12F βy (1)F αx (0)∇y ×G(rαβ),MαβUS =12F βy (1)F αx (0)(∇yG(rαβ) + ∇yG(rαβ)T)]. (B.25)208Appendix CJeffery orbits in shear-thinningfluidsC.1 Spheroidal multipolesHere we give the solution to the Stokes equations for a spheroid with the aspectratio of λ in a shear flow, following Einarsson et al. [75], in terms of a finite multi-pole expansion [39].To this end the Green’s function, G, of the Stokes equations and derivativeswill be utilized, these are, in component formGi j =δi jr+xix jr3, Stokeslet,Gdi jk = Gi j,k =δ jkxi + δikx j − δi jxkr3− 3 xix jr5, dipole,GDi j = Gi j,ll = 2δi jr3− 6 xix jr5, potential doublet,GRi jk =12(Gi j,k −Gik, j) = δikx j − δi jxkr3 , rotlet,GSi jk =12(Gi j,k +Gik, j) =δk jxir3− 3 xix jxkr5, stresslet,GQi jk = Gi j,llk = −6δ jkxi + δikx j + δi jxkr5+ 30xix jxkr7, potential quadrupole.(C.1)Representation of the flow around a spheroidal particle requires a weighted distri-bution of the above multipoles. Spheroidal multipoles are found by employing aline distribution of Stokeslets and derivatives between the foci ξ = −c to c given byQi j =∫ c−cdξGi j(x − ξp),QDi j =∫ c−cdξ(c2 − ξ2)GDi j(x − ξp),209C.1. Spheroidal multipolesQRi jk =∫ c−cdξ(c2 − ξ2)GRi jk(x − ξp),QSi jk =∫ c−cdξ(c2 − ξ2)GSi jk(x − ξp),QQi jk =∫ c−cdξ(c2 − ξ2)2GQi jk(x − ξp), (C.2)where c2 = a2(λ2 − 1)/λ2.Explicit expressions for spheroidal multipoles in terms of integrals of the stresslet,rotlet, and quadrupole are given in Einarsson et al. [75] and we add higher-ordermultipoles as needed hereQi j = δi jI01 + xix jI03 − (xi p j + x j pi)I13 + pi p jI23 ,QDi j = 2δi jJ03 + 6[− xix jJ05 + (xi p j + x j pi)J15 − pi p jJ25],QRi jk = (δikx j − δi jxk)J03 + (δi j pk − δik p j)J13 ,QSi jk = δ jkxiJ03 − δ jk piJ13+ 3[− xix jxkJ05 + (xixk p j + x jxk pi + xix j pk)J15− (xk pi p j + xi p j pk + x j pi pk)J25 + pi p j pkJ35],QQi jk = 6[− (δ jkxi + δikx j + δi jxk)K05 + (δ jk pi + δik p j + δi j pk)K15]+ 30[xix jxkK07 − (xixk p j + x jxk pi + xix j pk)K17+ (xk pi p j + xi p j pk + x j pi pk)K27 − pi p j pkK37],QRi jk,m = (δikδ jm − δi jδkm)J03+ 3(δikx j − δi jxk)(pmJ15 − xmJ05)+ 3(δi j pk − δik p j)(pmJ25 − xmJ15),QSi jk,m = δ jkδimJ03 + 3δ jkxi(pmJ15 − xmJ05) − 3δ jk pi(pmJ25 − xmJ15)+ 3[− (δimx jxk + δ jmxixk + δkmxix j)J05 − 5xix jxk(pmJ17 − xmJ07)+ (δimxk p j + δkmxi p j + δ jmxk pi + δkmx j pi + δimx j pk + δ jmxi pk)J15+ 5(xixk p j + x jxk pi + xix j pk)(pmJ27 − xmJ17)210C.1. Spheroidal multipoles− (δkm pi p j + δim p j pk + δ jm pi pk)J25 + 5pi p j pk(pmJ47 − xmJ37− 5(xk pi p j + xi p j pk + x j pi pk)(pmJ37 − xmJ27)],QQi jk,m = 6[− (δ jkδim + δikδ jm + δi jδkm)K05 − 5(δ jkxi + δikx j + δi jxk)(pmK17 − xmK07 )+ 5(δ jk pi + δik p j + δi j pk)(pmK27 − xmK17 )]+ 30[(δimx jxk + δ jmxixk + δkmxix j)K07 + 7xix jxk(pmK19 − xmK09 )− (δimxk p j + δkmxi p j + δ jmxk pi + δkmx j pi + δimx j pk + δ jmxi pk)K17− 7(xixk p j + x jxk pi + xix j pk)(pmK29 − xmK19 )+ (δkm pi p j + δim p j pk + δ jm pi pk)K27 − pi p j pkK37+ 7(xk pi p j + xi p j pk + x j pi pk)(pmK39 − xmK29 )], (C.3)whereInm =∫ c−cdξξn|x − ξp|m , (C.4)Jnm = c2Inm − In+2m , (C.5)Knm = c2Jnm − Jn+2m = c4Inm − 2c2In+2m + In+4m . (C.6)The integrals Inm satisfy the relationship∂∂xiInm = mpiIn+1m+2 − mxmInm+2. (C.7)To simplify integration one may employ an auxiliary coordinate system, (x′, y′, z′),with p aligned with x′ such thatInm =∫ c−cdξξn[(x′ − ξ)2 + (y′)2 + (z′)2]m/2 =∫ c−cdξξn[(x′ − ξ)2 + R2]m/2 , (C.8)where R2 = (y′)2 + (z′)2. Integrating one obtainsI0−1 =12[R2 log(R2 − (x′ − c)R1 − (x′ + c))+ c(R1 + R2) − (R2 − R1)x′],I01 = log(R2 − (x′ − c)R1 − (x′ + c)),211C.1. Spheroidal multipolesI11 = R2 − R1 + x′ log(R2 − (x′ − c)R1 − (x′ + c)),I21 =12[(2x′2 − R2)log(R2 − (x′ − c)R1 − (x′ + c))+ c(R1 + R2) + 3(R2 − R1)x′],I03 =1R2[x′ + cR1− x′ − cR2],I13 =1R1− 1R2+x′R2[x′ + cR1− x′ − cR2],I05 =13R4 (x′ + c)(2 (x′ + c)2 + 3R2)R31−(x′ − c)(2 (x′ − c)2 + 3R2)R32 ,I15 =13R4[ (2c3 + 3cR2) x′ + 3 (2c2 + R2) x′2 + 6cx′3 + R4 + 2x′4R31+(2c3 + 3cR2)x′ − 3(2c2 + R2)x′2 + 6cx′3 − R4 − 2x′4R32],I07 =115R6[(x′ + c)R51(8c4 + 8(4c3 + 5cR2)x′ + 4(12c2 + 5R2)x′2+ 20c2R2 + 32cx′3 + 15R4 + 8x′4)− (x′ − c)R52(8c4 − 8(4c3 + 5cR2)x′ + 4(12c2 + 5R2)x′2+ 20c2R2 − 32cx′3 + 15R4 + 8x′4)],I17 =115R6[1R52[− 3R6 +(8c5 + 20c3R2 + 15cR4)x′ − 5(8c4 + 12c2R2 + 3R4)x′2+ 20(4c3 + 3cR2)x′3 − 20(4c2 + R2)x′4 + 40cx′5 − 8x′6]− 1R51[− 3R6 −(8c5 + 20c3R2 + 15cR4)x′ − 5(8c4 + 12c2R2 + 3R4)x′2− 20(4c3 + 3cR2)x′3 − 20(4c2 + R2)x′4 − 40cx′5 − 8x′6]],I09 =135R8[1R71[c(16c6 + 56c4R2 + 70c2R4 + 35R6)+ 7(16c6 + 40c4R2 + 30c2R4 + 5R6)x′+ 14c(24c4 + 40c2R2 + 15R4)x′2 + 70(8c4 + 8c2R2 + R4)x′3212C.1. Spheroidal multipoles+ 280c(2c2 + R2)x′4 + 56(6c2 + R2)x′5 + 112cx′6 + 16x′7]+1R72[c(16c6 + 56c4R2 + 70c2R4 + 35R6)− 7(16c6 + 40c4R2 + 30c2R4 + 5R6)x′+ 14c(24c4 + 40c2R2 + 15R4)x′2 − 70(8c4 + 8c2R2 + R4)x′3+ 280c(2c2 + R2)x′4 − 56(6c2 + R2)x′5 + 112cx′6 − 16x′7]],I19 =135R8[1R71[5R8 +(16c7 + 56c5R2 + 70c3R4 + 35cR6)x′+ 7(16c6 + 40c4R2 + 30c2R4 + 5R6)x′2+ 14(24c5 + 40c3R2 + 15cR4)x′3 + 70(8c4 + 8c2R2 + R4)x′4+ 280c(2c2 + R2)x′5 + 56(6c2 + R2)x′6 + 112cx′7 + 16x′8]+1R72[ − 5R8 + (16c7 + 56c5R2 + 70c3R4 + 35cR6) x′− 7(16c6 + 40c4R2 + 30c2R4 + 5R6)x′2+ 14(24c5 + 40c3R2 + 15cR4)x′3 − 70(8c4 + 8c2R2 + R4)x′4+ 280c(2c2 + R2)x′5 − 56(6c2 + R2)x′6 + 112cx′7 − 16x′8]],(C.9)where on the surface of the particle we haveR1 =√(x′ + c)2 + R2,R2 =√(x′ − c)2 + R2,R =√(1 − e2) (a2 − x′2). (C.10)The integrals also satisfy the relationshipInm = x′In−1m +(n − 1)In−2m−2m − 2 −cn−1((−1)nR2−m1 + R2−m2)m − 2 . (C.11)Other integrals Jnm, and Knm can be calculated easily from equations (C.5) and (C.6).213C.2. A prolate spheroid in Stokes flowC.2 A prolate spheroid in Stokes flowFollowing Einarsson et al. [75] we use the following ansatz for the disturbance flowfield due to a prolate spheroid in a linear shear flowu′i = QRi jkϵ jkl[− {AR pl pm + BR(δlm − pl pm)}Ω′m +CRϵlmn pmE∞no po]+(QSi jk + αQQi jk)[(AS nAjklm + BS nBjklm +CS nCjklm)E∞lm+CR(ϵ jlm pk pm + ϵklm p j pm)Ω′l], (C.12)where AR, BR,CR, AS , BS ,CS , and α are seven unknown scalar coefficients andare calculated by enforcing the no-slip boundary condition on the surface of thespheroid (see Einarsson et al. [75] for further details). AlsonAjklm =(p j pk −13δ jk)(pl pm − 13δlm),nBjklm =p j pmδkl + pk pmδ jl + p j plδkm + pk plδ jm − 4p j pk pl pm,nCjklm = − δ jkδlm + δ jlδkm + δklδ jm + pl pmδ jk + p j pkδlm − p j pmδkl− pk pmδ jl − p j plδkm − pk plδ jm + p j pk pl pm. (C.13)The constants for a prolate spheroid areα =1 − e28e2,AR =e2 − 12L (e2 − 1) + 4e ,BR =e2 − 22L (e2 + 1) − 4e ,CR = − e22L (e2 + 1) − 4e ,AS = − e22(L (e2 − 3) + 6e) ,BS =e2(L(e2 − 1)− 4e3 + 2e)4(−3L (e2 − 1) + 4e3 − 6e) (L (e2 + 1) − 2e) ,CS =e2 − e43L (e2 − 1)2 + 2e (5e2 − 3) ,L = log(−e + 1e − 1). (C.14)214C.3. Shear-thinning correctionC.2.1 Jeffery OrbitsThe torque on a spheroid can be calculated by linearly superposing the contribu-tions from all the contained rotlets, ui = QRi jkϵ jklBl, asL0 = −16π∫ c−c(c2 − ξ2)dξB = −64πc33B, (C.15)where from (C.12) the rotlet strength isB = −{ARpp + BR(I − pp)} ·Ω′0 +CRp × (E∞ · p). (C.16)Ultimately, because the torque on the body is zero then the rotlet strength mustbe zero and hence B = 0. The Jeffery orbit solution immediately follows as thisrequires thatΩ0 = Ω∞ + Λp × E∞ · p,where Λ = CR/BR.The disturbance flow field may now be written more simply asu0i = A∞i j x j +(QSi jk + αQQi jk)D jklmE∞lm, (C.17)whereD jklm = AS nAjklm + BS nBjklm +CS nCjklm +CRΛ(ϵ jqt pk pt + ϵkqt p j pt)ϵpql pp pm.(C.18)From this solution of the Newtonian disturbance flow, the strain-rate tensor maybebe calculatedγ˙0 = M : E∞, (C.19)whereMimsl = 2δilδms +(QS Ti jkm + αQQTi jkm)D jkls, (C.20)and we’ve defined QS Ti jkm = QSi jk,m + QSm jk,i, and QQTi jkm = QQi jk,m + QQm jk,i.C.3 Shear-thinning correctionIn order to calculate the correction in the orientational dynamics of the prolatespheroid due to shear thinning given by (7.16),Ω1 =12(1 − β)(1 − n)Rˆ−1LΩ ·∫V|γ˙0|2γ˙0 : EˆΩ dV,we need both the strain-rate tensor of the Newtonian solution,γ˙0, given by (C.19)and the operators Rˆ−1LΩ and EˆΩ from the rigid-body motion problem that we nowshow.215C.3. Shear-thinning correctionRigid-body motionThe flow field due to a prolate spheroid rotating with Ωˆ in a quiescent flow, ob-tained from (C.12), isuˆi =[− ϵ jklQRi jk{AR pl ps + BR(δls − pl ps)}+(QSi jk + αQQi jk)CR(ϵ jsm pk pm + ϵksm p j pm)]Ωˆs. (C.21)The strain-rate tensor is thenˆ˙γ = 2EˆΩ · Ωˆ, (C.22)whereEˆΩims =12[− ϵ jklQRTi jkm{AR pl ps + BR(δls − pl ps)}+(QS Ti jkm + αQQTi jkm)CR(ϵ jsm pk pm + ϵksm p j pm)], (C.23)and QRTi jkm = QRi jk,m + QRm jk,i.The torque exerted on the particle can be found by integrating the rotlet densitysuch thatLˆ = 16π∫ c−cdξ(c2 − ξ2)(ARpp + BR(I − pp))· Ωˆ,=64πc33(ARpp + BR(I − pp))· Ωˆ. (C.24)Using the definition of the resistance, Lˆ = −RˆLΩ · Ωˆ, we obtainRˆ−1LΩ = −364πc3[1ARpp +1BR(I − pp)]. (C.25)216
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Elfring, Gwynn
Hydrodynamic interactions : microswimmers near boundaries
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