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Elastohydrodynamic interactions at small scales Nasouri, Babak 2018

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Elastohydrodynamic interactions at small scalesbyBabak NasouriB.Sc., Sharif University of Technology, 2012M.Sc., The University of Texas at Austin, 2014a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mechanical Engineering)The University of British Columbia(Vancouver)August 2018© Babak Nasouri, 2018The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:Elastohydrodynamic interactions at small scalessubmitted by Babak Nasouri in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Mechanical EngineeringExamining Committee:Gwynn J. Elfring, Mechanical EngineeringSupervisorGeorge M. Homsy, MathematicsSupervisory Committee MemberBrian Wetton, MathematicsUniversity ExaminerBoris Stoeber, Mechanical EngineeringUniversity ExaminerAdditional Supervisory Committee Members:Neil J. Balmforth, MathematicsSupervisory Committee MemberA. Srikantha Phani, Mechanical EngineeringSupervisory Committee MemberiiAbstractIn this dissertation, the effects of elasticity on hydrodynamic interactions at smallscales are investigated.In the microscale realm of microorganisms, inertia is irrelevant and viscousdissipation dominates the fluid motion and particles within it. As a result ofthis inertialess environment, microorganisms use non-reciprocal body distortionsto facilitate locomotion and exhibit nontrivial behaviors in interacting with theirsurroundings; behaviors that have been shown to be intimately correlated to theelasticity of the cell body, or its small appendages called flagella (or cilia). Motivatedby experimental observations, the effects of elasticity on hydrodynamic interactionsofmotile cells are investigated, using theoretical approaches. First, tomodel the flowfield induced by microswimmers, a framework is given to account for the effects ofthe higher-order force moments. Specifically, the contribution of the second-orderforce moments of the flow field is evaluated, and explicit formulas are reportedfor the stresslet dipole, rotlet dipole, and potential dipole for an arbitrarily shapedactive particle. For an elastic swimmer near a boundary, it is shown that the rotletdipole bends the swimmer and results in qualitatively different swimming behaviorsin comparison to the case of a rigid swimmer. Furthermore, it is demonstrated thatelasticity can be exploited to evade the kinematic reversibility of the field equationsin Stokes flow. A model elastic swimmer is proposed that despite the reversibleactuation, can propel forward due to its nonreciprocal body deformations. The effectof elasticity in the formation of metachronal waves in ciliated microorganisms suchas Paramecium and Volvox is also studied. Using a minimal model, it is shownthat elastohydrodynamic interactions of cilia attached to a curved body lead tosynchronization, with zero phase difference, thereby preventing the formation ofiiiwave-like behaviors unless an asymmetry is introduced to the system. Finally, thedynamics of capillary rise between two porous and elastic sheets are investigated.The liquid, as it rises, diffuses through the sheets and changes their properties.The significant drop in sheet bending rigidity due to wetting, causes the system tocoalesce faster, compared to the case of impermeable sheets, and also remarkablyreduces the absorbance capacity.ivLay SummaryAt microscopic scales, locomotion in fluid requires swimming techniques that areutterly different from the ones we use in our macroscopic world. Microorganismssuch as spermatozoa and bacteria, use elasticity to creep around, evade a predator,and to interact with their environment in fluids. Using mathematical models, weexamine the effects of body elasticity on the behaviors of microorganisms and showthat, indeed, elasticity can play a key role in motion of a cell in fluids. We furtherhighlight the importance of flexibility in interactions in fluids and study the liquidrise between two porous, flexible paper sheets to explain the liquid absorptivity ofmulti-ply papers.vPrefaceThe contents of this dissertation are the results of the research of the author, BabakNasouri, under the supervision of Professor Gwynn J. Elfring. The followingarticles has been published and/or are in progress:• B. Nasouri, G. J. Elfring, ‘Higher-order force moments of active particles’,Physical Review Fluids, 3 044101 (2018).The author of this dissertation was the principal contributor. G. J. Elfringsupervised the research and was involved in the concept formation, analysisand editing of the paper.• B. Nasouri, A. Khot, G. J. Elfring, ‘Elastic two-sphere swimmer in Stokesflow’, Physical Review Fluids, 2 043101 (2017).The author of this dissertation was the principal contributor. A. Khot con-tributed in the concept formation. G. J. Elfring supervised the research andwas involved in the concept formation, analysis and editing of the paper.• B. Nasouri, G. J. Elfring, ‘Hydrodynamic interactions of cilia on a sphericalbody’, Physical Review E, 93 033111 (2016).The author of this dissertation was the principal contributor. G. J. Elfringsupervised the research and was involved in the concept formation, analysisand editing of the paper.• B. Nasouri, B. Thorne, G. J. Elfring, ‘Dynamics of poroelastocapillary rise’,submitted.The author of this dissertation was the principal contributor. B. Thorneperformed the experiments. G. J. Elfring supervised the research and wasviinvolved in the concept formation, analysis and editing of the paper.• B. Nasouri, S. E. Spagnolie, G. J. Elfring, ‘Elastic active particles near awall’, in progress.The author of this dissertation was the principal contributor. S.E. Spagnolieand G. J. Elfring supervised the research and was involved in the conceptformation, analysis and editing of the paper.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiEpigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Lorentz reciprocal theorem . . . . . . . . . . . . . . . . . . . . . 52.2 Boundary integral equation . . . . . . . . . . . . . . . . . . . . . 73 Higher-order force moments of active particles . . . . . . . . . . . . 93.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Multipole expansion . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Evaluating the force moments of an active particle . . . . . . . . . 17viii3.3.1 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.2 Generalized squirmer . . . . . . . . . . . . . . . . . . . . 243.3.3 Axisymmetric slender rod . . . . . . . . . . . . . . . . . 273.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Elastic swimmer near a wall . . . . . . . . . . . . . . . . . . . . . . 304.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . 314.3 Toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Elastic two-sphere swimmer in Stokes flow . . . . . . . . . . . . . . 415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Translation of an elastic sphere . . . . . . . . . . . . . . . . . . . 435.2.1 Asymptotic analysis . . . . . . . . . . . . . . . . . . . . 465.3 Two-sphere swimmer . . . . . . . . . . . . . . . . . . . . . . . . 525.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 Hydrodynamic interactions of cilia on a spherical body . . . . . . . 566.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2 Motion of a single cilium . . . . . . . . . . . . . . . . . . . . . . 586.3 Interactions of two cilia . . . . . . . . . . . . . . . . . . . . . . . 606.4 Interactions of chain of cilia . . . . . . . . . . . . . . . . . . . . 656.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 Dynamics of poroelastocapillary rise . . . . . . . . . . . . . . . . . . 727.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 747.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . 777.4 Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . 807.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 817.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 88ixBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A Auxiliary flow field and stress field calculations of a spherical activeparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108xList of FiguresFigure 3.1 Schematic representation of an active particle of arbitrary shape.A point on the particle surface, ∂B, is denoted by y and x0 isa convenient reference point in the body. The instantaneousvelocity of a point on ∂B is given by rigid-body translation U,rigid-body rotationΩ× r and surface slip velocity us. . . . . . 12Figure 3.2 Flow fields induced by non-zero force moments of an axisym-metric squirmer of radius 1, using expressions given in (3.73)to (3.76): (a) Flow field due to a stresslet, for which we setB02 = 1 and other coefficients to zero. (b) Flow field due to astresslet dipole with B03 = 1. (c) Flow field induced by a po-tential dipole with B01 = 1. (d) Flow field due to a rotlet dipolewith C02 = 1. In (d), the color density indicates the magnitudeof the velocity where positive (negative) values indicate flowinto the plane (out of the plane) [145]. . . . . . . . . . . . . . 26Figure 4.1 The effect of velocity moments of the background flow on amodel swimmer. U∞ is the zeroth-order velocity moment, E∞andΩ∞ are the first-ordermoments of the background flow, andΓ∞, Λ∞, and e∞ together represent the second-order velocitymoments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34xiFigure 4.2 (a) Schematic of the considered swimmer. Spheres A and Brepresent the tail and the head, respectively. Sphere A is atdistance h from the wall, L characterizes the swimmer lengthscale and θ is the swimming angle. (b) Bending mechanism ofthe swimmer by the rotlet dipole. The swimmer bends from itscenter with angle δ which is positive when the swimmer bendstoward the wall and negative when it bends away. . . . . . . . 35Figure 4.3 Variation of bending angle δ with respect to the ratio of radii λ,using the expression given in (4.8) with f = 0.5, h(t = 0) = 10and k = 10−4. . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 4.4 Evolution of h and θ (inset) for rigid (a) pushers and (b) pullersusing expressions (4.10) and (4.11) evaluated at h(t = 0) = 10,λ = 0.1, f = 0.5 and k→∞. For each case, the initial swim-ming angles are θc±where θc = 0.0017β is the initial criticalangle and  = 0.002. The dotted line in the insets indicate thecritical angle and τ = 16h3(t = 0)/(3f ) is the characteristictimescale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 4.5 The time-evolving behavior of h and θ (inset) for elastic pusherswimmers. Plots are the numerical evaluation of Eqs. (4.10)and (4.11) with θ0 = −pi/8, k = 10−4, f = 0.5 and λ = 0.2,0.4and 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 4.6 The evolution of h for a puller for (a) λ = 0.5 (b) λ = 0.2 and(c) λ = 0.1 with three initial orientations of θ0 = 0.02,0.03and 0.05. For all cases k = 10−4 and f = 0.5. . . . . . . . . . 39Figure 5.1 Deformed shape of the translating elastic sphere when f = 1and  = 0.45. . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 5.2 One cycle of the two-step motion of the swimmer. Step (I):The rod shortens its length. Step (II): Spheres move away fromone another until they reach the initial distance. The steps ingrey color demonstrate the swimmer while it proceeds to thenext step and sphere B is deformed. . . . . . . . . . . . . . . 53xiiFigure 6.1 A schematic of the motion of a model cilium near a sphericalbody. The circular trajectory has a radius of Rˆ and its centeris at distance hˆ from the boundary. The cilium moves withvelocity Uˆ through a fluid with velocity uˆ∞. In this study, φindicate the instantaneous phase of the cilium and the vectorseφ and eR show the tangential and radial directions of themotion, respectively. . . . . . . . . . . . . . . . . . . . . . . 59Figure 6.2 A system of two cilia around a spherical body of radius Aˆ. Inthis figure, dˆ12 is the distance and θ12 is the angle betweencenter of the trajectories. . . . . . . . . . . . . . . . . . . . . 60Figure 6.3 Geometric terms (a) Θ and (b) Φ as functions of the anglebetween cilia. . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 6.4 The effect of the background flow field on the motion of eachcilia. The two cilia are orbiting clockwise, cilium (2) is ahead,thus its induced flow field pulls cilium (1) to a smaller radius oftrajectory which increases the instantaneous velocity of cilium(1). On the other hand, the velocity of cilium (2) decreasesas the flow field of cilium (1) pushes cilium (2) to a largertrajectory. In this figure arrows show the flow field induced byeach cilium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 6.5 The value of the coefficient α, which dictates the stability offixed points of a ciliary chain, is shown as a function of thegiven number of cilia N . . . . . . . . . . . . . . . . . . . . . 67Figure 6.6 Synchronization of a chain of (a) 10 and (b) 15 identical ciliadistributed uniformly around a spherical body, with the randominitial phases. Each line indicates the evolution of the phasedifference for each cilium i compared to cilium (1), ∆1i = φ1−φi , over the time T = t/(κ/ρ). These plots are the numericalevaluation of Eq. (6.18) at the characteristic values of ρ =3.6× 10−6, κ = 100 and ¯ˆω = 20pi rad.s−1 [27, 143]. . . . . . 68xiiiFigure 6.7 Phase differences of nearby cilia in a chain of 10 and 15 ciliaaround a spherical body when the intrinsic angular velocity ofcilium (1) is higher compared to the other cilia by ∆ω = 10−6for both cases. . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 7.1 The schematic of the system: (a) Twoporoelastic sheets clampedat the upper end are immersed into a liquid bath from the lowerend. (b) Three scenarios of the equilibrium. In regime I, sheetsonly slightly bend. In regime II, sheets lower end are in contactand in regime III, sheets coalesce over a finite length. . . . . . 75Figure 7.2 The time evolution of rescaled meniscus, z∗m =√Br /(2Ed)zm:(a) Permeable sheets with B = 2, Ed = 10 and Br = 0.1. (b)Two impermeable sheets with Ed = 10 and Ed = 100 and per-meable sheets with Ed = 10 (or Ew = 100). In (b), for all casesB = 3 and solid black line refers to the classical capillary rise[197]. Circles on each line indicate the time in which sheetsreach regime II (h(t, z = 0) = 0). . . . . . . . . . . . . . . . . 81Figure 7.3 The liquid absorption ratio, R, of permeable sheets comparedto impermeable ones of the same Ed for B = 1 and B = 10.Dashed lines separate the different combinations of equilibriumregimes for permeable and impermeable cases and symbolsdenote the regimes (e.g., II/I indicates that permeable sheetsreach regime II while impermeable sheets are at regime I). . . 82Figure 7.4 Time to reach the 99% of the equilibrium height versus B2Ew.Symbols represent the numerical results for the equilibriumtime scale of impermeable and permeable sheets. The dashedline indicates teq = 1. . . . . . . . . . . . . . . . . . . . . . . 84xivFigure 7.5 (a) Regime map for the equilibrium sate of permeable sheets ina logarithmic (B,Ed) space. Each shade refers to its specifiedregime and is obtained using the numerical scheme discussedin Section 7.4. Symbols are the equilibrium states observedexperimentally: Triangles (N) denote regime I, circles (•) in-dicate regime II and diamonds () refer to regime III. (b) Theexperimental apparatus. Sheets here are in regime III. . . . . 85xvAcknowledgmentsFirst and foremost, I’d like to express my sincere gratitude and appreciation to myadvisor Prof. Gwynn J. Elfring. Gwynn! Thanks for giving me the opportunityof being your PhD student. Thanks for your exceptional patience towards mypainfully-naive questions. Thanks for showing me the steps to become a scientist.And as importantly, thanks for being a knowledgeable soccer fan with whom I canrigorously discuss every game.I’d like to thank the members of my advisory committee: Prof. George (Bud)M. Homsy, Prof. Neil J. Balmforth, and Prof. A. Srikantha Phani. Bud! Itwas my true honor to have you as my committee member. Your deep (and in myopinion unbounded) knowledge of fluid dynamics inspired me dearly. Neil! Thisdissertation is heavily based on asymptotic analysis, and I was quite privileged tolearn the basics from you, and will be always thankful for that.I’d like to thank all my labmates in Complex Fluids Lab at UBC for making mygrad life more enjoy full and fun. I specially want to thank them for listening to(and probably surviving) my (seemingly) odd theories about almost everything.This journey could have not been possible without the unconditional supportand love of my parents, my brother and my (always) little sister. I am thankful toall of them and forever grateful for having them in my life.Last but certainly not least, I’d like to thank my new family for supporting meevery step of the way.Babak NasouriVancouver, CanadaAugust 2018xviDedicationTo my Yasaman.xviiEpigraphThere was a time that the pieces fit, but I watched them fall awayMildewed and smoldering, strangled by our covetingI’ve done the math enough to know the dangers of our second guessingDoomed to crumble unless we grow, and strengthen our communicationSchism, TOOLxviiiChapter 1IntroductionIn this dissertation, we intend to investigate the interactions between hydrodynamicforces and elasticity, and the phenomena their coupling creates. As the title ofthe dissertation suggests, we focus on such interactions at very small scales tocomprehend a realm that our intuition may not be accustomed to. A realm whereininertia is irrelevant and viscous dissipation governs the motion of the fluid.But why do elastohydrodynamic interactions matter at small scales? It is need-less to say that microorganisms, which are the overwhelming majority inhabitantsof our world, have a significant impact on our lives. Many of these micron-sizedcells are elastic themselves, and/or possess whip-like extensions, called flagella(or cilia), that are flexible thereby making these interactions ubiquitous in nature[65, 66, 153]. For instance, a spermatozoa uses its flagellum to swim toward theovum and bacteria such as E. coli buckle their flagella to change their swimmingdirection [177]. Another important example is the coordinated beating pattern ofcilia, often labeled as metachronal waves [78]. By forming these waves, cilia filterthe air flow channels in the human lung from the harmful inhaled material [171],play a crucial role in breaking the left-right symmetry in human embryonic devel-opment [87], and also help microorganisms such as Paramecium to evade predatorrotifers [109]. The contribution of elasticity to all of these phenomena, further high-lights the importance of understanding elastohydrodynamic interactions of motilecells and their surrounding environment. Indeed, we demonstrate that elasticity canqualitatively alter the behavior of cells: It can be exploited to generate propulsion, it1can lead to synchrony of cilia on a curved ciliate body, and also it is able to prevententrapment of swimmers by nearby obstacles.Furthermore, elasticity may also be coupled with capillary forces and exhibitastounding physical phenomena such as origami [158] and substrate wrinkling [90].In fact, this coupling, known as elastocapillarity, occurs quite frequently in our day-to-day life. When wiping a coffee spill, the liquid diffuses between the plies ofthe paper napkin, makes them softer and also drives them toward a coalescence.We show that such a coalescence is a direct consequence of the paper porosity andelasticity, and it can affect the absorbance capacity of the paper significantly.1.1 OutlineThis dissertation explores the elastohydrodynamic interactions in two different classof problems. In Chapters 3 to 6 the effects of elasticity onmicroorganisms is studied,while in Chapter 7 the dynamical behavior of elastic sheets due to capillary rise isinvestigated.We begin with a brief introduction to Stokes flow in Chapter 2. In this shortchapter, we re-derive two fundamental equations related to Stokes flow, namely thereciprocal theorem and the boundary integral equation, which are frequently usedthroughout this dissertation.In Chapter 3, we delve deeper into Stokes flow and use the boundary integralequation to express the disturbance flow field induced by motion of active particles.In particular, we evaluate the contribution of the second-order force moments to theflow field and, by the reciprocal theorem, present explicit formulas for the stressletdipole, rotlet dipole, and potential dipole for an arbitrarily shaped active particle.As examples of this method, we derive modified Faxén laws for active sphericalparticles and resolve higher-order moments for active rod-like particles.We characterize the effect of elasticity in interaction of a swimmer with a nearbywall in Chapter 4. We show that the elastic bending of the swimmer due to thepresence of the wall contributes significantly to the swimming direction and canlead to attraction or repulsion, depending on the propulsion mechanism.In Chapter 5, we inquire about the effect of elasticity on swimming in Stokesflow. We first look into pure translation of an weakly-elastic spherical particle and2show that its shape deformation is not front-back symmetric. We then propose aswimmer that can exploit this asymmetry to propel itself forward.We then study the effects of elasticity on coordinated beating pattern of cilia onthe surface of microorganisms in Chapter 6. Using a minimal model, we show thattwo cilia attached to a curved body synchronize purely through elastohydrodynamicinteractions. We also show that for a chain of cilia, the natural periodicity in thegeometry of the ciliate body prevents formation of any wave-like behavior unlessan asymmetry is introduced to the system.As the last step, we explore the coupling of elastic forces with capillary forcesin Chapter 7. We study the dynamical behavior of two elastic and porous sheetsimmersed into a liquid bath. Accounting for the change of sheets stiffness dueto wetting, we discuss the time-evolving behavior of the meniscus and the sheets’deflections as the system reaches the equilibrium.Finally in Chapter 8, we finish the dissertation with concluding remarks.3Chapter 2Stokes FlowThis dissertation is focused on motions in viscous fluids at very small scales.In problems discussed in Chapters 3 to 6, and also the capillary-induced motionstudied in Chapter 7, the scale of the problem is so small that the effect of inertia isnegligible and the field equations are governed by the Stokes equations∇ · σ = µ∇2u−∇p = 0, (2.1)∇ ·u = 0, (2.2)where σ is the stress field, u is the flow field, p is the pressure field, and µ is thedynamic viscosity. Given that these field equations are linear, it is insightful todetermine the Green’s function; the system response to a point force. We thenceprescribe a vector point force F applied on the fluid at x0 as,µ∇2u−∇p+Fδ (x− x0) = 0. (2.3)The induced pressure field, flow field, and stress field are respectivelyp =18piF ·P (x− x0) , (2.4)u =18piµF · J (x− x0) , (2.5)σ =18piF ·K (x− x0) , (2.6)4whereJ(x) =I|x| +xx|x|3 , (2.7)is the Green’s function of the Stokes equations, often referred to as the Oseen tensor,andP(x) =2x|x|3 , (2.8)K(x) =−6xxx|x|5 , (2.9)are its associated pressure and stress tensor, respectively.In the following, we re-derive two fundamental equations related to Stokes flowthat are frequently used throughout this dissertation. We begin with the reciprocaltheorem and then use that to derive the boundary integral equation. The followingderivations can be simply found in the classical references (see for instance [98] or[154]), but here are reproduced to set the foundation for the next chapters in thedissertation.2.1 Lorentz reciprocal theoremLet us now consider two flows characterized by{µ,p,u,σ}and{µ˜, p˜, u˜, σ˜}. We canwrite∇ · (u˜ · σ ) = ∇u˜ : σ + u˜ · (∇ · σ ) ,= ∇u˜ : σ , (2.10)where we have used ∇ ·σ = 0. The constitutive equation for a viscous fluid dictatesσ = −pI+ 2µE, (2.11)5where E = 12(∇u+∇uT)is the rate-of-strain tensor. By substituting σ from (2.11)in (2.10), we arrive at∇ · (u˜ · σ ) = −p∇ · u˜+ 2µ∇u˜ : E. (2.12)From continuity ∇ · u˜ = 0. Now, by exploiting the symmetry of E and definingE˜ = 12(∇u˜+∇u˜T), one can show ∇u˜ : E = 12 E˜ : E, which leads to∇ · (u˜ · σ ) = µE˜ : E. (2.13)Repeating the same procedure, we find∇ · (u · σ˜ ) = µ˜E : E˜. (2.14)But, due to their symmetry E : E˜ = E˜ : E, so from (2.13) and (2.14) we can concludeµ˜∇ · (u˜ · σ ) = µ∇ · (u · σ˜ ) . (2.15)It is useful to integrate Eq. (2.15) over a volume of fluid in domain B asµ˜∫B∇ · (u˜ · σ )dV = µ∫B∇ · (u · σ˜ )dV . (2.16)Provided that flow fields and stress fields are not singular within the considereddomain, we may use the divergence theorem to convert the volume integrals (B) tosurface integrals (∂B) asµ˜∫∂Bn · σ · u˜dS = µ∫∂Bn · σ˜ ·udS, (2.17)where n is the surface normal of boundary ∂B. Expressions given in Eqs. (2.15)to (2.17) are different forms of the reciprocal theorem and can be used to simplifythe calculations in Stokes flow, as we illustrate in Chapter 3.62.2 Boundary integral equationWe may now use the reciprocal theorem to derive the boundary integral equation.Let us again take{µ,p,u,σ}and{µ˜, p˜, u˜, σ˜}as two flows in the Stokes regime. Thistime, we pick one of them as the flow field due to a point force of arbitrary (andconstant) strength F˜ located at x0. Thus, we haveu˜ (x) =18piµ˜F˜ · J (x− x0) , (2.18)σ˜ (x) =18piF˜ ·K (x− x0) . (2.19)We note that J and K are singular at x = x0. Thus, the reciprocal theorem givenin Eq. (2.17) is no longer valid over the full domain B. To resolve this singularity,we define a new spherical domain B of vanishingly small radius  enclosing thesingular point x0 at its center. Now, the reciprocal theorem for the regular domainB −B can be written asµ˜∫∂B−∂Bn(x) · σ (x) · u˜(x)dS(x) = µ∫∂B−∂Bn(x) · σ˜ (x) ·u(x)dS(x), (2.20)and decomposed to− µ˜∫∂Bn(x) · σ (x) · u˜(x)dS(x) +µ∫∂Bn(x) · σ˜ (x) ·u(x)dS(x)= µ˜∫∂Bn(x) · σ (x) · u˜(x)dS(x)−µ∫∂Bn(x) · σ˜ (x) ·u(x)dS. (2.21)Now by using Eqs. (2.18) and (2.19) and discarding the arbitrarily chosen F˜ fromthe both sides, we find−∫∂Bn(x) · σ (x) · J (x− x0)dS +µ∫∂Bn(x)u(x) : K (x− x0)dS=∫∂Bn(x) · σ (x) · J (x− x0)dS −µ∫∂Bn(x)u(x) : K (x− x0)dS. (2.22)7As  → 0, σ (x) and u(x) on the left-hand side of Eq. (2.22) tend to σ (x0) andu(x0), respectively. Over ∂B, we can write x− x0 = n andJ (x− x0) = I +nn, (2.23)K (x− x0) = −6nnn2 . (2.24)The left-hand side of Eq. (2.22) is then reduced to−∫∂Bn(x) · σ (x) · J (x− x0)dS +µ∫∂Bn(x)u(x) : K (x− x0)dS= −σ (x0) :∫∂BnI+nnndS − 6µu(x0) ·∫∂Bnn2dS,= −8piµu(x0). (2.25)Substituting Eq. (2.25) to (2.22), we finally arrive at the boundary integral equationasu(x0) = − 18piµ∫∂Bn(x) · σ (x) · J (x0 − x)dS − 18pi∫∂Bn(x)u(x) : K (x0 − x)dS.(2.26)Equation (2.26) provides the flow field at an arbitrary point x0, using Oseen tensorJ and its associated stress tensor K over the boundary ∂B. In the next chapter, weshow how one can use the boundary integral equation to express the flow field ofactive particles in viscous fluids.8Chapter 3Higher-order force moments ofactive particles 13.1 IntroductionSelf-propulsion is ubiquitous in nature. Be it at the macroscopic scale of flyingbirds or the microscopic scale of swimming bacteria, the motion of active matterresults from converting internal or ambient energy into mechanical work withoutany other external input [162]. At sufficiently small scales in viscous fluids, inertiais irrelevant and viscous dissipation dominates the motion of the fluid and activeparticles within it [84]. In the absence of inertia, ‘reciprocal’ body distortionsare ineffective as a propulsion mechanism, and so active particles must propelthemselves by other means in this realm [157]. There exist several techniques toachieve net locomotion in the low-Reynolds-number regime [106, 109, 138]. Forinstance, microorganisms such as Paramecium and Volvox use small appendagescalled cilia to facilitate motion [124]. Cilia generate thrust through a coordinatedpattern of beating, which may arise from hydrodynamic [22, 136, 141] or basal[100, 159, 196] interactions. Propulsion can also be achieved synthetically bychemically-active particles with asymmetric non-uniform surface properties [2, 72,194]. In both of these examples, the effect of surface activity is confined to a narrow1A version of this chapter has been published [137].9region surrounding the particle and hence may be modeled using ‘apparent’ slipvelocities on the surface. This way, one can explicitly find the propulsion speedand thereby the disturbance flow field, in terms of prescribed (or measured) slipvelocities [54].For the inertialess motion of sufficiently small particles in viscous fluids, theflow field is often approximated by far-field singularity solutions of the Stokesequations. To leading order, the flow field decays linearly by distance (∼ 1/ |x|)and, at this level of approximation, the particle is replaced by a point force (i.e.,zeroth-order force moment) that leads to flow [98]. The next-order correction tothe flow field, which decays quadratically (∼ 1/ |x|2), can be expressed using aforce-dipole (i.e., first-order force moment), which is decomposed into a torque(the antisymmetric part) and a stresslet (the symmetric part) [10]. In the absence ofan external force, the over-damped motion of the particle has no net hydrodynamicforce or torque and so the stresslet governs the leading-order flow field. Theimportance of the stresslet in characterizing the interactions of active particles[13, 76, 108], the rheology [168] and stability [169] of active suspensions, and thecollective locomotion of bacteria [40] is well documented. However, the stressletterm alone fails to explain behaviors such as the ‘dancing’ of two Volvox colonieswhen they are in proximity of one another [41], or the vortices induced due to themotion of C. reinhardtii [42, 75]. An emerging picture is that modeling the motionusing only terms up to the stresslet may limit understanding of how active particlesinteract with their environment and motivates investigation of higher-order forcemoments. In a recent study, Ghose and Adhikari [61] showed that the swirlingmotion of an active spherical particle only appears in the flow field decaying as∼ 1/ |x|3 and ∼ 1/ |x|4 and derived expressions for higher-order force moments ofa sphere. In this chapter, we generalize their results by investigating the effectsof higher-order force moments on an arbitrarily-shaped active particle, therebyextending recent general results for the stresslet term by Lauga and Michelin [108].Using the boundary integral equations, we express the flow field around an activeparticle through a multipole expansion up to the contribution of the second-orderforce moments. We then provide explicit formulas for these force moments byexploiting the reciprocal theorem using a framework developed in [50].The reciprocal theorem for low-Reynolds-number hydrodynamics has long been10an avenue to simplify calculations in Stokes flow [84, 86, 111, 161]. Its applicationhas ranged from the inertialess jet propulsion [178], to boundary-driven channelflow [131], toMarangonimotion of a droplet coveredwith bulk-insoluble surfactantsin a Poiseuille flow [147]. In particular, Stone and Samuel [184] showed that thekinematics of an active particle can be determined explicitly, using the flow fieldinduced by the rigid-body motion of a passive particle of the same instantaneousshape. Subsequently, the reciprocal theorem has been widely used to determinethe kinematics of active particles both in Newtonian [49, 109] and non-Newtonianfluids [35, 36, 105, 107, 146]. This approach was recently extended to determinethe stresslet of active particles [108]. More recently, a general framework has beendeveloped for finding the force moments (of any order) of an active particle in aNewtonian (or non-Newtonian) fluid [50]. Following that approach, in this chapter,we provide formulas for calculating the force moments up to the second order, forany arbitrarily-shaped active particle.The chapter is organized as follows. In Section 3.2, we employ the boundaryintegral equation to describe the disturbance flow field caused by an active particle.Using an asymptotic expansion of the far-field flow, we show how the forcemomentscontribute to the disturbance flow field. We then in Section 3.3, use the reciprocaltheorem to find general expressions for these force moments and, as examples,evaluate them explicitly for a spherical active particle, a generalized squirmer andan active slender rod.3.2 Multipole expansionWe consider a particle with boundary ∂B in an otherwise unbounded Newtonianfluid of viscosity µ and background flow field u∞, as shown in Fig. 3.1. Usingthe boundary integral equations, the disturbance flow field u′ = u − u∞ can beexpressed as a summation of single-layer and double-layer potentials [98, 154]u′(x) = − 18piµ∫∂Bf′(y) · J(x− y)dS(y)− 18pi∫∂Bu′(y)n(y) : K(x− y)dS(y),(3.1)11U+⌦⇥ r+ usyx0r@BFigure 3.1: Schematic representation of an active particle of arbitrary shape.A point on the particle surface, ∂B, is denoted by y and x0 is a convenientreference point in the body. The instantaneous velocity of a point on∂B is given by rigid-body translation U, rigid-body rotation Ω× r andsurface slip velocity us.where f′ = n ·σ ′ is the traction of disturbance stress tensor σ ′, y is the position thatis integrated over the particle surface and n is the surface normal pointing into thefluid. Here J(x) = I|x| +xx|x|3 is the Green’s function of Stokes equations (or the Oseentensor) and K(x) = −6xxx|x|5 is its associated stress tensor. Expanding in y about aconvenient point in the body x0 (for example the center of mass),J(x− y) = J(x− x0)− r · ∇J(x− x0) + 12rr : ∇∇J(x− x0) + · · · , (3.2)K(x− y) = K(x− x0)− r · ∇K(x− x0) + · · · , (3.3)where r = y− x0. Equation (3.1) then takes the formu′(x) =− 18piµ[〈f′〉 · J(x− x0)− 〈f′r〉 : ∇J(x− x0) + 12 〈f′rr〉∇∇J(x− x0) + · · ·]− 18pi[〈u′n〉 : K(x− x0)− 〈u′nr〉∇K(x− x0) + · · · ] . (3.4)For convenience, in this chapter, we denote the surface integral by∫∂B · · ·dS ≡ 〈· · · 〉.We also use to denote a k−fold contraction where k = min{a,b} and a and b are thetensorial orders of the contracted tensors, e.g., [〈u′nr〉 ∇K]i =〈u′jnkrm〉∇mKkji .We define u′(x) =∑i=1u′(i)(x), where u′(i)(x) is the flow field that decays as |x|−i .At leading order, one can recognize the net hydrodynamic force F = 〈f′〉 as the12zeroth-order force moment. The presence of particle at this order is represented bya point force of strength −F and the flow field is simply governed by a Stokeslet,u′(1)(x) = − 18piµF · J(x− x0). (3.5)To find the flow field decaying as |x|−2, it is useful to decompose f′r and u′n,to their symmetric and antisymmetric parts as f′r = f′r+rf′2 +f′r−rf′2 and u′n =u′n+nu′2 +u′n−nu′2 . Noting the symmetry of K(x) and also recalling∇kJij =xiδjk + xjδik − xkδij(xlxl)32− 3xixjxk(xlxl)52, (3.6)we haveu′(2)(x) = − 18piµ〈r× f′〉 ·C(x− x0)− 12〈f′r+ rf′2− 13(f′ · r)I−µ (u′n+nu′)〉: K(x− x0),= − 18piµ[L ·C(x− x0)− 12S : K(x− x0)], (3.7)where L = 〈r× f′〉 is the antisymmetric first-order force moment, i.e. torque andC(x) = I×x|x|3 is the associated rotlet (or couplet) tensor [155]. In using the crossproduct, we follow the convention [r×f′]i = ijkrjf ′k and [I×r]ij = jskδisrk , where is the third-order permutation tensor. The symmetric and deviatoric first-orderforce moment (i.e. f′r) along with the contribution of the double-layer potentiallead to S =〈f′r−2µu′n〉, namely the stresslet [10]. The over-bracket denotes the13fully-symmetric and deviatoric part of a tensor which are defined[ ]ij = (1/2)([ ]ij + [ ]ji)− (1/3)[ ]ssδij ,[ ]ijk = (1/6)([ ]ijk + [ ]ikj + [ ]jik + [ ]jki + [ ]kij + [ ]kji)− (1/15){([ ]ssi + [ ]sis + [ ]iss)δkj +([ ]ssj + [ ]sjs + [ ]jss)δik+ ([ ]ssk + [ ]sks + [ ]kss)δij}, (3.8)for the second and third-order tensors, respectively.To determine the flow field decaying as |x|−3, we decompose the third-ordertensors f′rr and u′nr to their irreducible parts (see [4, 5] for the decompositiontechnique). Now by taking the second gradient of the Oseen tensor∇m∇kJij =δimδjk + δjmδik − δkmδij(xlxl)32+15xixjxkxm(xlxl)72− 3(xmxiδjk + xjxmδik − xkxmδij + xjxkδim + xixkδjm + xixjδkm)(xlxl)52,(3.9)we obtain the next order correction for the flow fieldu′(3)(x) =− 132piµ[〈f′rr−4µu′nr〉∇K(x− x0)]− 124piµ[〈r(r× f′)〉: ∇ [∇ × J(x− x0)]]− 180piµ[〈2|r|2f′ − (r · f′)r+ 3µ[4(u′ ·n)r− (u′ · r)n− (r ·n)u′]〉 · ∇2J(x− x0)]. (3.10)We may then identify stresslet dipole SD =〈f′rr−4µu′nr〉, rotlet dipole CD =〈r(r× f′)〉and potential dipoled =〈2|r|2f′ − (r · f′)r+ 3µ[4(u′ ·n)r− (u′ · r)n− (r ·n)u′]〉 .14Finally, we find the flow field around the particle that decays slower than |x|−4 asu′(x) = − 18piµF · J(x− x0) +L ·C(x− x0)− 12S : K(x− x0) + 14SD ∇K(x− x0)−CD : Υ (x− x0) + 110d · ∇2J(x− x0), (3.11)whereΥijk =iksxsxj + jksxsxi(xlxl)52, (3.12)is the tensor associated with the rotlet dipole.Introducing a more compact notation throughS = [F,L,S,SD,CD,d, · · · ] ,=〈f′〉 ,〈r× f′〉 ,〈f′r−2µu′n〉 ,〈f′rr−4µu′nr〉 ,〈r(r× f′)〉 ,〈2|r|2f′ − (r · f′)r+ 3µ[4(u′ ·n)r− (u′ · r)n− (r ·n)u′]〉 , · · ·, (3.13)andJ =[J,C, (−1/2)K, (1/4)∇K,−Υ , (1/10)∇2J, · · ·], (3.14)we obtainu′(x) = − 18piµS J. (3.15)We note in particular thatS = F′ +µ〈D′〉 , (3.16)represents the strengths of the multipoles of J which contains force moments from15the single-layer integral,F′ =[〈f′〉 ,〈r× f′〉 ,〈rf′〉 ,〈rrf′〉 ,〈r(r× f′)〉 ,〈2|r|2f′ − (r · f′)r〉 , · · ·] , (3.17)and terms due to surface disturbance velocity from the double-layer,D′ =[0,0,−2nu′ ,−4u′nr,0,12(u′ ·n)r− 3(u′ · r)n− 3(r ·n)u′ , · · ·]. (3.18)Finally, we note S = S′ because all terms in S∞ are zero as the boundary integralequation vanishes identically in that case [154].To summarize, the second-order force moment is decomposed into a rotletdipole, a stresslet dipole and a potential dipole, with contributions from the double-layer potentials. To better understand the physical interpretation of these forcemoments, consider an active particle that propels itself forward using flagella (e.g.,E. coli). The force exerted by the flagella is completely balanced by the drag forceon the body and since the distribution of these two forces is separated in position(e.g., tail and head), the induced flow field is captured by a stresslet. However, thelength scales of the cell body and the flagella differ, often by orders of magnitude,and so this asymmetry gives rise to a stresslet dipole. For a case wherein flagellause rotation to generate thrust, the cell body must counter-rotate to maintain thetorque-free motion thereby generating a rotlet dipole. Finally, the finite size of thecell body accounts for the presence of a potential dipole (further discussion may befound in Refs. [175, 179]).We should emphasize that the multipole expansion given in (3.15) is valid foran active (or passive) particle of any arbitrary shape, in a viscous fluid. Giventhat J is generic for any unbounded single particle, determining the flow field isreduced to finding the strengths S. Thus, both traction f′ and disturbance surfacevelocity u′ (x ∈ ∂B) are needed. Although the latter may be explicitly prescribed(e.g., through slip velocity), finding the traction generally requires solving theflow field in full. However, one can avoid such calculations by using the Lorentzreciprocal theorem for the Stokes flow [49, 108, 184] to calculate moments of f′.In the following, we employ the general framework given in [50] to find the forcemoments and hence themultipole strengthsS. Recovering the expressions for force,16torque and stresslet, we report explicit formulas for the stresslet dipole, rotlet dipoleand potential dipole for an arbitrarily-shaped active particle.3.3 Evaluating the force moments of an active particleWe are interested in the motion of an active particle with boundary conditionsu (x ∈ ∂B) = U+Ω× r+us, (3.19)where U and Ω are the rigid-body translation and rotation of the particle while usis a velocity due to surface activity, such as diffusiophoretic slip or a swimminggait [52, 130]. As a dual (or auxiliary) problem, here denoted by a hat, we take thepassive motion of a rigid body of the same instantaneous shape,uˆ (x ∈ ∂B) = Uˆ+ Ωˆ× r. (3.20)The reciprocal theorem indicates that the virtual power of the motion of thesetwo bodies is equal [84], namelyµˆ〈n · σ ′ · uˆ′〉 = µ〈n · σˆ ′ ·u′〉 , (3.21)where uˆ′ = uˆ− uˆ∞ and σˆ ′ = σˆ − σˆ∞ are the disturbance flow and stress field for theauxiliary problem, respectively. As we will show below, by using (operators of)the dual problem, the force moments of the active particle may be obtained withoutresolution of neither the disturbance field u′ nor traction f′.Following Elfring [50], we expand the background flow of the auxiliary problemaround a point in the body x0 asuˆ∞(x ∈ ∂B) = Uˆ∞ (x0) + r · ∇uˆ∞ (x0) + 12rr : ∇∇uˆ∞ (x0) + · · · , (3.22)which by decomposing ∇uˆ∞ and ∇∇uˆ∞ to their irreducible parts, can be rewrittenuˆ∞(x ∈ ∂B) =Uˆ∞ + Ωˆ∞ × r+ r · Eˆ∞+ rr : Γˆ∞+ ( · r)r : Λˆ∞ +(2|r|2I− rr)· eˆ∞ + · · · . (3.23)17Here, Uˆ∞ and Ωˆ∞ indicate the translation and rotation of the background flowof the auxiliary problem at x0. Eˆ∞ = ∇uˆ∞ and Γˆ∞ = (1/2)∇∇uˆ∞ are the fullysymmetric and deviatoric (Eˆ∞ii = 0, Γˆ∞iij = Γˆ∞iji = Γˆ∞jii = 0) second and third-ordertensors, respectively. Λˆ∞ = ∇ (∇ × uˆ∞) is a second-order symmetric tensor andeˆ∞ = (1/10)∇2uˆ∞. Using this expansion and relying on the linearity of the Stokesequations, we can writepˆ′ = µˆPˆ Uˆ′ , (3.24)uˆ′ = Gˆ Uˆ′ , (3.25)σˆ ′ = µˆTˆ Uˆ′ , (3.26)Fˆ′= −Rˆ Uˆ′ , (3.27)wherein the velocity gradientsUˆ′=[Uˆ− Uˆ∞,Ωˆ− Ωˆ∞,−Eˆ∞,−Γˆ∞,−Λˆ∞,−eˆ∞, · · ·], (3.28)are linearly mapped to the disturbance pressure, velocity, and stress fields byPˆ =[PˆU , PˆΩ, PˆE , PˆΓ , PˆΛ, Pˆe, · · ·], (3.29)Gˆ =[GˆU ,GˆΩ,GˆE ,GˆΓ ,GˆΛ,Gˆe, · · ·], (3.30)Tˆ =[TˆU , TˆΩ, TˆE , TˆΓ , TˆΛ, Tˆe, · · ·], (3.31)which are functions of the position in space and the geometry of the particle andeach term maintains the symmetry of the term, against which it operates. Rˆ is thegrand resistance tensor that linearly maps velocity moments to force moments in thedual problem. The symmetry of the stress tensor implies that all components of Tˆare symmetric in their first two (or non-contracted) indices. Under these definitions,the reciprocal theorem can be rewritten as〈(n · σ ′) · Gˆ〉 Uˆ′ = µ〈u′ ·(n · Tˆ)〉 Uˆ′ . (3.32)Importantly, given the boundary condition of the dual problem and the decomposi-tion of the background field, we know the terms in the operator Gˆ(x ∈ ∂B) on the18boundary of the particle, which may be defined asGˆU,ij = δij , (3.33)GˆΩ,ij = ijsrs, (3.34)GˆE,ijk =jkδijrk , (3.35)GˆΓ ,ijkm =jkmδijrkrm, (3.36)GˆΛ,ijk =jkijsrsrk , (3.37)Gˆe,ijkm = 2δijrsrs − rirj , (3.38)...where the over-brackets are identical to (3.8) but only operate over the specifiedindices. In this way, the operator acts precisely to map the traction f′ = n ·σ ′ to theforce momentsF′ =〈f′ · Gˆ〉=[〈f′〉 ,〈r× f′〉 ,〈rf′〉 ,〈f′rr〉 ,〈r(r× f′)〉 ,〈2|r|2f′ − (r · f′)r〉 , · · ·] .(3.39)Now, given that Uˆ′ is arbitrarily chosen, we may discard it from both sides ofEq. (3.32). Applying the boundary conditions on u′, and also expressing the rigidbody translation and rotation asU = [U,Ω,0, · · · ], Eq. (3.32) can be reduced to [50]F′ = −µµˆRˆU+µ〈(us −u∞) ·(n · Tˆ)〉. (3.40)Then by way of (3.16), one obtains the multipole strengths asS = −µµˆRˆU+µ〈(us −u∞) ·(n · Tˆ)+D′〉, (3.41)19where the double-layer potential contribution,D′ =[0,0,−2n (us −u∞),−4(us −u∞)nr,0,12[(us −u∞) ·n]r− 3[(us −u∞) · r]n− 3(r ·n) (us −u∞) , · · ·], (3.42)is simplified as the terms associated with rigid-body motion integrate to zero [154].Equation (3.41) provides the tensorial relationship between the boundary mo-tion and the strength of multipoles for any arbitrarily-shaped active particle inStokes flow. We note that the multipole strengths are split into terms arising fromthe rigid-body motion of the particle,U, and those associated with the (disturbance)surface activity us −u∞. Using this equation, one can derive explicit formulas forS provided Tˆ is known, as we illustrate in the following.We begin with F and L. In self-propulsion, in the absence of any external forceand torque, the net force and torque on the particle are strictly zero. However, tofind the net translational and rotational velocity (which are unknown at this point),we may use the reciprocal theorem for the force and torque. Upon setting F = 0and L = 0 in Eq. (3.41), we may solve for U directly UΩ = µˆµ RˆFU RˆLURˆFΩ RˆLΩ−1 ·  FsLs , (3.43)where RˆFU = −µˆ〈n · TˆU〉, RˆFΩ = −µˆ〈n · TˆΩ〉, RˆLU = −µˆ〈r×(n · TˆU)〉andRˆLΩ = −µˆ〈r×(n · TˆΩ)〉are the components of the grand resistance tensor Rˆ asso-ciated with rigid-body motion. Here Fs = µ〈(us −u∞) ·(n · TˆU)〉is the hydrody-namic force arising solely from the surface activities (often referred to as the thrust)and Ls = µ〈(us −u∞) ·(n · TˆΩ)〉is surface activity driven torque. Equation (3.43)simply illustrates the balance between the force and torque generated by the surfaceactivities and the hydrodynamic drag.We may now determine other components of S by using (3.41) at higher orders.20We obtainS = −µµˆ(RˆSU ·U+ RˆSΩ ·Ω)+Ss, (3.44)SD = −µµˆ(RˆSDU ·U+ RˆSDΩ ·Ω)+SsD, (3.45)CD = −µµˆ(RˆCDU ·U+ RˆCDΩ ·Ω)+CsD, (3.46)d = −µµˆ(RˆdU ·U+ RˆdΩ ·Ω)+ds, (3.47)where the resistance tensors may be written in terms of Tˆ as followsRˆSU,ijk = −µˆ〈 ijnsTˆU,sikrj〉, (3.48)RˆSΩ,ijk = −µˆ〈 ijnsTˆΩ,sikrj〉, (3.49)RˆSDU,ijkm = −µˆ〈 ijknsTˆU,simrjrk〉, (3.50)RˆSDΩ,ijkm = −µˆ〈 ijknsTˆΩ,simrjrk〉, (3.51)RˆCDU,ijk = −µˆ〈 ijnsTˆU,snkjmnrirm〉, (3.52)RˆCDΩ,ijk = −µˆ〈 ijnsTˆΩ,snkjmnrirm〉, (3.53)RˆdU,ij = −µˆ〈2nsTˆU,sijrlrl −nsTˆU,smjrmri〉, (3.54)RˆdΩ,ij = −µˆ〈2nsTˆΩ,sijrlrl −nsTˆΩ,smjrmri〉, (3.55)21and the contributions of the surface activities are likewiseSs = µ〈(us −u∞)n : TˆE −2(us −u∞)n〉, (3.56)SsD = µ〈(us −u∞)n : TˆΓ −4(us −u∞)nr〉, (3.57)CsD = µ〈(us −u∞)n : TˆΛ〉, (3.58)ds = µ〈(us −u∞)n : Tˆe + 12[n · (us −u∞)]r− 3[r · (us −u∞)]n− 3(r ·n) (us −u∞)〉. (3.59)We should emphasize that all components of Tˆ are unique for a given particlegeometry. Therefore by finding them once, we can determine the force momentsfor any prescribed surface activity provided the shape does not change.3.3.1 SphereWe now resolve the force moments of an active spherical particle, using the expres-sions reported in the previous section. We take x0 to be the center of the spherer = an, where a is the radius. Details of the auxiliary flow field and stress fieldcorresponding to each force moment (i.e., Pˆ, Gˆ and Tˆ) are reported in Appendix A.Having TˆU and TˆΩ at hand, the rigid-body resistance tensors can be evaluated.We find RˆFU = 6piµˆaI, RˆLΩ = 8piµˆa3I, RˆdU = 10piµˆa3I and[RˆFΩ, RˆLU , RˆSU , RˆSΩ, RˆSDU , RˆSDΩ, RˆCDU , RˆCDΩ, RˆdΩ]= 0.The force and torque are respectivelyF = −6piaµU− 3µ2a〈us −u∞〉 = −6piaµ(U− (1 + a26∇2)U∞)− 3µ2a〈us〉, (3.60)L = −8pia3µΩ+ 3µ〈(us −u∞)×n〉 = −8pia3µ (Ω−Ω∞) + 3µ〈us ×n〉 , (3.61)where U∞ and Ω∞ are the velocity and rotation rate of the background flow at thecenter of the sphere. If the particle is passive, us = 0, we recover Faxén’s first and22second laws as expected. In the absence of an external force and torque, the rigid-body translation and rotation of spherical active particle with surface velocities usare given byU = − 14pia2〈us −u∞〉 , (3.62)Ω = − 38pia4〈r× (us −u∞)〉 , (3.63)as first shown by Anderson and Prieve [3] and later generalized [49, 184]. Usingthe expression for stresslet given in Eq. (3.44), we findS = −5µ〈(us −u∞)n〉− 23µ〈(us −u∞) ·n〉I,=20piµa33(1 +a210∇2)E∞ − 5µ〈usn〉− 23µ〈us ·n〉I, (3.64)where E∞ = ∇u∞ (x0). We note that this expression for the stresslet amends atypographical error in the results of Lauga and Michelin [108] (Eq. (10) in theirreference). When the sphere is passive, Eq. (3.64) recovers Faxén’s third law. Byusing TˆΓ , we determine the stresslet dipoleSD = −354 µa〈(us −u∞)nn〉,=14µpia53(1 +a214∇2)Γ∞ − 354µa〈usnn〉, (3.65)where Γ∞ = (1/2)∇∇u∞ (x0). The rotlet dipole is then similarly foundCD = 4µa〈[(us −u∞)×n]n〉,=16µpia515Λ∞ + 4µa〈[us ×n]n〉, (3.66)23where Λ∞ = ∇ (∇ ×u∞) (x0). Finally, for the potential dipole, we arrive atd = −10piµa3U+ 15aµ2〈2[(us −u∞) ·n]n−us +u∞〉 ,= −10piµa3 (U−U∞) + 30µpia5d∞ + 15aµ2〈2(us ·n)n−us〉 , (3.67)with d∞ = (1/10)∇2u∞ (x0). In total, for a spherical active particle, we haveS =[− 6piaµU− 3µ2a〈us −u∞〉 ,−8pia3µΩ+ 3µ〈(us −u∞)×n〉 ,− 5µ〈(us −u∞)n〉− 23µ〈(us −u∞) ·n〉I,− 354µa〈(us −u∞)nn〉,4µa〈[(us −u∞)×n]n〉,− 10piµa3U+ 15aµ2〈2[(us −u∞) ·n]n−us +u∞〉 , · · ·]. (3.68)3.3.2 Generalized squirmerWe now examine the expressions obtained above for the specific case of a spherewith purely tangential surface activity, i.e., a squirmer [16, 120, 145]. One maythen express us = usθeθ +usφeφ in spherical coordinates (r,θ,φ) as [145]usθ =∞∑n=1n∑m=0− 2sinθPmn ′(ξ)nan+2 (Bmn cosmφ+ B˜mn sinmφ)+mPmn (ξ)an+1 sinθ(C˜mn cosmφ−Cmn sinmφ), (3.69)usφ =∞∑n=1n∑m=0sinθPmn ′(ξ)an+1 (Cmn cosmφ+ C˜mn sinmφ)+2mPmn (ξ)nan+2 sinθ(B˜mn cosmφ−Bmn sinmφ), (3.70)where Pmn (ξ) is a Legendre function of order m and degree n, and the prime inPmn′(ξ) indicates differentiationwith respect to ξ = cosθ. Here, Bmn, B˜mn,Cmn and24C˜mn are constant coefficients representing different modes of the surface activity.We find the net translational and rotational velocities in terms of these coefficientsand Cartesian unit vectors ex,ey and ez asU = − 43a3(B01ez −B11ex − B˜11ey), (3.71)Ω = − 1a3(C01ez −C11ex − C˜11ey). (3.72)Stresslet, stresslet dipole, rotlet dipole and potential dipole can be similarly deter-minedS =− 12piµa2[B02 ezez−2B12 exez−2B˜12 eyez+ 2B22(exex−eyey)+ 4B˜02 exey], (3.73)SD =− 8piµa2[53B03 ezezez+B13(exeyey −4exezez+exexex)+ B˜13(eyexex−4eyezez+eyeyey)+ 10B23(exexez−eyeyez)+ 20B˜23 exeyez+10B33(3exeyey −exexex)− 10B˜33(3eyexex−eyeyey)],(3.74)CD =− 48piµ5[C02 ezez−2C12 exez−2C˜12 eyez+ 2C22(exex−eyey)+ 4C˜22 exey], (3.75)d =− 80piµ3(B01ez −B11ex − B˜11ey). (3.76)By only keeping B0n and C0n terms in Eqs. (3.69) and (3.70) and setting theother coefficients to zero, the solution reduces to the axisymmetric motion of asquirmer [145]. In this case, by symmetry, all the force moments generated bythe surface activity are invariant by rotation with respect to ez. Thus, as one cansee from (3.73) to (3.76), they must be of form ez, ezez, ezezez, · · · which arethe irreducible traceless rotation-invariant tensors of ez. The contribution of non-zero force moments of an axisymmetric squirmer to the flow field is illustrated inFig. 3.2. Note that, to express the axisymmetric solutions in terms of tangentialsquirming modes Bn used in Lighthill [120] and Blake [16], one can simply set25C0n = 0 and substitute B0n by −an+2n+1Bn in Eqs. (3.71) to (3.76).    3 3030z33 3030z33 303z 011033 3030z3(a) (b)(c) (d)xxx xFigure 3.2: Flow fields induced by non-zero force moments of an axisym-metric squirmer of radius 1, using expressions given in (3.73) to (3.76):(a) Flow field due to a stresslet, for which we set B02 = 1 and othercoefficients to zero. (b) Flow field due to a stresslet dipole with B03 = 1.(c) Flow field induced by a potential dipole with B01 = 1. (d) Flow fielddue to a rotlet dipole with C02 = 1. In (d), the color density indicatesthe magnitude of the velocity where positive (negative) values indicateflow into the plane (out of the plane) [145].263.3.3 Axisymmetric slender rodLet us now consider a slender rod, whose orientation is given by a unit vector p,with an axisymmetric swimming gait along its length us = α(s)p in an otherwisequiescent fluid. Here s parameterizes (by arclength) the centerline of the rod, e.g.s ∈ [−l/2, l/2] where l is the length.To find the force moments, we first decompose a surface integral into an inte-gration around the perimeter (denoted by R) in a plane with normal p and one overthe length of the rod (denoted by s) so that 〈· · · 〉 = 〈〈· · · 〉R〉s. Using the resistiveforce theory for slender rods [109], we may approximate the force density per unitlength〈n · σˆ ′〉R = − [ζˆ‖pp+ ζˆ⊥ (I−pp)] · uˆ′ , (3.77)where ζˆ‖ and ζˆ⊥ are the parallel and perpendicular drag coefficients. Under thisapproximation, finding Tˆ does not require details of the auxiliary flow field as weillustrate in the following. Recalling that uˆ′ = Gˆ Uˆ′ and σˆ ′ = µˆTˆ Uˆ′, one canwriteµˆ〈n · Tˆ〉= −〈[ζˆ‖pp+ ζˆ⊥ (I−pp)]· Gˆ〉s. (3.78)We note that Gˆ(x ∈ ∂B) is known from Eqs. (3.33) to (3.38).To find the resistance tensors, from Eq. (3.78) we haveµˆ〈n · TˆU〉= −〈ζˆ‖pp+ ζˆ⊥ (I−pp)〉s, (3.79)thusRˆFU = −µˆ〈n · TˆU〉=〈ζˆ‖pp+ ζˆ⊥ (I−pp)〉s=[ζˆ‖pp+ ζˆ⊥ (I−pp)]l. (3.80)Similarly, wefind[RˆFΩ, RˆLU , RˆSU , RˆSDΩ, RˆCDΩ, RˆdΩ]= 0, RˆLΩ = − l312 ζˆ⊥ (I−pp),27RˆdU = l312[ζˆ‖pp+ 2ζˆ⊥ (I−pp)]andRˆSΩ,ijk =l324ζˆ⊥(ikspj + jkspi)ps, (3.81)RˆCDU,ijk =l324ζˆ⊥(pikjm + pjkim)pm, (3.82)RˆSDU,ijkm =l312ζˆ‖pipjpk pm + ζˆ⊥ijkδimpjpk −pipjpk pm . (3.83)Now to find S, we substitute (3.78) in (3.41). Noting that 〈D〉 = 0 since〈n〉R = 0, we arrive atS = −µµˆRˆU+µ〈us ·(n · Tˆ)〉,= −µµˆRˆU−〈αp ·[ζ‖pp+ ζ⊥ (I−pp)]· Gˆ〉s,= −µµˆRˆU− ζ‖p ·〈αGˆ〉s. (3.84)Note that ζ‖ = (µ/µˆ) ζˆ‖ and ζ⊥ = (µ/µˆ) ζˆ⊥. From this equation, we find Ls = 0,CsD = 0 andFs = −ζ‖p ·〈αGˆU〉s= −ζ‖ 〈α〉sp, (3.85)Ss = −ζ‖p ·〈αGˆE〉s= −ζ‖ 〈sα〉spp, (3.86)SsD = −ζ‖p ·〈αGˆΓ〉s= −ζ‖〈s2α〉sppp, (3.87)ds = −ζ‖p ·〈αGˆe〉s= −ζ‖〈s2α〉sp, (3.88)which are in the form of irreducible traceless rotation-invariant tensors with regardto symmetry axis p, as expected.With no external force or torque acting on the rod, we can then determine thetranslational velocityU = −(1/l)〈us〉s = −(1/l)〈α〉sp, (3.89)as also shown by Leshansky et al. [116]. Recalling that Ls = 0, we findΩ = 0.283.4 ConclusionIn this chapter, we investigated the effects of higher-order force moments on theflow field induced by an active particle. Using the boundary integral equations,we expressed the flow as a multipole expansion and decomposed the contributionof second-order force moments into a stresslet dipole, rotlet dipole and a potentialdipole. Then, via the reciprocal theorem, we derived explicit formulas for theseforce moments which are valid for an active particle of arbitrary shape and thenevaluated them for a spherical particle, a squirmer and an axisymmetric slenderrod. We believe that by providing simple and explicit formulas for more accurateapproximations of the flow-fields generated by active particles, we may enhanceour understanding of how these particles interact with their surroundings. Giventhe generality of the employed framework, our results can be extended to capturethe effect of third (or higher) order force moments and also can be adapted to studythe hydrodynamic interactions between two [172] or many [148] active particles oractive particles near boundaries [179, 185].29Chapter 4Elastic swimmer near a wall4.1 IntroductionMicroorganisms often swim in complex trajectories when they are in proximityof obstacles. Some flagellated cells such as E. coli swim in circles of large radiinear solid surfaces [12, 110], while some others like spermatozoa accumulate nearthe walls, when swimming in a confined setting [56, 118, 200]. It has also beenobserved that microswimmers can be entrapped by the nearby boundaries, swimalongside a wall for a long period of time [43], and then escape toward otherobstacles [21, 182, 186]. Such nontrivial interactions between microorganisms andsurfaces have been shown to contribute to biofilm formation [144, 156], pathogenicinfection [85], and are also important in metabolite transportation [183].In theory, the studies on the swimming trajectory of a microswimmer near awall can be categorized into two groups: The swimmer is very close to the wall,hence lubrication equations govern the motion [9, 30], or its distance from thewall is large compared to its body size such that far-field approximations may beemployed [13, 45, 117]. Although the latter relies on large distances from the wall,far-field hydrodynamics have shown to provide accurate results even at distancesas small as a fraction of the cell body [180], and have been frequently utilizedto explain interactions of a swimmer and nearby surfaces [38, 125, 181]. Basedon this approximation, a wall reflects back the disturbance flow field caused bythe swimmer and results in a force and a torque on the body [13]. Thus, the30wall presence is captured by the zeroth-order and first-order force moments of thereflected flow field. Depending on the geometry of the wall and the swimmer, andalso the propulsion mechanism, the swimmer-wall interaction can lead to attractionor repulsion [13, 181]. However, this model cannot capture the effect of swimmerflexibility on its behavior near a wall. In particular, flagellum is elastic and itsbending can significantly alter its hydrodynamic interactions [39, 199, 201]. Ina computational study, Montenegro-Johnson et al. [132] showed that an elasticfilament is scattered by around 5 to 10 degrees when swimming past a back-stepin a microchannel. They found that depending on values of viscous forces andelastic forces, the swimmer may scatter toward the back-step or away from it. It hasalso been shown that a flagellum may buckle for reorientation [177], which stemsfrom the elasticity of the flagellum [63, 202, 204] or the buckling of an elastic hookconnecting the flagellum to the cell body [94, 139, 173]. In this chapter, by allowingthe swimmer to bend from its connecting hook, we show that such reorientationsoccur naturally when the swimmer approaches a boundary. We show that effectsof the elasticity of the swimmer is captured by the second-order force moments ofthe reflected flow field, namely the rotlet dipole. Then, using a minimal model, wedemonstrate that elasticity prevents attraction by the wall for pusher-type swimmers,while it directs puller-type swimmers toward the wall.4.2 Theoretical FrameworkWe consider an elastic swimmer located at x0. The motion of the swimmer isforce and torque free, thus, the thrust (Fs) generated by the swimmer is completelybalanced by the hydrodynamic drag (FD)Fs + FD = FsLs+  FDLD = 0, (4.1)where Fs and Ls are the driving force and torque, while FD and LD are the cor-responding hydrodynamic drag and torque. The swimmer is elastic and deformspassively in response to a background flow. But, given the reported values forthe typical bending rigidity of a swimmer [25, 135], we may employ a quasistaticassumption for its elastic bending. Thus, we can define the rigid-body motion31of the swimmer by U = [U,Ω,0, · · · ], where U and Ω are the translational androtational velocity. Due to presence of a wall, there exists a background flow fieldu∞(x). As shown in the previous chapter, at distances far from the swimmer, onecan expand the background flow field around a point in the body of the swimmer(e.g., x0) as u∞ = GU∞, where G is a linear operator mapping the velocity mo-ments U∞ = [U∞,Ω∞,E∞,Γ∞,Λ∞,e∞, · · · ] to u∞. In Chapter 3, we showed thatU∞ = u∞ (x0), Ω∞ = (1/2)∇ ×u∞ (x0), E∞ = ∇u∞ (x0), Γ∞ = (1/2)∇∇u∞ (x0),Λ∞ = ∇ [∇ ×u∞ (x0)], and e∞ = (1/10)∇2u∞ (x0). Using this expansion, thehydrodynamic drag can be then written asFD = −R (U−U∞) , (4.2)whereR = RFU RFΩ RFE RFΓ RFΛ RFe · · ·RLU RLΩ RLE RLΓ RLΛ RLe · · · , (4.3)contains all the resistance tensors associated with the force and torque. The velocityof the swimmer is therebyU = R−1FU · Fs −R−1FU ·RU∞, (4.4)whereRFU = RFU RFΩRLU RLΩ , (4.5)is the rigid-body resistance tensor. Note that Eq. (4.4) provides the instantaneousvelocity of the swimmer for any background flow, provided the resistance tensor isknown.Let us now further simplify the problem by considering an axisymmetric flagel-lated microorganism as our model swimmer, which propels forward by generatingFs = [f pˆ,0]T , where pˆ is a unit vector. We assume that the flagella (i.e., tail) andthe cell body (i.e., head) are connected by an elastic hook which bends linearly inresponse to a torque dipole by the background flow (e.g., a torsional spring). We32define unit vector rˆ as the axis of symmetry of the swimmer in the absence of anybackground flow. Note that when there is no background flow (i.e., no wall), thethrust is aligned with the swimmer body pˆ = rˆ. The angle between pˆ and rˆ thencharacterizes the bending which can be defined asδ = tan−1( |rˆ× pˆ|rˆ · pˆ). (4.6)But, how does the wall bend the swimmer? To answer this question, we needto further inspect the contribution of the background flow induced by the wall. Theschematic of the effects of the background flow is illustrated in Fig. 4.1. U∞ appliesa force on the swimmer which is then balanced by the drag, thereby changing thetranslational velocity of the swimmer. Similarly, the vorticity of the backgroundflow field Ω∞, exerts a torque that is balanced by the drag due to rotation. But,these forces and torques are not exerted on a single point on the swimmer bodyand, in fact, are non-uniformly distributed. E∞ and Γ∞ account for this effect bystraining the swimmer due to the inequality of the forces between the head and tail.The finite size of the body gives rise to a potential dipole e∞. Any difference in thetorques across the body leads to a bending moment on the connecting elastic hook.This bending moment, which is a result of the rotlet dipole Λ∞, is balanced by thetorque due to bending of the hook askδrˆ× pˆ|rˆ× pˆ| +RLΛ ·Λ∞ = 0, (4.7)where k is the bending rigidity of the hook. Due to this bending, the resistancetensor R changes but also we have pˆ , rˆ, thus from (4.4) we can determine the newswimming velocity. As finding R requires the details of the shape of the swimmer,in the following, using a toy model, we show how elastic bending of the swimmeralters the swimming trajectory near a wall.4.3 Toy modelWe now use a toy model to quantify the effects of elastic bending on interactionsof a swimmer and a wall. At distances far from the swimmer, the leading-order33U1⌦1E11⇤1e1Figure 4.1: The effect of velocity moments of the background flow on amodel swimmer. U∞ is the zeroth-order velocity moment, E∞ andΩ∞are the first-order moments of the background flow, and Γ∞, Λ∞, ande∞ together represent the second-order velocity moments.flow field induced by force- and torque-free motion of the swimmer is capturedby a symmetric force-dipole, whose strength depends on the swimmer geometryand its thrust-generating mechanism. When the swimmer generates impetus fromits front end (e.g., Chlamydomonas), it is referred to as a puller, whereas in apusher-type swimmer, thrust originates from the rear end (e.g., E. coli). Underthis force-dipole approximation, the tail generates the driving force, which in turnis balanced completely by the drag force on the cell body (i.e., head). Here, weconsider a two-sphere model, wherein spheres represent the head and the tail.Without any loss of generality, we assume sphere A generates the hydrodynamicthrust capturing the effect of the tail, while sphere B represents the head, as shownin Fig. 4.2. The flagella and cell body are separated in position, thus we take Las the characteristic distance between the head and tail (or the length scale of theswimmer). Furthermore, the length scales of the flagella and the cell body differ,and so we consider aA and aB as their characteristic sizes, noting that for a typicalswimmer aA aB. The swimmer is at distance h from the wall and we refer to theangle between rˆ and ex as the swimming angle θ.34exezBA✓Lhpˆ < 0rˆ > 0⇤1⇤1(b)(a)Figure 4.2: (a) Schematic of the considered swimmer. Spheres A and Brepresent the tail and the head, respectively. Sphere A is at distanceh from the wall, L characterizes the swimmer length scale and θ isthe swimming angle. (b) Bending mechanism of the swimmer by therotlet dipole. The swimmer bends from its center with angle δ which ispositive when the swimmer bends toward the wall and negative when itbends away.We now scale lengths with L, velocities withU , forces with µLU and use µL2Uas the characteristic torque, where µ is the dynamic viscosity of the fluid. We non-dimensionalize all the terms using these characteristic values and henceforth referto dimensionless quantities.Using the far-field approximations, the flow field induced by each sphere can bemodeled using a Stokeslet. To account for the presence of the wall, we may use animage Stokeslet of the same strength for each of the spheres. The no-slip conditionat the wall is then satisfied by setting these image Stokeslets within the wall, atthe same instantaneous distance as their corresponding particles. The backgroundflow is then a summation of these two image Stokeslets. Calculating Λ∞ using theBlake solution for image Stokeslets [17, 180], and also finding R for the deformed350 10.50.511.51.510⇥103✓ = 0⇡/43⇡/8⇡/2✓ = 0⇡/43⇡/80.50.25 0.75 = +1 = 1Figure 4.3: Variation of bending angle δ with respect to the ratio of radii λ,using the expression given in (4.8) with f = 0.5, h(t = 0) = 10 andk = 10−4.swimmer using the results of Chapter 3, we find the bending angleδ =8βf 4λ3 cosθ9kh4 (1 +λ)3 (1 +λ3), (4.8)where λ = aAaB . Here we use β to identify pushers and pullers asβ =+1 pushers−1 pullers . (4.9)In Fig. 4.3, we illustrate variation of the bending angle with respect to θ and λ. Asthe size of the tail increases (λ increses), the rotlet dipole becomes stronger and sodoes the bending angle. However, the resistance of the swimmer against bendingalso depends on the size of the tail. The competition between these two effectsgives a rise to non-monotonic behavior of δ with respect to λ. We note that theelastic bending is maximized exactly when λ = 34 .Now, noting that dh/dt = U · ez, dθ/dt = −Ω · ey , and using Eq. (4.4), we find36the time-evolution equation for h and θ asdhdt= sinθ +3βf16(1 +λ)h2(1− 3cos2θ) , (4.10)dθdt= − 3βf16(1 +λ)h3sin2θ +81βλ8192f h3(14sin2θ − sin4θ)+3βf 2λ38k (1 +λ)3 (1 +λ3)h4cosθ. (4.11)We may first look into the limiting case where k → ∞; the swimmer is rigid,hence δ = 0. Defining θ0 as the initial orientation angle, the thrust is pushing theswimmer toward the wall (when θ0 < 0) or away from it (when θ0 > 0). But,the hydrodynamic torque, exerted by the background flow, tries to reorient theswimmer. Depending on the strength of the hydrodynamic interactions and alsothe value of θ0, this reorientation may qualitatively change the swimming direction(e.g., from repulsion to attraction). When |θ|  1, one can find the critical anglefor such a change as θc =3βf8(1+λ)h2Aat which dh/dt = 0. When θ0 > θc, we finddh/dt > 0. The swimmer thereby swims away from the wall, almost linearly, sincethe background flow correction to the swimming speed decays quadratically by h,and so dh/dt ≈ sinθ. Similarly, θ0 < θc indicates that dh/dt < 0, thus pushers andpullers are both attracted by the wall, as illustrated in Fig. 4.4.We now allow the swimmer to bend and begin with the behavior of pushers.When β = 1, the bending angle is positive, thus the swimmer bends toward thewall. The realignment of the head and tail caused by this bending, exerts an‘elastohydrodynamic’ torquewhich is in the direction of−ey . We should emphasizethat, this elastohydrodynamic torque is generated due to the bending of the swimmer.In this specific example, one findsRLU = 6pi (aA + aB) rˆ×I. Recalling that Fs = f pˆ,and also noting pˆ , rˆ due to bending, we haveRLU ·Fs , 0 indicating that the thrustitself exerts a torque on the body which is the elastohydrodynamic torque.For θ0 > θc, the elastic bending enhances the repulsion of pushers by thewall. When θ0 < θc, as the body gets closer to the wall, the elastohydrodynamictorque becomes more dominant (since it decays faster), and so at certain value ofh, the swimmer escapes the wall with an equilibrium angle that balances these twotorques. This behavior indicates that, regardless of the values of λ and θ0 (except37468101214468101214t/⌧0 0.05 0.1t/⌧0 0.05 0.10.010.01✓0.010.01✓✓0 = ✓c  ✏✓0 = ✓c + ✏✓0 = ✓c  ✏✓0 = ✓c + ✏(a) (b) = +1  = 1hht/⌧ t/⌧Figure 4.4: Evolution of h and θ (inset) for rigid (a) pushers and (b) pullersusing expressions (4.10) and (4.11) evaluated at h(t = 0) = 10, λ =0.1, f = 0.5 and k →∞. For each case, the initial swimming anglesare θc ±  where θc = 0.0017β is the initial critical angle and  =0.002. The dotted line in the insets indicate the critical angle andτ = 16h3(t = 0)/(3f ) is the characteristic timescale.for θ0 = −pi/2), elastic pushers repel the wall. Expectedly, at higher values of λ,the effect of elasticity becomes stronger, thus the swimmer escapes the wall faster,as shown in Fig. 4.5.Conversely, for pullers, δ is negative and the elastohydrodynamic torque actsin +ey , directing the swimmer toward the wall. Thus, for θ0 < θc, elasticitystrengthens the attraction. As illustrated in Fig. 4.6(a) and (b), for θ0 > θc, theswimmer is initially propelling away, then it changes direction due to elastic bendingand eventually swims toward the wall. It should be noted that, the increase of h dueto the initial repulsion, weakens the effects of elastohydrodynamic torque. Thus,as shown in Fig. 4.6(c), for higher values of θ0 and smaller λ, the elastic bendingmay not be able to overcome the hydrodynamic torque and the swimmer escapes.Unlike the elastic pushers wherein introducing elasticity guaranteed a repulsion, forelastic pullers, depending on the values of the initial angle and also the ratio of theradii, the swimmer either goes toward or away from the wall.38✓0.50.5 = 0.2 = 0.4 = 1610142t/⌧0 0.03 0.06 = +1ht/⌧Figure 4.5: The time-evolving behavior of h and θ (inset) for elastic pusherswimmers. Plots are the numerical evaluation of Eqs. (4.10) and (4.11)with θ0 = −pi/8, k = 10−4, f = 0.5 and λ = 0.2,0.4 and 1.1020304020.3 0.6 0.900.07 0.14 0.2102101520252610140.004 0.0080t/⌧ t/⌧ t/⌧(a) (b) (c)✓0 = 0.02✓0 = 0.03✓0 = 0.05✓0 = 0.02✓0 = 0.03✓0 = 0.05✓0 = 0.02✓0 = 0.03✓0 = 0.05 = 0.1 = 0.2 = 0.5h h h = 1  = 1  = 1Figure 4.6: The evolution of h for a puller for (a) λ = 0.5 (b) λ = 0.2 and (c)λ = 0.1 with three initial orientations of θ0 = 0.02,0.03 and 0.05. Forall cases k = 10−4 and f = 0.5.394.4 ConclusionIn this chapter, we inquired about the dynamics of an elastic swimmer near a flatwall. Using far-field approximations, we showed that an elastic swimmer bends dueto the rotlet dipole generated by the wall. We quantified the effect of bending onthe trajectory of the swimmer by considering a two-sphere model. Unlike in pullerswherein the effect of elasticity is not always dominant, repulsion for pushers is foundto be unavoidable if we allow the swimmer to bend elastically. Our results highlightthe importance of elasticity in understanding the interaction of motile cells withnearby boundaries. Although here we neglect the hydrodynamic interactions inthe lubrication regime (i.e., h < 1), it has been shown (both computationally [180]and experimentally [13]) that far-field approximations can capture the cell-wallinteractions very well. We also note that further insight into effects of elasticity ona swimmer interactions with a surface can be gained by employing more accuratemodels for the bending mechanism of the hook [94, 139, 173] and also accountingfor bending of the flagella [64, 123, 165].40Chapter 5Elastic two-sphere swimmer inStokes flow 15.1 IntroductionIn the microscale realm of motile cells, inertia is unimportant and the effect of vis-cous dissipation dominates the fluid forces on swimming bodies [84, 98]. To propelforward in this regime, many microorganisms deform their bodies periodically byconverting cells’ chemical energy into mechanical work [166]. As a direct con-sequence of this inertialess environment, to achieve nonzero net locomotion, suchbody deformations cannot be invariant under time reversal [157]. This constraint,colloquially referred to as the scallop theorem, indicates that due to the kinematicreversibility of the field equations in the low Reynolds number regime, reciprocalbody distortions have no net effect.Theoretically, the scallop theorem can be eluded under two circumstances:non-reciprocal kinematics or a violation of the theorem’s assumptions (see [106]and the references therein). The latter exploits the fact that the scallop theorem issolely valid for inertialess single swimmers in quiescent viscous fluid. Therefore,hydrodynamic interactions [190], a non-Newtonian medium [105], or inertia [74]can all lead to propulsion. Non-reciprocal kinematics are employed by many motile1A version of this chapter has been published [138].41cells in nature to facilitatemotion [109, 121], and also become a key design principlefor model swimmers at small scales. In 1977, Purcell introduced a simple three-linkswimmer with two rotational hinges that can change its shape in a non-reciprocalfashion, leading to a locomotion [157]. Subsequently, several analytical modelswimmers have been devised wherein non-reciprocal shape change provides thepropulsive thrust [44, 70, 92, 135]. Notably, Najafi and Golestanian [134] proposeda simple three-sphere swimmer, in which spheres are identical and connected by twoslender rods. The connecting rods change their length in a four-stage cycle that is notinvariant under time reversal. After completion of one cycle, the swimmer recoversits original shape but has been translated forward (see also [71] and [115]). Avronet al. [8] suggested a more efficient, yet as simple, swimmer that consists of twolinked spherical bladders of different radii. To compensate for the third sphere, theyrelaxed the rigidity constraint by allowing instantaneous volume exchange betweenspherical bladders in each stroke. The shape change of the bladders along withthe periodic change in their distance, leads to a net displacement of the swimmer.Inspired by these two models, in this chapter we investigate a simple, but lessintuitive, two-sphere swimmer where one of the spheres is elastic. We proposethat the elastic deformation of the swimmer can be sufficient to escape the scalloptheorem, alter hydrodynamic interactions and eventually lead to propulsion.Elasticity, as an inevitable characteristic of motile cells, can significantly affectthe hydrodynamics of a motion. The propulsion of flexible bodies [104, 157, 198],synchronization of flagella [51, 69] and cilia [22, 136, 141] through elastohydro-dynamic interactions, and reorientation of uni-flagellated bacteria due to bucklingof the flagellum [95, 177] are well-studied examples of such behaviors. For anelastic body in a flow, the balance of viscous forces, external forces and internalelastic forces causes the body to deform and to alter the surrounding flow field,often in a complex fashion [58, 59, 119]. Li et al. [119] reported that for an isolatedsedimenting filament, elasticity can destabilize the motion and lead to a substantialbuckling. Furthermore, Gao et al. [59] showed that elastic spheres in a shear flowexhibit a ‘tank-treading’ motion wherein the particle shape is at steady state whilethe material points on the boundary are undergoing a periodic motion. However,though seemingly simple, sedimentation of spherical elastic particles in a viscousfluid is largely unexplored. The most recent, and to the best of our knowledge the42only, analysis on sedimentation of elastic spheres dates back to more than threedecades ago, when Murata [133] investigated the steady state shape deformationof a compressible, Hookean sphere. Using an asymptotic analysis, it was shownthat the elastic sphere settles faster and deforms to a prolate spheroid of a smallervolume. In this chpater, to further investigate the deformation of elastic spheres, werevisit this sedimentation problem but this time for an incompressible neo-Hookeansphere under a prescribed body force. We asymptotically describe the steady stateeffects of non-linear elastic deformations on the swimming behaviors of an isolatedelastic sphere.The chapter is organized as follows. In Section 5.2, we investigate the translationof a single neo-Hookean sphere in Stokes flow. Using an asymptotic approach, weshow that for a given body force, due to deformation, the translational velocity of theelastic sphere is smaller compared to a rigid sphere of the same size. Furthermore,we find that the shape deformation is not front-back symmetric and so neither isthe flow field generated in the surrounding fluid. In Section 5.3, we show that byexploiting this asymmetry, the proposed two-sphere model can indeed swim in alow Reynolds number regime. Finally, in the case where the distance between thespheres is relatively large, we determine the propulsion velocity.5.2 Translation of an elastic sphereWebegin our analysiswith considering the translation of an incompressible isotropicneo-Hookean sphere in an otherwise quiescent viscous fluid. The sphere has radiusR0 and is driven by body force f(t). In the fluid domain (Ωf), the flow field aroundthe sphere is governed by the Stokes equationsdiv σ f = 0, (5.1)div v = 0, (5.2)where v is the fluid velocity and σ f is the dynamical stress tensor in the fluid domaindefined by the constitutive relationσ f = −pfI+ ηf[grad v+ (grad v)T], (5.3)43where pf is the pressure and ηf is the viscosity of the fluid. We assume the sphereis translating with velocity U thus the no slip boundary condition dictates v = Uat the fluid-solid interface. In the solid domain (Ωs), the governing equations aredescribed in terms of material coordinates. Thus, to avoid any confusion, we writethe material gradient, divergence and Laplacian using ∇, ∇· and ∇2, respectively.The equilibrium momentum balance inΩs then yields∇ · σ s + f(t) = 0, (5.4)where σ s is the solid elastic stress and f is a body force density on the sphere.Since the motion is axisymmetric, we assume the elastic sphere reaches a stableequilibrium, wherein the velocity gradient field in the solid domain is zero and thesphere has a rigid motion thereafter [91, 193]. As we will show later, for a weakly-elastic sphere, the leading-order effect of elasticity does not lead to any change inshape. Thus, a higher-order analysis is necessary in order to understand the changein shape of a translating elastic sphere. Extending linear elasticity to higher ordersintroduces further complexity by involving more material properties [142], insteadhere we use a phenomenological neo-Hookean model to capture the higher-ordereffects.The constitutive relation for an isotropic incompressible neo-Hookean solidcan be expressed in terms of the displacement vector u as [81, 142]σ s = −psI+ ηs(F · FT − I), (5.5)where F = I+∇u is the deformation gradient tensor and ηs is the shear modulus. Forany material point, the displacement vector is defined u = χ (X, t)−X, where X isthe position vector in the reference configuration (in other words material point) andχ (X, t) is the deformation vector mapping each material point to its new location[81]. Here, ps serves only as a Lagrange multiplier to impose the incompressibilityof the solid throughdet(F) = 1, (5.6)where det(F) is the determinant of tensor F. The solid and fluid momentum balancesare coupled through the continuity of normal traction at the interface (∂Ω), which44dictatesσ s ·n = σ f ·n, (5.7)where n is the normal vector to the surface of the deformed sphere.Without any loss of generality, we will assume that the translational velocity,U = Uez, and the body force density, f = bf (t)ez, are oriented along ez. Forsimplicity we assume a spatially uniform body force where b is a positive constantdenoting the magnitude of the forcing while f is a dimensionless O(1) functionsuch that the elastic deformation may be considered quasistatic.Before going further, we non-dimensionalize all the equations defining dimen-sionless quantities ∇ˆ = R0∇, uˆ = u/R0, vˆ = v/Uch, Uˆ = U/Uch, tˆ = t/(R0/Uch),pˆf = pf/(ηfUch/R0), σˆ f = σ f/(ηfUch/R0), σˆ s = σ s/ηs, pˆs = ps/ηs and fˆ(t) =f(t)/(ηs/R0), where Uch = 2bR20/9ηf. Here Uch simply denotes the translationalspeed of a rigid sphere under a constant body force of magnitude b. Furthermore,for a forcing profile with frequency ω, we define ν = ωR0/Uch as a ratio of timescales. Now for convenience, we drop the ( ˆ ) notation and henceforth refer todimensionless variables. The dimensionless form of the boundary condition at thefluid-solid interface is then derivedσ s ·n = σ f ·n, (5.8)where σ f = −pfI + grad v + (grad v)T is the dimensionless stress in the fluid,σ s = −psI+∇u+∇uT +∇u ·∇uT is the dimensionless stress in the solid phase and = ηfUch/ηsR0 represents the ratio of the viscous forces to the elastic forces. Therelaxation time scale of the solid τrelax ∼ ηf/ηs which when non-dimensionlizedscales as O(). Thus, for   1, the time required for relaxation is asymptoti-cally shorter than the imposed time scale of motion, which justifies the quasistaticassumption.In order to develop a geometric relation between the displacement vector andthe surface deformation, we consider spherical coordinate systems (r,θ,φ) in thespatial configuration. Since the motion is axisymmetric, we can define the surface45as rs(θ) where θ is the polar angle. Thus, at the interface, this definition yields||X+u|| = rs, (5.9)providing a geometric relation between surface equation and the displacementvector. We should emphasize that the governing equations in Ωs are expressed ina material description. Thus, to obtain the deformation in the spatial variables, wetransform the results of Eq. (5.9), using the mapping χ.5.2.1 Asymptotic analysisHere we focus on the case wherein the elastic forces aremuch larger than the viscousforces, i.e.,  1. We expand all the parameters in terms of  and refer to the ithorder of any parameter using superscript (i) (e.g., pf = p(0)f + p(1)f + 2p(3)f + · · · ).Due to the linearity of the Stokes equations, at any order the flow field around thesphere is governed by−grad p(i)f + div(grad v(i))= 0, (5.10)div v(i) = 0, (5.11)where σ (i)f = −p(i)f I + grad v + (grad v)T and i ∈ {0,1,2, · · · }. We use the generalsolution given by Sampson for axisymmetric Stokes flow in the spherical coordinatesystem [84, 170]. The boundary conditions in the fluid domain thereby are v(i) = 0at r → ∞ and v = Uez at r = rs. In the solid domain, the nonlinear governingequations are linearized perturbatively, thus we treat the problem at each orderseparately. As one can notice from Eq. (5.8), there is no deformation at the zerothorder thus the leading-order elastic effects are of O(). Throughout the followinganalysis we first solve the solid domain equations using a material description, andthen map to the spatial configuration to enforce the interface boundary conditions.All formula given below for u and ps are reported in terms of spatial variables.46Zeroth order flow field (first order solid deformations)At the zeroth order in the fluid domain, the motion is simply the translation of arigid sphere in Stokes flow. Satisfying v(0) = f ez at r = 1, we findv(0)r =f2(3r− 1r3)cosθ, (5.12)v(0)θ = −f4(3r+1r3)sinθ, (5.13)p(0)f =3f2r2cosθ. (5.14)The leading-order deformation equations in the solid domain are in the form of theStokes equations as−∇p(1)s +∇2u(1) + f(t) = 0, (5.15)∇ ·u(1) = 0, (5.16)σ(1)s = −p(1)s I+∇u(1) +∇u(1)T . (5.17)Thus, here as well, we can employ Sampson’s general solution for an axisymmetricStokes flow. At this order, the interface boundary condition is σ (1)s,rr = σ(0)f,rr andσ(1)s,rθ = σ(0)f,rθ . Noting that at this order the reference and spatial configurationscoincide, we obtainu(1)r =f2(1− r2)cosθ, (5.18)u(1)θ =f2(−1 + 2r2)sinθ, (5.19)p(1)s = −f2 r cosθ. (5.20)To find the surface deformation, we define surface equation rs = 1 + s(θ) and usethe geometric relation in (5.9), which at this order leads to s(1) = u(1)r at r = 1.Therefore, we find s(1) = 0, indicating that the elastic sphere remains spherical withno surface deformation. We note that this result is similar to the sedimentationof a falling drop in a viscous fluid. Taylor and Acrivos [188] showed that wheninertia is neglected and the flow fields both inside and outside the drop are similarly47governed by the Stokes equations, the shape has to remain spherical to satisfy thecontinuity of the normal tractions at the interface.First-order flow field (second order solid deformations)At this order, the flow field at surface of the sphere satisfies v(1) =U (1)ez. Recallingthat s(1) = 0, we findv(1)r =U1f22(3r− 1r3)cosθ, (5.21)v(1)θ = −U1f24(3r+1r3)sinθ, (5.22)p(1)f =3U1f 2 cosθ2r2, (5.23)where the first correction for translational velocity U1 shall be determined bysatisfying the interface boundary condition. In the solid domain, the governingequations are−∇p(2)s +∇2u(2) +∇(∇ ·u(2))+∇ ·(∇u(1) · ∇u(1)T)= 0, (5.24)∇ ·u(2) + tr(∇u(1),c) = 0, (5.25)where tr( ) and ( )c indicate trace and cofactor of the tensor, respectively. Here thestress in the solid phase is definedσ(2)s = −p(2)s I+∇u(2) +∇u(2)T +∇u(1) · ∇u(1)T . (5.26)Now, by enforcing the interface boundary conditions σ (2)s,rr = σ(1)f,rr and σ(2)s,rθ = σ(1)f,rθat r = 1, we find U1 = 0 andu(2)r = − f2r304(23 + 27r2 + (69 + 5r2)cos2θ), (5.27)u(2)θ =f 2r304(69 + 97r2)sin2θ, (5.28)p(2)s =f 2152(190− 199r2 − 27r2 cos2θ), (5.29)48leading to p(1)f = 0, v(1)r = 0 and v(1)θ = 0. It is worthwhile to emphasize that theleading order corrections for the flow field (i.e., v(1)r and v(1)θ ) are imposed by theleading order deformation in the solid domain. Thus, s(1) = 0 indeed causes nodisturbance in the flow field at this order. Finally, to find s(2), we use the geometricrelations(2) = u(2)r +(u(1)θ)22, at r = 1, (5.30)leading to s(2) = −31f 2304 (1 + 3cos2θ), which indicates a shape deviation from asphere to an oblate spheroid of aspect ratio 1− 93152f 22.Second-order flow field (third-order solid deformations)The no slip boundary condition for the Stokes equations at this order is v(2) +s(2) ∂v(0)∂r =U(2)ez. The second order flow field around the sphere isv(2)r =U2f3 cosθ2(3r− 1r3)+93f 3 cosθ1520(2r+1− 15cos2θr3− 3− 15cos2θr5),(5.31)v(2)θ =−U2f3 sinθ4(3r+1r3)− 93f3 sinθ6080(4r+13 + 15cos2θr3− 27 + 45cos2θr5),(5.32)p(2)f =3U2f 3 cosθ2r2− 93f3 cosθ3040( 4r2+15− 75cos2θr4). (5.33)Similar to the previous order, to determine the correction for the translationalvelocity (i.e., U2), we need to solve the solid deformation equations at the thirdorder given by−∇p(3)s +∇2u(3) +∇(∇ ·u(3))+∇ ·(∇u(1) · ∇u(2)T +∇u(2) · ∇u(1)T)= 0,(5.34)∇ ·u(3) + det(∇u(1))+ T = 0, (5.35)49where T is the O(3) contribution of tr(∇uc). Here, the stress inside the solid isdefinedσ(3)s = −p(3)s I+∇u(3) +∇u(3)T +∇u(1) · ∇u(2)T +∇u(2) · ∇u(1)T . (5.36)Enforcing the third order interface boundary conditions at r = 1 asσ(3)s,rr + s(2)∂σ(1)s,rr∂r− ds(2)dθσ(1)s,rθ = σ(2)f,rr + s(2)∂σ(0)f,rr∂r− ds(2)dθσ(0)f,rθ , (5.37)σ(3)s,rθ + s(2)∂σ(1)s,rθ∂r− ds(2)dθσ(1)s,θθ = σ(2)f,rθ + s(2)∂σ(0)f,rθ∂r− ds(2)dθσ(0)f,θθ , (5.38)we finally find U2 = −31f3380 , andu(3)r =(7r2(310r2 − 2091)cos2θ − 5816r4 + 9645r2 − 30240) f 3 cosθ25536,(5.39)u(3)θ =(7r2(434r2 + 697)cos2θ + 5284r4 + 381r2 + 10080) f 3 sinθ8512, (5.40)p(3)s = −r(322r2 cos2θ + 3766r2 − 2991) f 3 cosθ1596. (5.41)Thence, we can determine the third order shape deformation using the geometricrelation (5.9), which reads s(3) = u(3)r + u(1)θ u(2)θ at r = 1. Notably, we find s(3) =−2777f 31824 cos3θ indicating a ‘egg-like’ deformation which exhibits a front-backasymmetry in the surface of the elastic sphere as shown in Fig. 5.1.It is also worthwhile to note that elastic capsules containing viscous fluids ex-hibit a similar asymmetry in their deformation under pure translation. In a numericalstudy, Ishikawa et al. [93] showed that at steady state, a weakly-elastic sphericalmicro-torque swimmer deforms to an egg-like shape. A similar deformation wasobserved experimentally for sedimenting vesicles as well [91].Third-order flow fieldTo quantify the effect of the shape asymmetry on the motion of the particle, weshall determine the third order correction for the flow field. Once again, we solve50zx−1.5−1−0.500.511.5−1.5 −1 −0.5 0 0.5 1 1.5ezFigure 5.1: Deformed shape of the translating elastic sphere when f = 1 and = 0.45.the Stokes equations, but this time with v(3) + s(3) ∂v(0)∂r =U(3)ez at r = 1. Thus, wecan find the third order correction for the fluid field and stress field in terms of thetranslational velocity U (3). Now to find U (3), instead of solving for the next ordersolid deformation (as we did in the previous orders), we employ an auxiliary casewherein a rigid sphere of the same radius is translating with the same driving force[149]. Since the motion is over-damped, regardless of the shape, the driving forceis always entirely balanced by the viscous surface forces. Thus, for a given drivingforce, the net drag force on both elastic and rigid spheres are the same. For theelastic sphere we haveFdr + 3∫∂Ωσ f ·ndS = 0, (5.42)where Fdr = fV is the total driving force on the sphere, V is the volume and Srepresent dimensionless area element of the sphere. On the other hand, for the rigidsphere case, the drag law dictates Fdr/V = 9f2  ez, thus1V∫∂Ωσ f ·ndS = −3f2 ez. (5.43)Since div σ f = 0, by the divergence theorem, one may alternatively integratethe traction over an undeformed sphere of radius larger than the mean radius of51the deformed sphere. Then by substituting σ f = σ(0)f + σ(1)f + 2σ(2)f + 3σ(3)f ,we determine the left-side of Eq. (5.43) as 1V∫∂Ωσ f · ndS = −32(f +U (3)3)ez,indicating that U (3) = 0. Thus, the final expression for the translational velocityU =[1− 31380f 22 +O(4)]f , (5.44)and the third order corrections in the flow field arev(3)r =2777f 417024( 1r2− 1r4)(3− 9cos2θ − 3− 30cos2θ + 35cos4θr2), (5.45)v(3)θ =2777f 434048(sin2θr4)(12− 14cos2θ − 12− 28cos2θr2), (5.46)p(3)f =2777f 442560( 1r4)(15− 45cos2θ − 21− 210cos2θ + 245cos4θr2). (5.47)To recover the solution for the case of translation under a constant body force(i.e. sedimentation), one can set f = 1 finding U = 1 − 313802 +O(4). We seethat the deformation of the incompressible neo-Hookean sphere causes a reductionin the settling speed compared to a rigid sphere. Our results are notably differentwhen compared with the sedimentation of a compressible Hookean solid for whicha faster-than-rigid settling speed and a reduction in volume are reported [133].5.3 Two-sphere swimmerWe consider a model swimmer which consists of two spheres: a rigid sphereA and a neo-Hookean isotropic incompressible elastic sphere B (identical to theelastic sphere defined in Section 5.2). The spheres are of equal radii and linkedby a rod of length L. To propel itself forward, the swimmer repeats a two-step,one-dimensional motion in which the connecting rod shortens its length in step(I), and then returns back to its original length in step (II) in a harmonic fashion(see Fig. 5.2). While advancing from one step to another, sphere B changes itsshape continuously and instantaneously, until it reaches its spherical shape again atthe end point of each step. We note that despite the reversible actuation, the flowfield induced by sphere B is not front-back symmetric. Thus, for sphere A, thecontribution of the background flow (induced by sphere B) is different between step52−1.5−1−0.500.511.5−1.5−1−0.500.511.5A BA B−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5A BBABA(I)(II)ezFigure 5.2: One cycle of the two-step motion of the swimmer. Step (I): Therod shortens its length. Step (II): Spheres move away from one anotheruntil they reach the initial distance. The steps in grey color demonstratethe swimmer while it proceeds to the next step and sphere B is deformed.(I) and (II). The net motion in each cycle thereby is not kinematically reversible andindeed the swimmer can propel with a velocity that we determine below.The connecting rod exerts driving forces FA and FB on spheres A and B,respectively. The force-free motion of the swimmer necessitates FA + FB = 0.Although, in practice, the driving forces are applied locally at the sphere-rodjunctions, here we neglect the effect of the rod and assume a spatially uniformforce density for both spheres, noting that such actuation forces can be imposed bymagnetic fields or optical tweezers. Thus, we prescribe the periodic motions bydefining FA/V = −FB/V = 92 sin(νt)ez. Assuming that spheres are well separatedat all times, we employ a far-field approximation to determine the flow field aroundthe swimmer. The velocity of each sphere, i.e. UA and UB, then follows the draglawUA = R−1A ·FA +FA[vB→A], (5.48)UB = R−1B ·FB +FB[vA→B], (5.49)where RA and RB are hydrodynamic resistance tensors for spheres A and B, FAand FB are the Faxèn operators, and vB→A (vA→B) is the background flow fieldon sphere A (B), induced by sphere B (A). Here, to focus only on the leadingorder propulsion velocity, we limit our calculations to the first reflection of the53flow fields. Therefore, we neglect the contribution of the background flow onthe deformation of sphere B. At each step, we take the average velocity of thetwo spheres as the instantaneous velocity of the swimmer, defining U(I) = U(I)A +U(I)B2and U(II) = U(II)A +U(II)B2 , where superscripts (I) and (II) refer to the quantities at thecorresponding steps. Thence, to find the net propulsion velocity we average theswimming velocities over one complete cycleU¯ =1τ(∫ τ/20U(I)dt +∫ ττ/2U(II)dt), (5.50)where τ = 2pi/ν is the period of the cycle. By making use of Eqs. (5.48) and(5.49) and noting that FA(t + τ2 ) = −FA(t) and FB(t + τ2 ) = −FB(t), Eq. (5.50) canbe reduced toU¯ =12τ{∫ τ/20(FA[v(I)B→A]+FB[v(I)A→B])dt +∫ ττ/2(FA[v(II)B→A]+FB[v(II)A→B])dt}.(5.51)From the general description of Faxèn operator [20, 97], we find FA = 1 + 16∇2and FB = 1 +(16 +312282)∇2 +O(3/l2). Now using the asymptotic descriptionsof the flow fields reported in Section 5.2, we arrive at the leading order propulsionvelocityU¯ =249931361923L2ez. (5.52)We note that the deformation of the sphere governs the propulsive thrust and thatthe magnitude of the change in distance between the spheres does not contribute tothe leading order motion, unlike the three sphere swimmer where the difference inarm lengths quantifies the asymmetry [134].5.4 ConclusionIn this chapter, we inquired about the effects of elasticity on swimming in Stokesflow. We started by addressing the pure translation of an elastic particle in viscousfluid. We asymptotically showed that under a body force the translational velocity54of an elastic sphere is slower, and also the shape deformation is not front-backsymmetric. The latter indicates an asymmetry in the surrounding flow field whichcan be exploited to evade the scallop theorem. To highlight the effect of thisdeformation on swimming, we proposed a very simple swimmer of two spheresthat can swim with a reversible actuation, solely due to elasticity of one of thespheres. Our results show that accounting for elasticity of bodies may be crucialto fully understand the dynamics of swimming cells and specifically can be usefulin designing microswimmers. Finally we note that while conceptually simple, ourelastic two-sphere swimmer is not very effective for small deformations, but inpractice one might use an elastic body which is already asymmetric to exacerbatethis effect.55Chapter 6Hydrodynamic interactions ofcilia on a spherical body 16.1 IntroductionTo propel themselves in a low-Reynolds-number regime [157], many microorgan-isms use small whip-like extensions, called flagella (when they possess one or two)or cilia (when they possess many) [19, 124]. The motion of cilia is controlled byATP-fuelled motor proteins which exert a driving force on the cilia by convertingchemical energy in the cell [166]. The cyclic motion of each cilium in a chaincan form a coordinated pattern of beating, wherein each pair of neighboring ciliaare orbiting with a constant, non-zero, phase difference [101]. As a result of thissynchrony, the tips of cilia form a moving wave, known as a metachronal wave [78].By forming metachronal waves, the microorganism minimizes the required energyfor beating [77], which enhances the efficiency of the motion [99]. In addition toproviding a means of locomotion, cilia in the human body filter air flow channelsin the lung from the harmful inhaled material [171], and also play a crucial role inbreaking the left-right symmetry in human embryonic development [87].Several analytical studies have shown that hydrodynamic interactions alone canlead to synchronization (zero phase difference) or phase-locking (constant non-zero1A version of this chapter has been published [136]56phase difference) for model systems of two flagella [51, 191, 192] or many cilia[31, 141]. Using minimal models, it was observed that certain conditions, be theyelastic deformations of trajectory [141], or shape [53], or a certain forcing profile[191], may be required to reach such phase-locking or synchronization. Recentexperimental studies have also confirmed the hydrodynamic synchronization ofmicro-scale oscillators in natural systems. Using high-speed video microscopy,it was shown that beating flagella on Chlamydomonas reinhardtii [67, 152] andVolvox carteri [22, 24], exhibit a synchronization due to hydrodynamic interactions.Similar synchronywas observed inmodel colloidal systemswhere two spheres wereoscillating on linear [26, 102] or circular [18] trajectories and each spherewas drivenby optical tweezers.In a ciliary array, the distribution of cilia as well as the details of the ciliatebody affect the behavior of the dynamical system [25, 54, 73]. Niedermayer etal. reported that introducing radial flexibility to the circular trajectory of twoorbiting beads leads to synchronization, but that a non-periodic array of suchbeads cannot reach stable collective phase-locking [141]. They also showed thatmarginally-stable metachronal waves are formed only when the cilia are distributedin a periodic fashion. More recently it was observed that an open-ended array ofcilia can indeed form robust metachronal waves if the cilia beat perpendicular to theciliate boundary [22]. It has also been shown that the presence of a large body nearan array of linearly oscillating beads is necessary for emergence of metachronalwaves [203]. The bounding surface restricts the range of hydrodynamic interactionsof the beads and leads the system to a collectively phase-locked state. The emergingpicture from the literature is that the stability and existence of metachronal wavesdepends on the geometry of the cilia and ciliate body. Notably however, in manyciliates in nature the cilia are continuously distributed about a closed curved bodysuch as on paramecia or volvox, and this imposes a natural periodicity to thedynamical system and mediates the hydrodynamic interactions of the ciliary chainsin a way that is yet to be understood.In this chapter we investigate the effects of a large curved ciliate body onthe hydrodynamic interactions of cilia in a viscous fluid. Following the work ofNiedermayer et al. who studied interactions of cilia above a flat wall [141], we usethe discrete-cilia model [16, 122] where each cilium is replaced by a single sphere57and assume that a constant tangential forcing is applied by the dynein motors. Wefirst present an analysis of the interactions of two cilia and then build up our modelof a chain of cilia around a large spherical body. We show that the radial flexibilityin the trajectories can lead the system to synchronize similar to the case of cilia neara flat boundary [141]. Furthermore, we show that with this model, the only stablefixed point for a chain of identical cilia is when all cilia are in phase (synchronized).Finally, we demonstrate an emergent wave-like behavior of the cilia in response toan imposed asymmetry in the beating rate of one cilium.6.2 Motion of a single ciliumWe model the cilium as a single sphere of radius aˆ undergoing a circular orbit, ofradius Rˆ, whose center is at distance hˆ from the ciliate body as shown in Fig. 6.1.Dynein motors drive the motion of the cilium and, in a viscous fluid at small scales,this forcing is balanced entirely by the hydrodynamic drag,Fˆm + FˆD = 0. (6.1)A simple model of the forcing stipulates a constant tangential driving force [113],Fˆdr, and an elastic restoring force that keeps the cilia moving along a preferred path(of radius Rˆ0) [141], such thatFˆm = Fˆdreφ − kˆ(Rˆ− Rˆ0)eR, (6.2)where kˆ is the stiffness of the cilia. The drag force, FˆD , for the rigid body translationof a sphere at velocity Uˆ is given by the drag lawFˆD = −RˆFU ·(Uˆ−F [uˆ∞]), (6.3)where u∞ is the background flow and the Faxén operator is F = 1 + aˆ26 ∇2[11]. The resistance tensor for a sphere moving parallel to a wall is RˆFU =6piµˆaˆ(I+O(aˆ3/hˆ3))[84]. In our analysis, we assume the thickness of a ciliumis much smaller than its length so that in our minimal model aˆ hˆ, therefore theeffect of wall on the hydrodynamic resistance shall be neglected.58hˆUˆeReRˆuˆ1Figure 6.1: Aschematic of themotion of amodel cilium near a spherical body.The circular trajectory has a radius of Rˆ and its center is at distance hˆfrom the boundary. The cilium moves with velocity Uˆ through a fluidwith velocity uˆ∞. In this study, φ indicate the instantaneous phase ofthe cilium and the vectors eφ and eR show the tangential and radialdirections of the motion, respectively.As a starting point, we examine the motion of a single cilium in the absence ofother cilia. The background flow field is then zero and the cilium orbits strictly onits preferred circular path. In this case, Eq. (6.2) leads to a steady state solution˙ˆφss =Fˆdr6piµˆaˆRˆ0≡ ωˆ, (6.4)˙ˆRss = 0, Rˆss = Rˆ0, (6.5)where the over-dot indicates differentiation with respect to time and ωˆ defined as theintrinsic angular velocity of the cilium. Using the reported values for the bendingrigidity of a cilium [27, 143], Niedermayer et al. [141] noted that radial relaxationis much faster than a period of rotation, namely Fˆdr /kˆRˆ0 1, and so a quasi-staticassumption may be employed for the radial dynamics of the system.59Odˆ12✓12(1) (2)Figure 6.2: A system of two cilia around a spherical body of radius Aˆ. Inthis figure, dˆ12 is the distance and θ12 is the angle between center of thetrajectories.6.3 Interactions of two ciliaWe now consider a system of two cilia around a spherical body, with trajectoriescentered at a distance dˆ12 as shown in Fig. 6.2. The velocity of each cilium can bewritten as Uˆi = Rˆi˙ˆφieφi +˙ˆRieRi , where i ∈ {1,2}. The motion of each cilium inthis case is affected by the background flow field induced by the other cilium. Theciliate body is considerably larger than the thickness of a cilium, aˆ Aˆ, where Aˆis the radius of the spherical body. We also assume the cilia are far apart from oneanother (dˆ12  hˆ, Rˆ0) [143], so that far-field approximations for the induced flowfields may be employed. Under these assumptions one can model the flow field dueto the motion of a sphere by a point force (or Stokeslet) to leading order while theno-slip boundary condition on the surface of the spherical body is satisfied by animage Stokeslet set in the body. The background flow field on cilium (1), inducedby cilium (2) isuˆ∞(xˆ1) =18piµˆ(Jˆ(xˆ1, xˆ2) + Jˆ∗(xˆ1, xˆ∗2))· Fˆm2 , (6.6)where Fˆm2 refers to the driving force of the cilium (2), xˆ1 and xˆ2 indicate thelocation of each cilium, Jˆ is the Oseen tensor and Jˆ∗ is the Blake’s solution for theimage Stokeslet [17, 84], at a point xˆ∗2 = (Aˆ2/ |xˆ2|2)xˆ2 located to satisfy the no-slip60condition at the spherical boundary [181].Before going further, we non-dimensionalize all equations by scaling lengthsby the radius of the spherical body, Aˆ, and rates by the average angular velocityof the two cilia, ¯ˆω. We drop the notation ( ˆ ) for the dimensionless quantitiesdefined by aˆ = Aˆa, Rˆj = AˆRj , hˆ = Aˆh, dˆ12 = Aˆd12, ωˆj = ¯ˆωωj and˙ˆφj = ¯ˆωφ˙j ,Uˆj = (Aˆ ¯ˆω)Uj , ˙ˆRj = (Aˆ ¯ˆω)R˙j , tˆ = (1/ ¯ˆω)t and RˆFU = (kˆ/ ¯ˆω)RFU . A dimensionlessparameter κ = kˆ/6piµˆaˆ ¯ˆω 1 which indicates the ratio of the elastic restoring forceto the hydrodynamic drag force then naturally arises. We assume the dimensionlesslength scales are ordered as follows, a {h,R0}  1 and take R0 = O(h) [143].Since the radius of the trajectory is small compared to the scale of the body, wemay write d12 = 2sin(|θ12|/2) +O(h).In these limits, by using the description for background flow field in Eq. (6.6),the motion Eq. (6.2) can be solved for case of two neighboring cilia, asymptotically.The evolution equations are then to leading orderφ˙1 =ω1 + ρω2S12 − ρω1ω2κ L12, (6.7)φ˙2 =ω2 + ρω1S21 − ρω1ω2κ L21, (6.8)R1 = R0 +ρR0ω2κL12, (6.9)R2 = R0 +ρR0ω1κL21, (6.10)where ρ = 9ah2/8 is the strength of the hydrodynamic interactions dictated by thefunctionsSij =4d3ij[Θij cos∆ij +Φij cosϕij], (6.11)Lij =4d3ij[Θij sin∆ij +Φij sinϕij]. (6.12)Here we’ve defined phase difference ∆ij = φi −φj and sum ϕij = φi +φj whilethe distance between cilium is given by dij = 2sin(|θij |/2) where θij is the angle610 π 2π01θ0 π 2π−11θ⇥ ✓ ✓(a) (b)⇡ 2⇡⇡ 2⇡Figure 6.3: Geometric terms (a)Θ and (b)Φ as functions of the angle betweencilia.between the cilia i and j. Finally the functionsΘij =cos |θij |+ sin(|θij |/2)1 + sin(|θij |/2) , (6.13)Φij =cos |θij | − sin(|θij |/2)1 + sin(|θij |/2) , (6.14)capture the effect of the geometry of the spherical body on the hydrodynamicinteractions, as shown in Fig. 6.3.We observe that as expected hydrodynamic interactions above a wall scale asO(d−3ij ) [113, 192], but now, due to the spherical shape of the ciliate body, both therelative position, ∆ and average position ϕ, of the two cilia around the ciliate bodyaffect the hydrodynamic interactions as well by way of the geometric functionsΘ and Φ respectively. The background flow velocity on each cilium (induced bythe other) directly impacts both the angular velocity of each cilium, through thefunction Sij , as well as its radial position via the function Lij , which in turn affectsthe phase-speed as well.Taking the difference of (6.7) and (6.8) we obtain an evolution equation for thephase difference∆˙12 = ∆ω(1− ρS12)− 2(ρ/κ)ω1ω2L12, (6.15)where∆ω =ω1−ω2. We see the phase difference evolves due to a difference in the62(1) (2)(1) (2)Figure 6.4: The effect of the background flow field on the motion of eachcilia. The two cilia are orbiting clockwise, cilium (2) is ahead, thus itsinduced flow field pulls cilium (1) to a smaller radius of trajectory whichincreases the instantaneous velocity of cilium (1). On the other hand,the velocity of cilium (2) decreases as the flow field of cilium (1) pushescilium (2) to a larger trajectory. In this figure arrows show the flow fieldinduced by each cilium.intrinsic phase-speed, due to hydrodynamic interactions directly but also indirectlybecause of elastic radial displacements. To illustrate the latter point, let us assumecilium (2) is ahead by a positive phase difference of ∆12, as shown in Fig. 6.4.In this case, the background flow drives a contraction of the orbit for cilium (1)(R1 < R0) and expansion of the orbit for cilium (2) (R2 > R0). Since the internaldriving forces of the cilia are constant, the changes in trajectories speed up cilium(1) and slow down cilium (2).If intrinsic velocities are different, ∆ω , 0, for an equilibrium phase differenceto arise, this difference must not overwhelm the elasto-hydrodynamic interactions,in other words ∆ω = O(ρ) for fixed points in phase-difference. The second termon the right-hand side of Eq. (6.15) is then O(ρ2) and shall be neglected.To provide further clarity we note that the individual phases evolve on a much63shorter time scale than the phase-differences, φ˙i =O(1) while ∆˙12 =O(ρ); hence,we use a multiple scale analysis and average over a period of the short-time scale,τφ= 2pi/ω1 +O(ρ) to focus on the long-time behavior of the phase difference(indicated with an overbar). The cycle-averaged evolution equation for the phase-difference is then an Adler equation˙¯∆12 = ∆ω −γΘ12 sin ∆¯12, (6.16)where the synchronization strength in the case of a flat wall γ = 8(ρ/d3κ)ω1ω2[141], is augmented by the geometric term Θ12. If the frequency mismatch issmall enough to be balanced by the elasto-hydrodynamic coupling, |∆ω| < γΘ12,a steady-state phase difference emerges given by ∆¯eq12 = sin−1(∆ω/γΘ12). In thelimiting case of a rigid cilium (κ→∞) hydrodynamic interactions do not lead toan evolution of phase, and no phase-locking can occur where ∆ω , 0. When thecilia are identical (i.e., ω1 =ω2 = 1) the phase-locking of the system is guaranteed(if θ12 , pi) as Equation (6.16) reduces to˙¯∆12 = −γΘ12 sin ∆¯12, (6.17)indicating that the equilibrium phase difference is zero.Unsurprisingly, the evolution equations for phase difference on a spherical bodyare largely similar to above a flat wall under the assumption that the cilia are muchsmaller than the ciliate. The difference is that the hydrodynamic interactions aremediated by the geometry of the body throughΘ12. We see that when the two ciliaare located at the opposite sides of the spherical body (θ12 = pi), radial interactionsare completely screened by the ciliate as Θ12 = 0. We also note that for the anglesnear zero (and 2pi), special care must be used as these limits force d12 → 0. Toevaluate the system at these angles we can rescale the evolution Eqs. (6.7) and (6.8)with distance dˆ12, thereby recovering the flat body solution reported in Ref. [141]in the limiting case where θ12→ 0 (or 2pi) and Aˆ→∞.646.4 Interactions of chain of ciliaNow we proceed to the system of N identical cilia around a spherical body whereN ≥ 3. Relying on the linearity of the Stokes equation, the flow field induced bya chain of cilia can be determined by summing the contributions of all the cilia aswell as their image points. Following the procedure outlined in the case of twocilia, the evolution equation for cilium (i) in a chain ofN cilia, to the leading order,isφ˙i =ωi + ρN∑j,iωjSij − ρωiκN∑j,iωjLij , (6.18)where {i, j} ∈ {1,2, . . . ,N }. For simplicity, we shall assume first that all cilia in thechain have the same intrinsic angular velocity, ωi = ωj = 1. The evolution of thephase difference on the long time scale is then˙¯∆1i = 4ρN∑m,1N∑j,i(d−31mΘ1m cos ∆¯1m − d−3ij Θij cos ∆¯ij)− 4ρκN∑m,1N∑j,i(d−31mΘ1m sin ∆¯1m − d−3ij Θij sin ∆¯ij), (6.19)where we have set cilium (1) as the reference phase, defining ∆1i = φ1 −φi . Wenote that unlike the case of two identical cilia where the average tangential terms donot contribute to synchronization (because of a pair-wise symmetry), in the case ofmany cilia the tangential terms (Sij) do not vanish and hence contribute to evolutionof the phase differences.To further simplify the system, we now consider a chain of cilia which areequally distributed around the body and hence the angle between any two cilia isθij = 2pi(i−j)/N . By direct substitution into Eq. (6.19), one can show that an equalphase-difference, ∆eq, between all neighboring cilia is a fixed point of Eq. (6.19).Because the system is periodic (in θ), the sum of the phase differences must be aninteger multiple of 2pi,∆eqN = 2piM, (6.20)65where M ∈ Z for which there are only N unique solutions (due to periodicity inφ), synchronized (whenM = 0) or metachronal waves (whenM , 0).The strength of the interactions between a pair of cilia decays rapidly as theirdistance increases, due both to the d−3ij term as well as the effect of the angle,thus, we perform a linear stability analysis of these equilibrium states of the systemconsidering only nearest neighbor interactions. Without loss of generality, weassume −pi ≤ ∆eq ≤ pi. Using Gaussian elimination, we determined the maximumeigenvalues of the Jacobian at ∆eq, asλ1 = 4d−312Θ12[(ρ/κ)(αsgn[cos∆eq]− 2)cos∆eq+αρ sin |∆eq|], (6.21)where α(N ) ∈ [0,2) is a constant which depends on the number of cilia as shown inFig. 6.5. As an example, when N = 3 the angle between each pair of cilia is 2pi/3.In this case α = 0 hence the system has a stable equilibrium only if cos∆eq > 0and of the possible solutions ∆eq = 0,±2pi/3 only ∆eq = 0 is stable. When thecilia are all in-phase, ∆eq = 0, λ1 < 0 for any N , indicating asymptotic stabilityof the synchronized state for any number of identical, evenly distributed cilia on aspherical ciliate provided the system has finite flexiblity. In the rigid limit, κ→∞,the largest eigenvalue is zero which causes a loss of stability of the synchronizedstate (as shown in numerical simulations).For a system to form metachronal waves, a non-zero equilibrium phase dif-ference between the cilia is required. However, one can show directly that ifpi/2 ≤ |∆eq| ≤ pi, λ1 > 0 while when 0 ≤ |∆eq| < pi/2 for stability one must have theintegerM <N2pitan−1(2−αακ), (6.22)which is satisfied only by M = 0 for κ > 1 (and here κ  1). Thus, all non-zero values of M (metachronal waves) are linearly unstable for any number ofidentical, evenly distributed cilia on a spherical ciliate. Unlike the reported resultsfor the chain of cilia near a flat boundary [141], this system cannot form a stablemetachronal wave and all the cilia eventually synchronize. The synchronization of663 20 40 60 80 100012Nα↵Figure 6.5: The value of the coefficient α, which dictates the stability of fixedpoints of a ciliary chain, is shown as a function of the given number ofcilia N .two chains of N = 10 and N = 15 cilia from random initial conditions has beenillustrated numerically in Fig. 6.6.In real biological examples one can hardly expect perfect symmetry and uni-formity in the system so it is important to understand the effect of an imposedasymmetry on the stability of this system. There are several well documentedsources of asymmetry, from biochemical noise [67, 195], to the different intrinsicproperties of a developing cilium [68, 83] or even the addition of a transverse ex-ternal flow [80] which have all been found to spontaneously affect the behavior ofa ciliary system. In particular, the beating rate of a developing cilium fluctuates asit grows, which can perturb the equilibrium state of the system [83]. To analyzethis phenomenon, we impose an asymmetry on the system by increasing the intrin-sic velocity of cilium (1) to 1 +∆ω, where we assume ∆ω  1. The evolution670 0.5 1−π−π/20π/2πT∆i10 0.5 1−π−π/20π/2πT∆i1(b)(a)1i1iTFigure 6.6: Synchronization of a chain of (a) 10 and (b) 15 identical ciliadistributed uniformly around a spherical body, with the random initialphases. Each line indicates the evolution of the phase difference foreach cilium i compared to cilium (1), ∆1i = φ1 − φi , over the timeT = t/(κ/ρ). These plots are the numerical evaluation of Eq. (6.18) atthe characteristic values of ρ = 3.6×10−6, κ = 100 and ¯ˆω = 20pi rad.s−1[27, 143].680 −π/2 π 3π/2 2π0π/4π/23π/4θi1  N = 15N = 10✓1ieq1iFigure 6.7: Phase differences of nearby cilia in a chain of 10 and 15 ciliaaround a spherical body when the intrinsic angular velocity of cilium(1) is higher compared to the other cilia by ∆ω = 10−6 for both cases.equations for the phase differences are then˙¯∆1i = ∆ω+ 4ρN∑m,1N∑j,i(d−31mΘ1m cos ∆¯1m − d−3ij Θij cos ∆¯ij)− 4ρκN∑m,1N∑j,i(d−31mΘ1m sin ∆¯1m − d−3ij Θij sin ∆¯ij). (6.23)Now, due to the imposed asymmetry, the system no longer converges to a synchro-nized state where phases are equal. There must be a non-zero equilibrium phasedifference between cilium (2) and cilium (1) (for example) to balance the differencein the intrinsic velocities. However, the effect of the imposed asymmetry becomesweaker as the distance from cilium (1) increases and, therefore, the phase differencebetween a two adjacent cilia decreases. These phase differences form a coordinatedsystem of beating, which is illustrated in Fig. 6.7 for two sample cases of N = 10and N = 15.69These results indicate that the system responds to this asymmetry through anemergent wave-like behavior. Since this asymmetry arises from any developingcilium in the chain, these waves can originate from different parts of the ciliateand vanish once the beating rate of the developing cilium reaches the frequencyof the other cilia. Here we should note that unlike metachronal waves which have∼ 7 − 10 cilia per wavelength [127, 203], the asymmetry-induced behavior has acharacteristic wavelength which spans the entire chain of N cilia. Furthermore,as N increases, the strength of the imposed asymmetry becomes weaker and theequilibrium phase differences of the cilia decrease. Thus, the amplitude of suchwaves is inversely proportional to the number of cilia in the chain, as shown inFig. 6.7.6.5 ConclusionIn this chapter, we used a minimal model to capture the dynamics of cilia on aspherically shaped microorganism. We showed that, similar to the case of ciliaabove a flat wall, elasto-hydrodynamic interactions can lead to synchronization,however here the interactions are additionally mediated due to the geometry of theciliate body.For a chain of identical cilia uniformly distributed around a spherical boundary,we showed that the natural periodicity in the geometry of the ciliate leads the systemto synchronize. We also showed that in this system, metachronal waves are strictlyunstable fixed points of the dynamical system unlike in the case of interaction abovea flat wall. This result suggests that the geometry of the ciliate plays a crucial rolein the behavior of the ciliary chain and it has to be accounted for when analyzingmicroorganisms with curved bodies and suggests that a natural extension of thisanalysis would be to investigate a distrubtion of cilia over the whole surface of theciliate. Our results also suggest that to form stable metachronal waves, rotationand translation of the ciliate [57], elasticity of the cell-internal fibers connectingthe cilia [159], or motion of the cilia perpendicular to the ciliate body [22], may benecessary in such microorganisms. We also reported a wave-like response of thesystem when one of the cilia is intrinsically faster. In this case, the neighboringcilia display stable phase-locking with the faster cilium with a phase difference70that decreases with distance from the asymmetry. Although the characteristics ofthis asymmetry-induced phenomenon do not match metachronal waves, we shouldnote that in real ciliary chains there are likely many cilia of differing lengths orbiochemical noisewhichmay lead tomore complex dynamics in biological systems.71Chapter 7Dynamics of poroelastocapillaryrise7.1 IntroductionInteractions of capillary forces and elastic materials are abundant in nature: Bundleformations in bristles of a wet painting brush or in wet eyelashes [14, 46], stronghydrophobic interactions of the feathers of aquatic birds [164], or the fluid-mediatedadhesion of a beetle to a substrate [48] are just a few examples amongst the many.These interactions, referred to as elastocapillary effects, have shown to be a keyfactor in collapsing [126] or fabricating [32, 114] engineered microstructures, in thelubrication of soft materials [89, 176] and can be exploited for ultra-thin whitening[29]. Our understanding of capillary rise dates back to experiments of Newton(1704), Jurin (1712), and the analysis of Laplace (1805). When a small tube is incontact with a wetting fluid, capillary forces drive the liquid into the tube until theyare balanced by the gravitational forces. In a seminal work, Washburn [197] showedthat in capillary rise the balance of surface tension forces and viscous dissipationgoverns the rate of fluid motion.The coupling of surface tension forces and elastic forces can lead to surfacedeformations when liquid is in contact with elastic media, for instance by formingwrinkles [90] or a ridge [150]. In particular, capillary-induced self-assembly of thinflexible materials has attracted extensive attention, owing to recent developments72in micro- and nano-engineering (see [15] and the references therein). To shed somelight on such interactions, Kim and Mahadevan [96] notably studied the capillaryrise of a liquid between two flexible sheets, clamped at one end and free at theother, and analytically characterized the equilibrium configurations. Duprat et al.[47] then complemented this analysis by looking into sheet deformation prior toequilibrium and developed a framework for capturing the dynamical behavior ofthis elastocapillary rise (see also [7]). Subsequently, several studies have furtherextended this model by, for instance, investigating its multiple equilibria [187],considering a series of sheets [60, 174] or by employing the model to enhance thecapillary flow in microchannels [6]. In all of these studies, the sheets have beentaken as impermeable entities whose properties remain constant upon wetting.For example, paper sheets are often permeable as liquid may diffuse through andchange their properties significantly [112]. For instance, water softens a papernapkin as it flows between the plies, which may affect its absorbency. Given thatsuch absorbency is important in paper products used in household and diagnosticapplications [128], here we try to quantify the effect of sheet permeability onelastocapillary rise, as a simple model of, for instance, flow between plies of apaper towel.Paper consists of multiple layers of cellulose fibers. Each fiber has an internalcavity of half of its size, and the surrounding wall is closely packed with hydrophilicmicrofibrils [189]. When infiltrated by water, before filling the cavities, the liquiddiffuses within the microfibrils and causes expansion and swelling [55]. Imbibitionof water into cellulose sponges [82, 103], swelling of two parallel sheets submergedinto a liquid bath [88], and self-rolling of a piece of paper immersed into water [163]are all examples of this phenomenon. In a recent study, Lee et al. [112] characterizeddeformation and stiffness of a strip of a paper when it imbibes a stain of water dueto surface tension forces. They showed that by absorbing water, the paper sheetswells, increasing its thickness by ∼ 25%, while simultaneously decreasing itsYoung’s modulus from Eˆ = 828 MPa to Eˆ = 24 MPa. This significant change ofstiffness, which occurs due to imbibition of water by fiber-based materials, can alterthe papers absorptivity, and so is quite important in painting [28] and diagnosticapplications [129]. Thus, to understand the effect of this permeability on capillaryrise, in this chapter, we consider the elastocapillary rise of a liquid (e.g., water)73between two paper sheets. The paper sheets are permeable and they become softeras the liquid rises and permeates through. To study the system’s behavior, weclosely follow the work of Duprat et al. [47], but modify their model to incorporatethe permeability of the sheets by allowing the properties to change upon liquidimbibition. We discuss the dynamics of sheet deformation and the equilibriumstates, and compare them with those of impermeable sheets. We show that due tothe softening of the sheets, the absorbency of the system with permeable sheets isreduced compared to the system with impermeable ones.The outline of this chapter is as follows. In Section 7.2, we present the detailsof the system and perform a scaling analysis. In Section 7.3, we derive evolutionequations for the sheet deformation and the meniscus position. We then solvethese equations using a finite difference scheme given in Section 7.4. Finally, inSection 7.5, we discuss the results, compare them with those of impermeable sheetsand our experimental observations.7.2 Problem StatementWe are interested in the capillary rise of a viscous fluid between two permeableelastic paper sheets. We consider sheets of length lˆ, thickness bˆ, width wˆ andassume they are separated initially by distance 2hˆ0, as shown in Fig. 7.1(a). Thesheets are clamped at the upper end (i.e., zˆ = lˆ) and immersed into the liquid bathfrom the lower free end (i.e., zˆ = 0). The liquid then rises vertically (in ez) and werefer to the meniscus position as zˆm. The sheets are elastic and deform as the liquidrises and we quantify the deflections by hˆ(zˆ, tˆ) which varies from hˆ0 (no deflection)to zero (coalescence). The behavior of the system is dominated by three forces:surface tension forces drive the flow, gravitational forces resist the rise of the liquid,and finally the elastic forces account for the sheet deformations. We characterizethese forces using two dimensionless groups, namely the elastocapillary number E,which compares surface tension forces to elastic forces, and the Bond number B,74Initial State Regime I Regime II Regime IIIClamped End Wet Sheets Dry Sheets Liquid(b)(a)lˆwˆezexbˆwbˆdhˆzˆ2hˆ0zˆmFigure 7.1: The schematic of the system: (a) Two poroelastic sheets clampedat the upper end are immersed into a liquid bath from the lower end.(b) Three scenarios of the equilibrium. In regime I, sheets only slightlybend. In regime II, sheets lower end are in contact and in regime III,sheets coalesce over a finite length.which compares the effects of gravity and surface tension. We defineE = γˆ lˆ4Bˆhˆ20, (7.1)B = ρˆgˆ lˆhˆ0γˆ, (7.2)where γˆ is the surface tension, ρˆ is density, gˆ is the magnitude of gravitationalacceleration, and Bˆ = 112 Eˆbˆ3 is the bending stiffness per unit width with Eˆ beingthe Young’s modulus of the sheet. Duprat et al. [47] showed that depending onvalues of B and E, the system can exhibit three different configurations: the sheetsbend but do not touch (hˆ(zˆ = 0) > 0, regime I), they deflect such that they touch butdo not coalesce (hˆ(zˆ = 0) = 0 and ∂hˆ∂zˆ (zˆ = 0) , 0, regime II), or they coalesce overa finite length (regime III), as shown in Fig. 7.1(b).Since the sheets are permeable, as liquid rises due to surface tension forces, italso permeates through the sheets and changes their properties. Thus, to accountfor such changes, in our model we consider different properties for wet and dryparts of the sheets and henceforth refer to them using subscripts ‘w’ and ‘d’ as75{Bˆw,Ew, bˆw, · · ·}and{Bˆd ,Ed , bˆd , · · ·}.Scaling analysisThis dynamical system involves several time scales and to simplify the problem itis important to compare them. The classic time scale over which fluid rises (τˆr), isfound by employing a steady unidirectional (Poiseuille) flow field u = ∆pˆhˆ20/(3µˆlˆ)and noting that at equilibrium pressure must balance gravity ∆pˆ = ρˆgˆ lˆ and thatthe pressure is set by surface tension ∆pˆ = γˆ/hˆ0 so that capillary rise time scaleτˆr = lˆ/ uˆ = 3µˆγˆ/hˆ30(ρˆgˆ)2. When the elastic deformation of the sheets is appreciable,Duprat et al. [47] argued that the pressure scale is instead set by the deflection ofthe sheets∆pˆ = Bˆhˆ0/ lˆ4 so that in this case the ‘visco-elastic’ time scale τve = lˆ/ uˆ =3µˆlˆ6/Bˆhˆ30 = B2E τˆr . Duprat et al. [47] found that this occurs when B2E & 10.Furthermore, at the moment when the sheets touch the liquid, inertia is clearly notnegligible, and in fact dominates the capillary rise [160]. Das et al. [34] showedthat this inertia-dominated regime prevails at tˆ < τˆc, where τˆc =√ρˆhˆ30/γˆ (seealso [33]). In our problem fluid also permeates through thickness of the sheetas well as through the sheet vertically. Considering Washburn-like behavior forthe fluid permeation in the sheet [197] the time scale for lateral fluid permeationτˆD,x = bˆ2/Dˆw and vertical fluid permeation τˆD,z = lˆ2/Dˆw, where Dˆw is the isotropicdiffusion coefficient of the paper.To compare these time scales, we use properties of a filter paper sheet reported in[112] and consider water as the viscous fluid. We also take lˆ = 1 cm and hˆ0 = 0.5mm as typical values for the sheet length and gap size, respectively. We findτˆc/τˆr ∼ 10−2, so we may neglect the effect of inertia since it vanishes very fast. Wealso find τˆD,x/τˆr ∼ 10−2 indicating that fluid diffuses through the thickness of thesheet considerably faster than the capillary rise. Thus, we assume liquid permeatesthe sheet (in its thickness), instantly. We also find τˆr /τˆD,z ∼ 10−3, implying thatfluid diffusion along the length of the sheet is very slow and so is negligible at thetime scale of the rise. Relying on these scalings, one can assume that paper sheetis wet in the liquid-filled region and dry in the liquid-free region. Thus, given ameniscus position zˆm, we consider properties of the wet paper sheet for 0 ≤ zˆ ≤ zˆmand dry one for zˆm < zˆ ≤ lˆ.767.3 Governing EquationsHere, we take lˆ as the characteristic vertical length scale, hˆ0 as the characteristicdeflection of the sheet, and τˆve as the characteristic time scale of the problem. Wethereby non-dimensionalize all the quantities defining z = zˆ/ lˆ, zm = zˆm/ lˆ, h = hˆ/hˆ0,t = tˆ/ τˆve and p =pˆBˆwhˆ0/ lˆ4. Note that we have dropped the caret notation for thedimensionless parameters. We assume that a reflection symmetry between thesheets is maintained and so we derive the governing equations for one sheet (e.g.,the one on the right in Fig. 7.1(a)) and the other side shall be identical. In modelingthe system, we closely follow the works of Stone et al. [7, 47], but incorporatethe change of properties due to wetting. Since the properties of the sheet have adiscontinuity at the meniscus, we treat the wet and dry parts separately and enforceboundary conditions at the interface.Noting that hˆ0  lˆ, we employ the lubrication approximation to express thefluid motion. The one-dimensional momentum equation then dictatesu = −h2w(∂p∂z+BEw), (7.3)where u is the vertical component of the meniscus velocity and all variables areaveraged across the gap. Although the deflection of the sheet is a continuousfunction, we differentiate the deflection of the wet (hw, when z ≤ zm) and dry(hd , when z > zm) parts for clarity. As noted earlier, the time scale of the liquidpermeation through the sheet thickness is considerably smaller than the one of therise. Thus, one can assume that the liquid saturates the sheets thickness instantlyas the meniscus rises. Furthermore, because the sheet is very thin (bˆw  hˆ0),we neglect the mass of liquid permeated within the sheet. One-dimentional massconservation then yields∂hw∂t+∂∂z(hwu) = 0. (7.4)Provided the sheet is sufficiently thin and long (i.e., bˆ wˆ lˆ), we may use linear77elasticity to approximate the quasi-static sheet deflectionp =∂4hw∂z4. (7.5)Substituting pressure (7.5) to (7.3), we findu = −h2w(∂5hw∂z5+BEw), (7.6)which at z = zm gives the time evolution of the meniscus since u(z = zm) =dzmdt .Now by making use of Eq. (7.6), we can rewrite the continuity equation (7.4) interms of hw and its derivatives as∂hw∂t= h2w(3∂hw∂z∂5hw∂z5+ 3BEw∂hw∂z + hw∂6hw∂z6). (7.7)For z > zm, there is no pressure gradient across the sheet and so we have0 =∂4hd∂z4. (7.8)Boundary conditionsTo account for the fixed end at z = 1, we set hd(z = 1) = 1 and∂hd∂z (z = 1) = 0. Atthe lower end (z = 0), we note that there exists no net pressure or moment on thesheet so ∂4hw∂z4(z = 0) = ∂2hw∂z2 (z = 0) = 0. Furthermore, in regime I, wherein sheetscannot apply any force on each other ∂3hw∂z3 (z = 0) = 0. In regime II, the sheets areno longer force free at z = 0 but instead we have hw(z = 0) = 0. For regime III, thisboundary condition changes to hw(z = zc) = 0 and∂hw∂z (z = zc) = 0, where zc is thelength of the coalescence (i.e., hw(z < zc) = 0).At the interface z = zm, continuity of sheet deflection and its slope are enforcedby hw(z = zm) = hd(z = zm) and∂hw∂z (z = zm) =∂hd∂z (z = zm). Force and momentmust also be continuous across the interface, thus Bw∂3hw∂z3 = Bd∂3hd∂z3 , and Bw∂2hw∂z2 =Bd∂2hd∂z2 . Finally, due to surface tension forces, there exists a pressure jump at themeniscus, which can be found using the Young-Laplace equation [37] and can be78written in dimensionless form as∂4hw∂z4− ∂4hd∂z4= −Ewhw, (7.9)where we assume the contact angle is zero and neglect the effects of a dynamiccontact angle.In summary, we have a coupled system of sixth-order non-linear Partial Differ-ential Equations (PDEs) subjected to two boundary conditions at the clamped end,three (or four in regime III) conditions at the free end and one condition account-ing for the pressure jump at the meniscus. Additionally, we have four continuityconditions that need to be satisfied at the interface.To solve this problem numerically, it is convenient to incorporate the dynamicsof the liquid-free region into the boundary conditions for the liquid-filled region,given the simplicity of the governing equations in the liquid-free region [7, 47]. Toillustrate, from Eq. (7.8), one can find the deflection and the slope at the liquid-freeregion ashd = A3z3 +A2z2 +A1z+A0, (7.10)∂hd∂z= A6z2 +A5z+A4, (7.11)where A0 to A6 are determined using the boundary conditions at the fixed end(z = 1) and the interface (z = zm). hd and∂hd∂z are then found in terms of hw and itsderivatives at z = zm ashd(z) = 1 +Br6(1− z)2[3∂2hw∂z2+∂3hw∂z3(z − zm) + 2∂3hw∂z3(1− zm)], (7.12)∂hd∂z(z) = −Br2(1− z)[2∂2hw∂z2+∂3hw∂z3(z − zm) + ∂3hw∂z3(1− zm)], (7.13)79where Br =BwBd. Noting that hw = hd and∂hw∂z =∂hd∂z at the interface, we then findhw(z = zm) = 1 +Br3∂3hw∂z3(1− zm)3 + Br2∂2hw∂z2(1− zm)2, (7.14)∂hw∂z(z = zm) = −Br2∂3hw∂z3(1− zm)2 −Br ∂2hw∂z2(1− zm), (7.15)which provides two boundary conditions for hw at z = zm. Now, the governingequations in the liquid-filled region are independent of hd , and so we can determinethe behavior of the sheet by solely solving the system for z ≤ zm. Once hw is found,we use Eq. (7.12) to find the deformation for the whole sheet.7.4 Numerical ApproachAt early times, t  1, one may consider a quasi-static deformation of the sheet(∂hw∂t = 0) since sheet deflections are very small and time enters the problem onlythrough boundary conditions [47]. We expand the deflection hw(z) =∑n=5n=0Cnzn+O(z6m), where zm  1. From the pressure drop at the meniscus given in (7.9), wefind C5 = − Ew120zm . Boundary conditions at z = 0 dictate C4 = C2 = 0 and, recallingthat in this limit sheets can only reach regime I, we have C3 = 0. Finally, from theboundary conditions given in (7.14) and (7.15), we arrive athw = 1− Ew120zm z5 +(Ewz3m24+Edz3m12− Edz2m3+Edzm4)z+(−Ewz3m30+Edzm6− Ed6)zm +O(z6m), (7.16)which governs the sheet deflection in the liquid-filled region for t 1. Substitut-ing hw from (7.16) in (7.6), we find the leading-order evolution equation for themeniscus asdzmdt=Ewzm−BEw. (7.17)To solve the dynamical system in full, we use Eqs. (7.16) and (7.17) for early times,and then to determine the behavior of the system at later times, we follow thework ofAristoff et al. 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(b) Twoimpermeable sheets with Ed = 10 and Ed = 100 and permeable sheetswith Ed = 10 (or Ew = 100). In (b), for all cases B = 3 and solid blackline refers to the classical capillary rise [197]. Circles on each lineindicate the time in which sheets reach regime II (h(t, z = 0) = 0).order accurate in space and first-order accurate in time. To resolve the nonlinearterms in Eqs. (7.6) and (7.7) (e.g., h2w∂hw∂z∂5hw∂z5 ), we discretize the higher-order term(∂5hw∂z5 ) and then use the results of the previous time-step for the lower-order terms(hw and∂hw∂z ). We repeat the procedure iteratively until the relative convergenceerror reaches below 10−5. We discretize the sheet length in the liquid-filled regionusing 30 points and take ∆t = 10−3 as the typical time step. Recalling that in thescale of the considered problem we found τˆc/τˆr ∼ 10−2, we neglect inertia and takehw(t = 0) = 1 (zero deflection) and zm(t = 0) = 10−3 (negligible inertial meniscusrise) as initial conditions.7.5 Results and DiscussionAt early times (t  1) when the sheet deflection is not yet significant, elasticitydoes not contribute to the dynamics of the meniscus, nor does the permeability ofsheet. Thus, regardless of the values of Ed andBr(= BwBd =EdEw), themeniscus strictlyfollows the classical behavior of a simple capillary rise in a rigid channel given by8115 25 355B = 1B = 100.60.810.40.2EdRIII/IIII/III/IIII/IIII/IIFigure 7.3: The liquid absorption ratio, R, of permeable sheets comparedto impermeable ones of the same Ed for B = 1 and B = 10. Dashedlines separate the different combinations of equilibrium regimes forpermeable and impermeable cases and symbols denote the regimes (e.g.,II/I indicates that permeable sheets reach regime II while impermeablesheets are at regime I).zm =√2Ewt (note that in the dimensional form Ew disappears). Defining a rescaledmeniscus position as z∗m =√1/(2Ew)zm (or equivalently z∗m =√Br /(2Ed)zm), onecan see from Fig. 7.2(a) that z∗m =√t predicts the initial behavior of z∗m quite accu-rately. This behavior can also be explained using the asymptotic expressions givenin (7.16) and (7.17). At leading order, we find hw = 1 indicating no deflection andso the problem is reduced to capillary rise between two rigid sheets. Furthermore,since at early times zm  1, meniscus dynamics can be approximated to leadingorder as dzm/dt = Ew/zm (or dz∗m/dt = 1/(2z∗m)) confirming the diffusive behavior.At later times, the sheet deformation becomes appreciable and the meniscus posi-tion no longer follows the classical capillary-rise predictions. Once the lower endsof the sheets are in contact (denoted by a circle in Fig. 7.2(a)), after a short periodof almost stationary position, the meniscus rises with z∗m ∼ t1/13, and then finallyreaches the equilibrium.82Wenowcompare the dynamics of the permeable sheetswith impermeable cases.As noted earlier, for t 1, one can neglect the effect of elasticity and permeabilityand so all the cases identically follow the classical behavior. For impermeablesheets, as the elastocapillary number increases, the effect of bending becomesmore dominant and the meniscus position deviates from z∗m =√t sooner. Butin permeable sheets, the time evolution of meniscus is dictated by elastocapillarynumbers of both the dry region (e.g., Ed = 10) and the wet region (e.g., Ew = 100).Meniscus deviation from the classical behavior thereby lies within two cases ofimpermeable sheets with bounding elastocapillary numbers (Ed = 10 and 100),as can be seen in Fig. 7.2(b). However interestingly, the permeable sheets reachregime II (hw(z = 0) = 0) faster than both bounding cases (denoted by circles inFig. 7.2(b)).Note that the system can only collect liquid while hw(z = 0) > 0; once the lowerends of sheets touch (hw(z = 0) = 0), further rise of the liquid is purely due tosheet deflection as the liquid in the bath can no longer flow into the system. Thisbehavior may indicate that permeable sheets have less time to capture the liquid.To better highlight this point, in Fig. 7.3 we have reported the absorption ratio ofpermeable sheets to impermeable sheets, R, by determining the area of the risenliquid between the sheets at equilibrium. As noted earlier, for small values ofelastocapillary number, for which both cases reach regime I, the sheets only slightlybend. The effect of permeability is thereby not significant and the absorption ratio isnearly one. For permeable sheets, as the value of elastocapillary number increases,the sheets deflect more readily in response to the capillary rise. Thus, the softeningof the sheets due to wetting facilitates this bending and leads the sheets to coalescefaster, thereby decreasing the absorption ratio. For instance, when Ed = 35, thisratio drops to ∼ 40% for B = 10 and ∼ 20% for B = 1 in regime III.Recall that Duprat et al. [47] found that for impermeable sheets whenB2Ed & 10the time scale for the capillary rise is set by the deformation of the sheets, τˆve. Wefind here, surprisingly, that despite the discontinuity of E at themeniscus, permeablesheets exhibit the same behavior for B2Ew & 10, as shown in Fig. 7.4 (note thatin our dimensionless units t = 1 is tˆ = τˆve). This result indicates that when thesheets are sufficiently flexible, the equilibrium time scale for permeable sheets isdominantly dictated by the properties of the liquid-filled region, and the dry region83teqB2EwFigure 7.4: Time to reach the 99% of the equilibrium height versus B2Ew.Symbols represent the numerical results for the equilibrium time scale ofimpermeable and permeable sheets. The dashed line indicates teq = 1.has no appreciable contribution.In Fig. 7.5(a), the regime map for permeable sheets with 1 < B < 10 and1 < Ed < 102 is illustrated. Unlike the case of impermeable sheets wherein forsome values of elastocapillary number (i.e., 10 . Ed . 30) regimes I and II coexist[47], here the three regimes are distinct, which may be caused by further softeningof the sheets due to wetting and their higher tendency to coalesce. Recall thathigher Bond numbers can indicate larger gaps, so one can argue that sheets needto bend more to touch (regime II) or coalesce (regime III). Thus, reaching regimeII and III is more difficult when the Bond number (B) is large. Indeed, as shownin Fig. 7.5(a), our numerical results show that when the gap (or similarly B) islarge, reaching coalescence requires longer sheets which indicates higher values ofelastocapillary number. We note that this is in contrast to impermeable sheets forwhich when the gap is large, coalescence is more easily obtained for shorter sheets[47].841010211 10BEdRegime IRegime IIRegime III(a) (b)10 mmFigure 7.5: (a) Regime map for the equilibrium sate of permeable sheets ina logarithmic (B,Ed) space. Each shade refers to its specified regimeand is obtained using the numerical scheme discussed in Section 7.4.Symbols are the equilibrium states observed experimentally: Triangles(N) denote regime I, circles (•) indicate regime II and diamonds () referto regime III. (b) The experimental apparatus. Sheets here are in regimeIII.Finally, in deriving the governing equations, we neglected the vertical liquidpermeation within the sheet as it occurs on a longer time scale than the capillaryrise. But, once the system reaches the equilibrium, the contribution of this upwardpermeation becomes significant, and the system then further evolves until the sheetsare fully wet.Experimental observationsWe compare our numerically-obtained regimemapwith experimental observations.We use filter papers (Whatmann grade 1) that we clamp at the upper end using rare-earth magnets, and set the initial gap using steel shims of varying thicknesses,as shown in Fig. 7.5(b). We slowly immerse the sheets into a water bath andcapture the equilibrium state using a digital camera. We report our observations forvarious combinations of Bond numbers and elastocapillary numbers in Fig. 7.5(a)85(symbols in the figure). We see that our numerical model is in good agreementwith the experiments in predicting regime III. However, the model predictions andthe experimental results deviate from each other at lower values of elastocapillarynumber for which the numerical model predicts regime I, while in the experiments,sheets surprisingly reach regime II. We believe that this discrepancy can arise fromthe inertial effects which were neglected in our model. To examine regime I, thesystem should have a low elastocapillary number. Noting that in our experiments wecan only tune hˆ0 and lˆ, we experimentally obtain low values for the elastocapillarynumber by increasing the initial gap between the sheets (see Eq. (7.1)). But, thetime scale of inertial effects τˆc scales as ∼ hˆ3/20 , and so it becomes more importantwhen hˆ0 increases. Specifically, for gap values of order ∼ 2 mm, the time scale ofinertial effects is of the same order as that of the capillary rise. Therefore, for thecases with a large value of the initial gap (which often are in regime I), the effectsof inertia is no longer negligible and may contribute to the equilibrium state. In ourexperiments, we observe that in this regime sheets rapidly (and significantly) bendtoward each other at the very beginning and then exhibit the expected elastocapillaryrise dynamics thereafter. Due to this initial effect, sheets’ lower ends meet morereadily and so the system proceeds to regime II.7.6 ConclusionIn this chapter, we studied the dynamics of capillary rise of a liquid between twoflexible permeable paper sheets. Accounting for the change of sheet stiffness dueto wetting, we discussed the motion of the meniscus and the sheet deflection asthe system reaches the equilibrium. As the liquid rises, it permeates within thesheets and softens them significantly. Noting that the dynamics of the system isgoverned by both dry and wet regions of the sheets, as a direct consequence of thisfurther softening in the wet region, permeable sheets evolve toward coalescencemore readily, compared to impermeable ones. We also showed that the equilibriumtime scale of the system is quite similar to those of impermeable sheets, however,the volume of fluid captured between the permeable sheets can be notably lower.The lower ends of permeable sheets meet sooner which means the system has lesstime to draw liquid in from the bath, and also the sheets are softer and so they86deform significantly in response to the liquid rise. Recalling that, for instance,multi-ply paper towel is a collection of compressed multi-layer permeable sheets,our results indicate that permeability of fibers should be accounted for in modelingcapillary-based systems, such as in sorption of oil spills [79, 140] and in designingmicrofluidic paper-based analytical devices [128, 129]. Given the generality of theconsidered model, our results can be extended to capture the behavior of a seriesof permeable sheets [60, 174] and be adapted to study the buckling of papers whenthey are fixed at both ends.87Chapter 8Concluding RemarksIn this dissertation, using theoretical approaches, we investigated the effects ofelasticity on hydrodynamic interactions at small scales. We started by evaluatingthe force moments of an arbitrarily-shaped active particle and provided explicitformulas for the stresslet dipole, rotlet dipole, and potential dipole, using thereciprocal theorem. It was then shown that for an elastic swimmer near a boundarythe rotlet dipole of the background flow governs the elastic bending of the swimmer,thereby playing a key role in directing the swimmer toward (or away from) the wall.We also demonstrated that an elastic sphere deforms asymmetrically under a bodyforce, and proposed a two-sphere swimmer that can exploit this asymmetry to propelitself forward. Moreover, using aminimalmodel, we investigated the hydrodynamicinteractions of a ciliary chain around a curved body and showed that no wave-likebehavior emerges from their interactions unless an asymmetry is introduced to thechain. Finally, we looked at the capillary rise between two elastic porous sheetsand quantified their dynamical behavior as the liquid rises and diffuses through thesheets. We demonstrated that, by imbibing the liquid, sheets tend to coalesce fasterand so the system absorbs less liquid.From all these studies, we can conclude that elasticity plays a key role in thedynamical behavior of the system and can alter the hydrodynamic interactionssignificantly. Our results highlight that elasticity should be accounted for whenmodeling hydrodynamic interactions in inertialess environments. Nevertheless,there are, inevitably, limitations to the taken methodologies which are worthwhile88to be discussed and can motivate future directions.In the vast world of microorganisms, the effect of elasticity can be far morecomplex than what we considered in our studies. The linear elasticity modelsemployed in Chapters 4 and 6 have proven useful in recovering some essentialbehaviors of microorganisms and revealing the “leading-order” effects of the elas-ticity. However, they are founded on the assumption of weak elasticity which maynot be always accurate. For instance, single-celled eukaryotes such as Euglenahave large body deformations as they swim [167], and many bacteria buckle theirflagella, substantially, to change direction [177]. These two phenomena exemplifya class of problems wherein understanding the elastohydrodynamic interactionsmay require a geometric or constitutive nonlinearity. But, one should note that, aswe showed in Chapter 5, using non-linear elastic constitutive equations to expressdeformations even in a problem as (seemingly) simple as a sedimenting sphere canbe quite challenging.Despite the growing attention devoted to ciliary motion, our understanding ofthe dynamics of the ciliated microorganisms is still limited. Interactions betweenthe cilia (and flagella) and the formation of wave-like beating patterns have beenspeculated to be strongly depended on hydrodynamic interactions. Several studies,both theoretically [26, 73] and experimentally [22, 23], have verified that hydro-dynamic interactions can indeed lead to the phase-locking of adjacent cilia andflagella. However, in an experimental work, Quaranta et al. [159] revealed that inunicellular interactions of flagella, the coupling forces may be an order of mag-nitude larger than hydrodynamic forces, indicating a non-hydrodynamic origin forflagellar synchronization [100] such as intracellular elastic basal coupling [196].This observation triggers the question of how do cilia and flagella interact within thecell body and also what is the contribution of the substrate elasticity to their phaselocking? Answering these questions requires a careful examination of the cell bodyelasticity, which is often neglected in the minimal models used to understand theciliary motion.Minimal models may also fail to explain the behavior of more complex ciliatedorganisms such as larval forms of marine invertebrates. Ciliary bands, other thangenerating the propulsive thrust [1], have shown to also contribute to the larvaefeeding [62, 151]. In a starfish larvae, the body deformation along with the local89vortices generated by the cilia, entrain the nearby prey and facilitate the feedingmechanism [62]. Given that, in such organisms, cilia are closely packed aroundthe ciliate body and may ‘tangle’ [62], the bead-spring model [141] and far-fieldapproximations can no longer be employed to capture the cilium-cilium interactions.Furthermore, modeling the active shape changes of the larvae is also challenging asit requires a thorough understanding of the internal impetus-generating mechanism.Given the complexity of the poroelastocapillary rise phenomenon, several as-sumptions and approximations were made in Chapter 7 that can be lifted to providea better quantitative model. For instance, the liquid diffusion into the length ofthe sheets was neglected due to the relatively slow evolution timescale, but shouldbe accounted for when analyzing the equilibrium state of the system. 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(A.6)Relevant to the stresslet calculations, we impose uˆ′E(x ∈ ∂B) = r ·(−Eˆ∞)in whichEˆ∞ is a symmetric and deviatoric second-order tensor and soPˆE,ij =5a3(xlxl)52xixj , (A.7)GˆE,ijk =a5(xlxl)52jkδijxk +52 a3(xlxl)52− a5(xlxl)72xi xjxk , (A.8)TˆE,ijkm =a5(xlxl)52(δkjδim + δkiδjm)+ 5 7a5(xlxl)92− 5a3(xlxl)72xixjxkxm+5a32(xlxl)52(δkjxixm + δkixjxm + δmjxixk + δmixjxk)− 5a5(xlxl)72(δimxkxj + δjmxkxi + δijxkxm + δjkxixm + δikxjxm). (A.9)In determining the stresslet dipole, we have uˆ′Γ (x ∈ ∂B) = rr :(−Γˆ∞). Noting thatΓˆ∞ is a fully symmetric, deviatoric third-order tensor, we findPˆΓ ,ijk =354a5(xlxl)72xixjxk , (A.10)GˆΓ ,ijkm =18 15a7(xlxl)72− 7a5(xlxl)52 jkmδimxjxk +358 a5(xlxl)72− a7(xlxl)92xi xjxkxm,(A.11)109andTˆ Γ ,ijkmn =158a7δniδjkxm + δmjxk(xlxl)72− 7xjxkxm(xlxl)92+ δnj δikxm + δimxk(xlxl)72− 7xixkxm(xlxl)92− 78a5δniδjkxm + δmjxk(xlxl)52− 5xjxkxm(xlxl)72+ δnj δikxm + δimxk(xlxl)52− 5xixkxm(xlxl)72+354a5−7xixjxkxmxn(xlxl)92− 354 a7δijxmxnxk(xlxl)92− 9xixjxmxnxk(xlxl)112+358 a5(xlxl)72− a7(xlxl)92[δjkxixmxn + δjmxixkxn + δjnxixkxm+ δikxjxmxn + δimxjxkxn + δinxjxkxm]. (A.12)For a sphere with boundary condition uˆ′Λ(x ∈ ∂B) = ( · r)r :(−Λˆ∞), wherein Λˆ∞is a second-order symmetric and deviatoric tensor, we havePˆΛ,ij = 0, (A.13)GˆΛ,ijk = a5jkijmxmxk(xlxl)52, (A.14)TˆΛ,ijkm =a5(xlxl)52(iksδmjxs + jksδmixs + kmixj + kmjxi)− 5a5(xlxl)72xmxs(iksxj + jmsxi). (A.15)110Finally, uˆ′e(x ∈ ∂B) =(2|r|2I− rr)· (−eˆ∞) yieldsPˆe,i =5a32(xlxl)32xi , (A.16)Gˆe,ij = 3a54(xlxl)32+5a34(xlxl)12δij −  9a54(xlxl)52− 5a34(xlxl)32xixj , (A.17)Tˆe,ijk =152 3a5(xlxl)72− a3(xlxl)52xixjxk − 9a52(xlxl)52(δijxk + δikxj + δjkxi).(A.18)111

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