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Dynamics of small particles, passive and active, in complex fluids Datt, Charu 2019

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Dynamics of small particles, passiveand active, in complex fluidsbyCharu DattB.Tech., National Institute of Technology, Kurukshetra, 2012M.S., École Polytechnique, 2014A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2019© Charu Datt 2019The following individuals certify that they have read, and recommendto the Faculty of Graduate and Postdoctoral Studies for acceptance, thedissertation entitled:Dynamics of small particles, passive and active, in complex fluidssubmitted by Charu Datt in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Mechanical EngineeringExamining Committee:Gwynn J. Elfring, Mechanical EngineeringSupervisorIan A. Frigaard, Mathematics, and Mechanical EngineeringSupervisory Committee MemberSavvas G. Hatzikiriakos, Chemical and Biological EngineeringSupervisory Committee MemberAnthony Wachs, Mathematics, and Chemical and Biological EngineeringUniversity ExaminerJames J. Feng, Mathematics, and Chemical and Biological EngineeringUniversity ExaminerAdditional Supervisory Committee Member:George M. Homsy, Mathematics, and Mechanical EngineeringiiAbstractThe focus of this thesis is on small non-Brownian particles in fluids thatshow deviations from standard Newtonian fluids. We study the motion ofswimmers and sedimenting particles in Newtonian fluids with viscosity gra-dients, in shear-thinning fluids, and in fluids with viscoelasticity. The workis theoretical; its aim is to study the first effects of non-Newtonian rheologyon particle motion and towards this end uses the reciprocal theorem of lowReynolds number hydrodynamics and methods of perturbation expansion.We find that the dynamics of the particles is often qualitatively changed dueto the rheological properties of the fluid, and such changes are difficult topredict a priori.iiiLay SummaryThis thesis focusses on understanding the motion of swimmers that are onlya few micrometers in size in fluids that are complex. Water is an exampleof a simple (Newtonian) fluid. But many fluids where microswimmers swimshow both fluid-like and solid-like properties. Sperm cells in human cervicalmucus and H. pyroli in gastric mucus are examples of microswimmers incomplex fluids.We study such swimmers theoretically, and demonstrate how their dy-namics in complex fluids is different from that in Newtonian fluids. Infact, we find that often the swimming behaviour in complex fluids is totallyunexpected beforehand. Noting that designing artificial microswimmers inNewtonian fluids can be challenging, we also show how complexity of thefluid medium may be used to design simpler swimmers. Our work has poten-tial applications in aspects of biomedical engineering focussed at developingmicrorobots for targeted drug delivery and minimally invasive surgery.ivPrefaceParts of this thesis, with minor changes, have been previously published asresearch articles. These parts are listed below. I have underlined my nameamong the names of authors to guide the eye.• Chapter 4 has been previously published as ‘C. Datt & G.J. Elfring,Dynamics and rheology of particles in shear-thinning fluids, Journal ofnon-Newtonian Fluid Mechanics 262 (2018) 107-144’.C.D. and G.J.E. designed the research. C.D. performed the calcu-lations. C.D. and G.J.E. analyzed the results and wrote the paper.C.D. is the principal contributor to this work. C.D. was supervised byG.J.E.• Chapter 5 has been previously published as ‘C. Datt, L. Zhu, G.J.Elfring, & O.S. Pak, Squirming through shear-thinning fluids, Journalof Fluid Mechanics Rapids, 784 (2015) R1’.All authors designed the research. C.D. performed the theoreticalcalculations. L.Z. performed the numerical simulations. All authorsanalyzed the results. C.D., G.J.E., and O.S.P. wrote the paper withinputs from L.Z. C.D. is the principal contributor to this work. C.D.was supervised by G.J.E.• Chapter 6 has been previously published as ‘C. Datt, G. Natale, S.G.Hatzikiriakos, & G.J. Elfring, An active particle in a complex fluid,Journal of Fluid Mechanics, 823 (2017), 675-688’.This work is one part of a simultaneously running two-part project.The second part, mentioned later in the preface, is not included in thethesis. All authors contributed to discussion of the project.C.D. performed the calculations in this work. C.D. and G.J.E. ana-lyzed the results and wrote the paper. C.D. is the principal contributorto this work. C.D. was supervised by G.J.E.vPreface• Chapter 8 has been previously published as ‘C. Datt, B. Nasouri, &G.J. Elfring, Two-sphere swimmers in viscoelastic fluids, Physical Re-view Fluids 3, 123301 (2018)’.C.D. designed the research. C.D. performed all calculations in thiswork except for the section on swimmers with elastic spheres. B.N.performed the calculations for the section on swimmers with elasticspheres. All authors analyzed the results and wrote the paper. C.D. isthe principal contributor to this work. C.D. was supervised by G.J.E.Chapter 7 has been submitted for publication as ‘C. Datt & G.J. Elfring,A note on higher-order perturbative corrections to squirming speed in weaklyviscoelastic fluids’. C.D. designed the research and performed the calcula-tions. C.D. and G.J.E. analyzed the results and wrote the paper. C.D. isthe principal contributor to this work. C.D. was supervised by G.J.E.Chapter 3 may soon be submitted for publication as ‘C. Datt & G.J.Elfring, Swimming in viscosity gradients’. C.D. and G.J.E. have designedthe research. C.D. has performed the calculations. C.D. and G.J.E. haveanalyzed the results and written the article. C.D. is the principal contributorto this work. C.D. was supervised by G.J.E.Related or directly stemming from the thesis, but not included in it, aretwo articles that are listed below.• ‘G. Natale, C. Datt, S. Hatzikiriakos, & G.J. Elfring, Autophoreticlocomotion in weakly viscoelastic fluids at finite Péclet number, Physicsof Fluids 29, 123102 (2017).’This is the second part of the two-part project mentioned above.G.N. wrote the numerical code for this work. C.D. provided help withthe code, contributed to the analyses of results and writing of thepaper. C.D. was supervised by G.J.E.• ‘K. Pietrzyk, H. Nganguia, C. Datt, L. Zhu, G.J. Elfring, & O.S. Pak,Flow around a squirmer in a shear-thinning fluid, submitted.’C.D. contributed to the design of the research, in calculations and inanalyses. C.D. was supervised by G.J.E.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The swimmer, the fluids, and the reciprocal theorem . . . 42.1 The swimmer . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The reciprocal theorem . . . . . . . . . . . . . . . . . . . . . 83 Swimming in viscosity gradients . . . . . . . . . . . . . . . . 123.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Theoretical formulation . . . . . . . . . . . . . . . . . . . . . 133.2.1 Reciprocal theorem . . . . . . . . . . . . . . . . . . . 143.2.2 Asymptotic analysis . . . . . . . . . . . . . . . . . . . 153.2.3 Remarks on the asymptotic analysis . . . . . . . . . . 163.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 163.3.1 Passive sphere . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Squirmer . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Dynamics and rheology of particles in shear-thinning fluids 234.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Reciprocal Theorem . . . . . . . . . . . . . . . . . . . . . . . 254.3 Shear-thinning fluid . . . . . . . . . . . . . . . . . . . . . . . 264.4 Sedimenting spheres . . . . . . . . . . . . . . . . . . . . . . . 28viiTable of Contents4.4.1 Single sphere . . . . . . . . . . . . . . . . . . . . . . . 284.4.2 Two spheres . . . . . . . . . . . . . . . . . . . . . . . 294.5 Sedimentation of a rotating sphere . . . . . . . . . . . . . . . 334.6 Sphere under an external force in a linear flow of shear-thinning fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.7 Suspension of spheres in a shear-thinning fluid . . . . . . . . 364.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Squirming through shear-thinning fluids . . . . . . . . . . . 415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . 435.2.1 Shear-thinning rheology: the Carreau-Yasuda model . 435.2.2 Asymptotic analysis . . . . . . . . . . . . . . . . . . . 455.2.3 The reciprocal theorem . . . . . . . . . . . . . . . . . 465.2.4 Numerical solution . . . . . . . . . . . . . . . . . . . 475.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 475.3.1 Drag and thrust . . . . . . . . . . . . . . . . . . . . . 495.3.2 Addition of other squirming modes . . . . . . . . . . 515.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 An active particle in a complex fluid . . . . . . . . . . . . . 546.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 Modelling active particles . . . . . . . . . . . . . . . . . . . . 566.3 Swimming in a background flow of a weakly non-Newtonianfluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.4 Janus particle in non-Newtonian fluids . . . . . . . . . . . . 616.4.1 Viscoelasticity: second-order fluid . . . . . . . . . . . 616.4.2 Shear-thinning rheology: Carreau model . . . . . . . 666.5 Conclusion and future work . . . . . . . . . . . . . . . . . . . 687 A note on higher-order perturbative corrections to squirm-ing speed in weakly viscoelastic fluids . . . . . . . . . . . . . 707.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . 727.2.1 The squirmer model . . . . . . . . . . . . . . . . . . . 727.2.2 The second-order fluid model . . . . . . . . . . . . . . 727.2.3 The reciprocal theorem . . . . . . . . . . . . . . . . . 737.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 747.3.1 Giesekus fluid . . . . . . . . . . . . . . . . . . . . . . 757.3.2 A fluid of grade three . . . . . . . . . . . . . . . . . . 78viiiTable of Contents7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 Two-sphere swimmers in viscoelastic fluids . . . . . . . . . 818.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2 Swimmer in a viscoelastic fluid . . . . . . . . . . . . . . . . . 848.2.1 Two-sphere swimmers . . . . . . . . . . . . . . . . . . 848.2.2 Theory for swimming in complex fluids . . . . . . . . 858.2.3 Constitutive equation . . . . . . . . . . . . . . . . . . 878.2.4 Small amplitude expansion . . . . . . . . . . . . . . . 888.2.5 Results and discussion . . . . . . . . . . . . . . . . . 908.3 Swimmer with elastic spheres . . . . . . . . . . . . . . . . . . 928.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . 969.1 The findings . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.2 The limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 989.3 The future . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101AppendicesA Some expressions for motion of spheres . . . . . . . . . . . 119B Linear viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . 121ixList of Figures3.1 The initial orientation of the swimmers einitial is along thepositive x-axis. The viscosity gradient is along the positivey-axis. After sometime all swimmers swim antiparallel to theviscosity gradient. Note that in this orientation, pushers arethe fastest, while pullers are the slowest. The trajectories areplotted for time t = 0 to t = 100. . . . . . . . . . . . . . . . 193.2 Planar trajectories of pushers (left), neutral swimmers (cen-tre), and pullers (right). The initial position of the swimmers,(x = 1, y = 1), is marked by the red dot. The swimmers orig-inally point in the positive x-axis. The swimmers are in aradial viscosity gradient: viscosity increases radially outwardfrom the point (0, 0), δη =√x2 + y2. Pushers find a stableorbit. The trajectory of neutral swimmers is bounded at longtimes. Pullers perform ‘unstable’ motion about the ‘origin’of viscosity gradient. Note the farthest the swimmers havetravelled from this origin after equal times (t = 4000). . . . . 204.1 a) Variation of non-Newtonian drag F 1 as a function of thedistance of separation between the two spheres. b) Normal-ized drag as a function of this distance. Note that the varia-tion is non-monotonic and that drag reduction is minimizedat h/d ≈ 2.35. . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1 In (a) we show results from numerical simulations for a neu-tral squirmer (α = 0, l), puller (α = 5, s) and pusher(α = −5, t) for ε = 0.99 and n = 0.25. (b) The non-monotonic variation in velocity is well captured by asymp-totics for a neutral squirmer (solid line), and a pusher/puller(dashed line) at ε = 0.1 and n = 0.25. . . . . . . . . . . . . . 48xList of Figures5.2 In (a), the symbol ∆ denotes the difference between drag,or thrust, in a shear-thinning fluid and a Newtonian fluid:the dash-dot curve represents the drag reduction in a shear-thinning fluid compared with a Newtonian fluid, while thesolid and dashed curves represent loss in thrust for squirm-ers with α = 0 and α = ±5 respectively (n = 0.25, ε =0.1). In (b) we show that the difference between drag andthrust is positive both for a neutral squirmer (solid) and apusher/puller (dashed) in a shear-thinning fluid. All quanti-ties are dimensionless. . . . . . . . . . . . . . . . . . . . . . . 505.3 In (a) we show the level set curve below which faster swim-ming occurs for specific values of α and ζ = B3/B1 whenCu ≪ 1. In (b) we show the swimming speed of a neu-tral squirmer (solid line) and a pusher/puller (dashed line) atvalues of ζ chosen below the curve in (a) so that the swim-ming speed is larger than the Newtonian value at small Cu(the upper solid line: α = 0, ζ = −15 ; the upper dashedline: α = ±5, ζ = −4 ); conversely, with values of ζ abovethe curve in (a) the swimmers always swim slower than in aNewtonian fluid as shown (lower solid line: α = 0, ζ = 15;lower dashed lines: α = ±5, ζ = 4). Here ε = 0.1, n = 0.25. . 526.1 Self-phoretic particle with two compartments of different ac-tivity, Af and Ab. We consider particles with a constant uni-form mobility over the surface. When θd = pi/2, the particlehas compartments of equal cover, which we call a symmetricJanus particle. . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2 Variation of the scaled first-order swimming velocityU(M)1 /UN with θd obtained for the first M +1 modes (dashedlines), and b = 0.2. U (∞)1 corresponds to the convergencevalue (M = 99) and is depicted by the solid line. Inset plotshows the relative error. . . . . . . . . . . . . . . . . . . . . . 656.3 Variation of the scaled first-order swimming velocityU(M)1 /UN (obtained for M + 1 modes) with Cu for twovalues of µ ≡ cos θd. Solid lines correspond to M = 30, andM = 28 for µ = 0 (symmetric) and µ = ±0.9 respectively(additional modes lead to negligible differences). Dashed-lines correspond to the swimming velocity with just the firsttwo modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67xiList of Figures7.1 Swimming speeds in the Giesekus fluid as a function of De.The solid lines include corrections up to O (De3). The dashedlines are Padé approximations to the series for the speeds inthe text. The dotted lines include only O (De) corrections.The addition of the higher order modes decreases the speedsof the squirmers. As seen here, all squirmers at large valuesof De swim slower than in a Newtonian fluid, as found in thenumerical work of Zhu et al. [217]. . . . . . . . . . . . . . . . 777.2 Swimming speeds in fluids of grade three. Solid lines: P =3/2, Q = −7. Dashed lines: P = 3/2, Q = 5. Solid anddashed lines for β = 0 overlap. Depending on the values of Qfor a given P, either of the puller or pusher can swim fasteror slower than in a Newtonian fluid at small De. . . . . . . . 798.1 Schematic of the two-sphere swimmer. The spheres, labeledB1 and B2, have radii aα and a, respectively (α > 1). Thespheres are (on average) a distance d0 apart and e∥ is the unitvector pointing from B1 to B2. . . . . . . . . . . . . . . . . . 848.2 Schematic showing one complete cycle for the two swimmers:(a) The in-phase swimmer maintains the distance between thespheres as it moves forward. (b) In the anti-phase swimmer,the spheres converge and diverge. The steps in grey show thetransition from one half cycle to the next. The red dot marksthe position of the swimmer. . . . . . . . . . . . . . . . . . . 858.3 The swimming speed coefficient U is plotted with variationin the clearance ∆c between the two spheres for different sizeratios α. The square symbols (connected by dashed lines)represent the anti-phase swimmer and the circles (connectedby solid lines) represent the in-phase swimmer. All quantitiesare dimensionless. . . . . . . . . . . . . . . . . . . . . . . . . . 90xiiChapter 1IntroductionThis thesis is about the dynamics of small particles in non-Newtonian fluids.A non-Newtonian fluid does not obey Newton’s law of viscosity, i.e., shearstresses in the fluid are not linearly proportional to the rates of strain, andoften demonstrates properties like viscoelasticity and shear-thinning viscos-ity [5, 16]. With viscoelasticity, the fluid retains memory of its flow historyand exhibits stress anisotropy, whereas a shear-thinning rheology means thatthe fluid viscosity decreases with increasing shear rates. We focus on the twoproperties in incompressible fluids here. We study only small particles, sotheir motion is dominated by viscous forces, and inertia may be neglected—the motion obeys equations of low Reynolds number hydrodynamics [93].These particles can be passive or active. Active particles can convert storedor ambient free energy into systematic motion [139, 180]; passive particlesare the ones that are not active.The following is a brief overview of the literature, and an outline of thethesis. The chapters in the thesis that describe our research are written inthe format of research articles and have their own literary introduction.Plastic microbeads in toothpastes, microvesicles in biofluids, and duringtertiary oil recovery, oil drops in polymer solution are examples of particlesin complex fluids. Such systems range from natural to industrial settings[33], and therefore, it becomes interesting, from both fundamental and engi-neering point of view, to understand them. The interest in motion of smallparticles in complex fluids is not new; in fact, one finds review articles byLeal [132, 133] and Brunn [29] in the late 70’s. Even so, sedimentationof a spherical particle in viscoelastic shear flow [64], effects of viscoelastic-ity on sedimenting anisotropic particles [40], Einstein viscosity for a dilutesuspension in viscoelastic fluids [65]—some seemingly canonical problems—were worked out (by others) during the period of this doctorate study. Lestthis suggest that progress has not been made in complex fluids, we referthe reader to the recent review articles by Zenit & Feng [216] and D’Avino& Maffettone [49], which also suggest directions for the future of the field.Importantly, it must be pointed out that dealing with complex fluids (inparticular, viscoelastic fluids) also meant, and means, tackling some fun-1Chapter 1. Introductiondamental issues [55], like slip or no slip? or the high Weissenberg numberproblem, which are not present in Newtonian fluids. A sustained, and in factgrowing interest in understanding complex fluids, in context of this thesis, isalso due to the fast emerging field of microfluidics, where ‘microchannels de-signed to focus, concentrate, or separate particles suspended in viscoelasticliquids are becoming common’ [47].The relevant works in the foregoing discussion were centred on passiveparticles. The focus in this thesis is on “active particles”, a term often inter-changeably used with self-propelled particles and microswimmers. Bacteria,which represent the bulk of world’s biomass, are often found in environments,such as gastric mucus (H. pyroli) and biofilms, which display non-Newtonianrheological properties [127]. Bacteria are examples of active particles. Mam-malian sperm cells swimming through cervical mucus [72] are also examplesof active particles in complex fluids. Our understanding of active particleshas developed enough that we can describe self-sustained turbulent struc-tures in living fluids [60], and also use bacteria in fluids as work horsesfor turning microscopic gears [185]. This is hardly surprising consideringsuch giants of yesteryears as G. I. Taylor, M. J. Lighthill and E. M. Purcellcontributed to the field of microswimming [137, 170, 200]. However, muchof this understanding is in Newtonian fluids [129], and understanding incomplex fluids is still nascent [164].Research in swimming in complex fluids has burgeoned after the the-oretical work of Lauga [121] where he considered a simple swimmer andcompared its swimming speed and its power consumption to those in anequivalent Newtonian fluid: questions regarding the swimming speed andthe power consumption of microswimmers in non-Newtonian fluids are nowbeing asked and answered for various model and real swimmers [70, 195]. Animportant motivation for these studies, other than the fundamental curios-ity, remains of creating autonomous devices for targeted drug delivery [79]and other biomedical applications [208] which will swim in non-Newtonianfluids—what most biological fluids are [120, 142]. The work in this thesiswas driven by such questions as: how does a small change in rheologicalproperty of the fluid affect the swimming motion? can intuitions devel-oped in Newtonian fluids lead to a wrong footing in complex fluids? canthe complexity of the fluid be used to our advantage in modelling artificialswimmers?The work in the thesis is done in the spirit of Leal’s work on passive par-ticles [133] where ‘the particle-motion problem can be treated theoreticallyif the deviations from the “standard conditions” are small.’ Chapter 3 talksabout a swimmer in a Newtonian fluid with small gradients in viscosity.2Chapter 1. IntroductionChapter 4 is about the dynamics and rheology of passive particles in weaklyshear-thinning fluids. In Chapter 5, we look at a swimmer in a weakly shear-thinning fluid, and in Chapter 6 and 7, in weakly viscoelastic fluids. Chapter8 is about creating an artificial swimmer using the rheological property ofthe fluid. We end with talking about what we learnt, and the things thatinterest us for the future in Chapter 9.Although chapters in this thesis are written in the format of articles, andtherefore aimed to be self-contained, we present, in Chapter 2, additionaldetails of the overarching theoretical models and methods that we use forstudying swimmers and fluids; the reader not familiar with the field mayfind these details useful.3Chapter 2The swimmer, the fluids, andthe reciprocal theorem2.1 The swimmerOur theoretical microswimmer is a squirmer. As a spherical squirmer, a mi-croswimmer is represented as a sphere with prescribed velocities on its sur-face [17, 137]. It is only the surface velocities that propel the squirmer. Thesquirmer was proposed by Lighthill [137] as a finite-body model swimmerthat could swim in Newtonian fluids at zero Reynolds number. Recently,the works of Chisholm et al. [36], Khair & Chisholm [110], Wang & Ardekani[209] have explored the effect of inertia on the squirmer’s motion.The description below is of an axisymmetric spherical squirmer of radiusa, as described in the works of Lighthill [137] and Blake [17] (Blake [17]corrected Lighthill’s solution [137]). The surface velocities on such squirmersare represented as(u)r=a =∞∑n=0An (t)Pn (cos θ) , (v)r=a =∞∑n=1Bn (t)Vn (cos θ) , (2.1)where u and v are the radial and azimuthal components of the velocityfield, respectively. θ is the polar angle (in the spherical coordinate system)measured from the axis of symmetry, and Pn represents the ordinary Leg-endre polynomial, whereas Vn = −2/ (n (n+ 1))P 1n (cos θ), P 1n being theassociated Legendre function of the first kind. In incompressible Newtonianfluids, the surface velocities lead to a propulsion speed (along ez, the axisof symmetry) of [17, 137]U = 13 (2B1 −A1) , (2.2)with the flow field around the squirmer in the lab frame (fluid at infinity is42.1. The swimmerat rest) given by [17]u = A0a2r2P0 +23 (A1 +B1)a3r3P1+∞∑n=2[(n2anrn−(n2 − 1) an+2rn+2)AnPn +(an+2rn+2− anrn)BnPn],(2.3)v = 13 (A1 +B1)a3r3V1+∞∑n=2[(n2an+2rn+2−(n2 − 1) anrn)BnVn +n2(n2 − 1)(anrn− an+2rn+2)AnVn],(2.4)where r is measured from the centre of the squirmer. When the squirmerdoes not deform, and its surface impermeable, the radial surface velocitiesare zero i.e., for all n, An = 0. With only tangential surface velocities, Bnmay be unambiguously referred to as the nth squirming mode. In this work,we restrict our attention to steady squirmers with just tangential surfacevelocities, and, in the remaining section, will consider only such squirmers.From equation (2.2), it is clear that, of all the squirming modes, only thefirst mode B1 contributes to the swimming speed. From equations (2.3), and(2.4), one also observes that the strongest contribution to the flow field farfrom the squirmer is due to the second mode B2, the flow field due to whichdecays the slowest, as 1/r2. As a consequence, often in Newtonian fluids,the surface velocities are truncated after only the first two modes [165], i.e.,Bn = 0 ∀n > 2. The ratio α = B2/B1 decides the type of squirmer [165].When α < 0, the squirmer’s centre of thrust is behind its centre of drag,and the squirmer is a pusher-type swimmer. When α > 0, the squirmer’scentre of thrust is in front of its centre of drag, and the squirmer is a puller-type swimmer. When α = 0, the thrust and drag centres coincide, and thesquirmer is a neutral-type swimmer. In the absence of any external force,such as weight due to gravity, or torque, the total hydrodynamic force andtorque on a squirmer is zero.Note that both Lighthill [137] and Blake [17] did not consider axisym-metric surface velocities in the azimuthal direction eϕ in the squirmer model.These were considered only recently by Pak & Lauga [159], who also provideresults for non-axisymmetric squirming motion.52.2. The fluids2.2 The fluidsThe motion of squirmers, and in general microswimmers, in Newtonian flu-ids is understood reasonably well [129, 165]. The motivation of this thesisis to understand their dynamics in fluids that are not Newtonian in a ‘stan-dard’ sense; the definition includes not just non-Newtonian fluids, but alsoNewtonian fluids where the viscosity depends on spatial coordinates due to,for e.g., temperature or concentration gradients. In non-Newtonian fluids,we focus on two common properties, shear-thinning viscosity, and viscoelas-ticity. Shear-thinning fluids are fluids in which the shear viscosity decreasesas applied shear-rates increase. These fluids come under the wider class ofgeneralized Newtonian fluids, for which the deviatoric stress ττ = η (γ˙) γ˙, (2.5)where η (γ˙) is the viscosity, and γ˙ is the strain-rate tensor, andγ˙ =√12∑i∑j γ˙ij γ˙ji is the magnitude of the strain-rate tensor [16].We model shear-thinning viscosity using the Carreau model [16], whereη (γ˙) = η∞ + (η0 − η∞)[1 + λ2t |γ˙|2](n−1)/2. (2.6)Here η0 is the zero shear-rate viscosity, and η∞ is the infinite shear-rateviscosity. The parameter λt has units of time and gives the strain rate 1/λtat which non-Newtonian effects start becoming important. n is the powerlaw index and is less than 1 for shear-thinning fluids. Note that we do not usethe power-law model η = mγ˙n−1, primarily because of the model’s inabilityto describe viscosities at small strain-rates [16]. The inability can give riseto large errors, even altogether incorrect results, in problems relevant to thisthesis (for example, see the discussion of creeping flow around a sphere inshear-thinning fluids by Chhabra [33]).Viscoelastic fluids show both viscous and elastic properties. Looselyspeaking, the response to stress of such fluids is like solids at short times,but as liquids at long times; ‘short’ and ‘long’ are relative to some charac-teristic timescale of the fluid [150]. Viscoelasticity may be studied underthe theory of simple fluids, a lucid description of which can be found in thebook by Astarita & Marrucci [5]. The principles that govern simple fluidsare determinism of stress, local action, non-existence of a natural state, andfading memory [5]. Determinism of stress means that ‘the stress at a giventime is independent of future deformations, and only depends on past de-formations’ [5]. Local action means ‘the stress at a given point is uniquely62.2. The fluidsdetermined by the history of deformation of an arbitrarily small neighbor-hood of that material point’ [5]. Non-existence of a natural state implies that‘every simple fluid is isotropic’ [5]. Fading memory means ‘the influence ofpast deformations on the present stress is weaker for the distant past thanfor the recent past’ [5]. The principle of fading memory gives rise to the ideaof a ‘natural time’ of the fluid; natural time quantifies the memory span ofthe material [5]. Natural time is often referred to as the relaxation time.In this thesis, we study incompressible fluids under two different types ofmotion, slow flows, and small deformations. In these two types of motions,the general constitutive equation for constant density simple fluids can beexpressed asN th order approximations [5]. For slow flows, a general constantdensity simple fluid at zeroth order is represented byσ = −pI, (2.7)where σ is the stress and p is the pressure [5]. Equation (2.7) is the con-stitutive equation for an incompressible ideal fluid [5]. The first-order ap-proximation gives the constitutive equation for incompressible Newtonianfluidsσ = −pI + ηγ˙. (2.8)The popular second-order fluid is the second-order approximation withσ = −pI + η0γ˙ + α1A2 + α2A21, (2.9)where η0, α1 and α2 are constants, andA1 ≡ L+ L⊤,An ≡ DAn−1Dt + L⊤An−1 + An−1L,(2.10)with L⊤ =∇u, and D/Dt representing the material derivative [5, 198]. Theterm α1A2 includes the first effects of memory to an otherwise purely viscousapproximation [5]. One can proceed in a similar manner to obtain higher-order approximations (and fluids). In this thesis, we restrict our attentionto slow flows of only second-order and third-order fluids.Under small deformations, say, realized with oscillations of small am-plitude of some fluid boundary, simple fluids of constant density at thefirst-order approximation are represented asσ = −pI +∫ ∞0f (s)Gt ds, (2.11)72.3. The reciprocal theoremwhere Gt is the Cauchy strain tensor, and s is the time lag; distant pastcorresponds to large values of s, recent past to small values of s [5]. Equa-tion (2.11) is the constitutive equation for a linearly viscoelastic fluid. Thesecond-order approximation givesσ = −pI +∫ ∞0f (s)Gt ds+∫ ∞0∫ ∞0{α (s1, s2)Gt (s1) · Gt (s2) + β (s1, s2) tr[Gt (s1)]Gt (s2)}ds1 ds2,(2.12)where functions f (), α () and β () depend on the fluid [5]. Details of usingequation (2.12) for problems involving oscillating boundaries can be foundin the pedagogical description by Böhme [18].It is to be noted that many viscoelastic models that adequately representexperimental results, and are obtained using kinetic theory of polymers donot come under the umbrella of simple fluids (‘equations …[that] do not allowstrain impulses are not included in the general theory of simple fluids withfading memory’ [5]). One example is an Oldroyd-B fluid [150]τ + λ ∇τ= η(γ˙ + λr∇γ˙)(2.13)where the total viscosity η = ηs+ηp is the sum of viscosity of the Newtoniansolvent and that due to the polymer contribution. λ is the Maxwell relax-ation time, and the retardation time λr = λ (ηs/η). Other common usefulmodels like FENE-P and Giesekus are not simple fluids either. However, wenote that, for flows that are slow or due to small deformations, such modelsgive qualitatively the same constitutive relations as one would obtain usingequations (2.9) and (2.12), respectively, up to at least the first effects ofnon-linear rheology.2.3 The reciprocal theoremIn this thesis, we extensively use the Lorentz reciprocal theorem [93] toobtain results for motion of particles in complex fluids. The form of thetheorem for complex fluids has been discussed by Leal [133] for passiveparticles, and by Lauga [125] for active particles. The formulation below isthat of Elfring & Lauga [70], and Elfring [68].We consider the motion of a particle B, with surface ∂B,u (x ∈ ∂B) = U +Ω × x+ uS , (2.14)82.3. The reciprocal theoremin a complex fluid, which is expressed in the formτ = ηγ˙ + τNN , (2.15)where the deviatoric stress τ is written as a sum of ‘Newtonian’ and ‘non-Newtonian’ terms. Non-linearities of the constitutive equation are embeddedin τNN . η represents ‘some’ viscosity of the fluid, γ˙ is the strain-rate. Uand Ω are the unknown translational and angular velocities of the particle,respectively, and uS represents the prescribed (known) surface velocities onthe particle. If the particle is passive, uS = 0. For the purpose of the deriva-tion here, we consider the particle to be in an otherwise quiescent fluid. Thecalculation of velocities U and Ω, if not for the reciprocal theorem, requiressolving the Stokes equation (for velocity and pressure fields, u and p, re-spectively) with the constitution equation (2.15), and the relevant boundaryconditions. Solving for the flow fields in complex fluids analytically is, ingeneral, difficult. The reciprocal theorem provides a shortcut to obtainingU and Ω, if the solution for the same geometry is known in a Newtonianfluid (or precisely, the resistance or mobility matrices for the geometry areknown in a Newtonian fluid [70]). Consider a passive particle, of the sameshape as the particle in complex fluid, in an otherwise quiescent Newtonianfluid with viscosity ηˆ, i.e.,uˆ (x ∈ ∂B) = Uˆ + Ωˆ × x. (2.16)The hat quantities represent quantities in Newtonian fluids, and are known.We know that at zero Reynolds number∇ · σ =∇ · σˆ = 0, (2.17)where σ and σˆ represent the stress in the two fluids. Equation (2.17) isused, rather straightforwardly, to write the reciprocal relationuˆ · (∇ · σ) = u · (∇ · σˆ) = 0. (2.18)Equation (2.18) can be rewritten, making use of the fact that the stresstensor is symmetric, as∇ · (σ · uˆ)− σ :∇uˆ =∇ · (σˆ · u)− σˆ :∇u = 0. (2.19)Incompressibility of the fluids demands−pI :∇uˆ = −p∇ · uˆ = 0, (2.20)−pˆI :∇u = −pˆ∇ · u = 0, (2.21)92.3. The reciprocal theoremso that equation (2.19) simplies to∇ · (σ · uˆ)− ηγ˙ :∇uˆ− τNN :∇uˆ =∇ · (σˆ · u)− ηˆ ˆ˙γ :∇u = 0, (2.22)on replacing σ = −pI + τ and τ = ηγ˙ + τNN (2.15). From the property ofa symmetric tensor in a doubly contracted product, we haveˆ˙γ :∇u = γ˙ :∇uˆ. (2.23)Using equation (2.23) in the right half of (2.22), and then substituting forthe doubly contracted product in the left half gives∇ · (σ · uˆ)− ηηˆ∇ · (σˆ · u)− τNN :∇uˆ = 0. (2.24)Taking integral of equation (2.24) over the entire fluid volume, and usingthe divergence theorem, we get∫Sn · σ · uˆ dS − ηηˆ∫Sn · σˆ · udS +∫VτNN :∇uˆ dV = 0, (2.25)where n points inside the fluid volume, the surface S includes the surfaceof the particle and the bounding surface at infinity, and V is the fluid vol-ume. For sufficiently fast spatial decay of the integrands above, the surfaceintegrals at infinity do not contribute—this is indeed the case for problemsthat we consider in the thesis. Neglecting the surface integral at infinity andsubstituting for surface velocities from equations (2.14) and (2.16), we getF · Uˆ +L · Ωˆ − ηηˆ(Fˆ ·U + Lˆ ·Ω +∫∂Bn · σˆ · uS dS)+∫VτNN :∇uˆdV = 0,(2.26)where F and Fˆ , and L and Lˆ represent the total hydrodynamic forces andtorques, respectively, on the particles. Using the linearity of the Stokesequations in Newtonian fluids, we write, for compactness, uˆ = GˆU · Uˆ, σˆ =TˆU · Uˆ, and Fˆ = −RˆFU · Uˆ, where Uˆ =[Uˆ Ωˆ]⊤, and Fˆ = [Fˆ Uˆ]⊤, andconsequently, GˆU , and TˆU are linear operators that map the particle velocityUˆ to the velocity and the stress fields, respectively, whereas the resistancetensorRˆFU =[RˆFU RˆFΩRˆLU RˆLΩ](2.27)102.3. The reciprocal theoremlinks Uˆ to the hydrodynamic force and torque on the particle. Rewritingequation (2.26), we getF · Uˆ − ηηˆ(Fˆ · U +∫∂Bn · uS ·(TˆU · Uˆ)dS)+∫VτNN :∇(GˆU · Uˆ)dV = 0,(2.28)which on rearranging terms, cancellation of the arbitrary Uˆ on both sides,and using the symmetry of the resistance tensor [93], givesU = ηˆηRˆ−1FU · [−F + FT + FNN ] , (2.29)whereFT =ηηˆ∫∂BuS · (n · TˆU)dS, (2.30)is a Newtonian ‘thrust’ due to the surface velocity uS , and FNN is the non-Newtonian contribution (or contribution due to any deviation from Newto-nian fluids of uniform viscosity) given byFNN =−∫VτNN :∇GˆU dV,=−∫VτNN : EˆU dV,(2.31)where EˆU is defined as ˆ˙γ/2 = EˆU · Uˆ. The equivalence in (2.31) comes fromusing the property of the doubly contracted product when one of tensors issymmetric (here, τNN ) and noting that EˆU is the symmetric part (for innerindices) of ∇GˆU .One may easily extend the above derivation to include cases with parti-cles in a background flow [44, 68, 133]. Equation (2.29) is then formulatedin terms of disturbance quantities [68], and will be seen in Chapter 4. Notethat in the formulation of equation (2.29) as a mobility problem, FNN as ofyet is unknown, since τNN has not been calculated in the fluid. Avoidingits calculation was the very reason to use the reciprocal theorem. How-ever, as will become clear in the following chapters, asymptotic analysis insome small parameter, allows one to evaluate τNN from the solution in a‘standard’ Newtonian fluid.11Chapter 3Swimming in viscositygradientsMicroswimmers swimming in mucus layers and biofilms experience spatialgradients of viscosity. We model the swimmers using the squirmer model,and show how the effects of viscosity gradients on the swimmer motion,leading to the phenomenon of viscotaxis, depend on the swimming gait ofthe swimmers. We also show how such gradients in viscosity may be used tosort swimmers based on their swimming style.3.1 IntroductionCells often swim in environments, such as biofilms and mucus layers, thathave spatial gradients of viscosity [194, 211]. Much like the effect of othergradients, such as light (leading to phototaxis [14]), chemical stimuli (chemo-taxis [15]), magnetic fields (magnetotaxis [206]), temperature (thermotaxis[7]), or gravitational potential (gravitaxis [176]), gradients of viscosity canlead to viscotaxis in microswimmers. Bacteria like Leptospira and Spiro-plasma are known to move up the viscosity gradients (positive viscotaxis)[41, 196], where as Escherichia coli demonstrates negative viscotaxis [182].It is suggested that viscotaxis plays an adaptive role in microorganisms; itprevents them from being stuck in regions where they are poor swimmers[41, 182, 196]. This migration across regions of different viscosity affectsorganisms’ population distribution, and possibly their virulence [167]. Theaggregation of microswimmers in specific regions of viscosity may also beused for sorting of cells [214].In a recent work, Liebchen et al. [136] studied the physical mechanism ofviscotaxis. Using assemblies of one, two and three spheres as model swim-mers, they showed how viscotaxis can emerge from a mismatch of viscousforces on different parts of the swimmer, thereby demonstrating the possi-bility of both positive and negative viscotaxis. In this work, we study visco-taxis using a general model to study microswimmers—the squirmer model123.2. Theoretical formulation[17, 137, 165]. Squirmers can be used to model different types of swimmerswithin the same theoretical framework, and have been used in understandingswimming at small scales in both Newtonian (e.g., see [165] and referenceswithin) and non-Newtonian fluids (e.g., [44, 45, 51, 122, 135, 217]). Thesquirmer model allows us to study the response of different classes of mi-crowswimmers, namely, pushers, pullers and neutral swimmers, to spatialgradients in viscosity. We find that the three types of swimmers behave dif-ferently in viscosity gradients, and their different swimming dynamics canbe used to sort them based on their swimming style.We present the theoretical formulation of the problem in the next section,followed by a section on results and discussion, where we briefly discuss howviscotaxis may be used as an effective mechanism for sorting cells.3.2 Theoretical formulationWe model the microswimmers as spherical squirmers [137]. As a squirmer,a microswimmer is represented as a sphere with some prescribed surface ve-locity; the surface velocity approximates the detailed propulsion mechanismof the swimmer [137]. We consider axisymmetric squirmers with steadytangential surface velocities (uS = useθ),uS = Σ∞l=1BlVl (θ) , (3.1)where Vl (θ) = −2P 1l (cos θ) / (l (l + 1)) with P 1l being the associated Legen-dre function of the first kind and θ the polar angle measured with the axisof symmetry [17, 137]. The coefficients Bl are called the squirming modes.In Newtonian fluids, the propulsion velocity of the squirmer is due to justthe first mode B1, whereas B2 gives the strongest contribution to the flowfar from the swimmer [17, 101]. For swimmers that generate thrust fromthe front, the puller type (like Chlamydomonas), the ratio α = B2/B1 isgreater than zero, and for those that generate thrust from the rear (likeEscherichia coli), α < 0. Swimmers in which the thrust and drag centrescoincide are modelled with α = 0 and are called neutral squirmers. Pak& Lauga [159] provide a detailed description of general, non-axisymmetric,squirming modes.We consider fluids which have spatial gradients of viscosity. These gra-dients may arise due to temperature or due to gradients in the concentrationof solute in real systems. In order to study swimming in such fluids theoret-ically, we consider only small variations in viscosity, representing the viscos-ity field as η (x) = η0 + εδη (x). Here ε is a small dimensionless parameter.133.2. Theoretical formulationIn physical terms, with this form of viscosity field, we have assumed that thedeviations in viscosity on the scale of the particle are small i.e., O (ε), whereε = a/L, a being the length scale of the particle, and L is some length overwhich the viscosity changes are considerable (O (1)). Our considerationsof only small variations in viscosity are in the spirit of the recent work byOppenheimer et al. [157], who also consider small viscosity variations dueto the temperature difference between the swimmer and its surroundings.The novelty of the present work lies in the fact that the viscosity variationsare externally imposed and are independent of the swimmer or its position.The small viscosity variations, of O (ε), allow us to obtain the first effectsof viscosity gradients on the swimmer motion with the relative ease of usingthe reciprocal theorem of low Reynolds number flows [93]. We primarilyconsider two types of gradients, ∇ (δη (x) /η0) ∼ ex (or a similar form), and∇ (δη (x) /η0) ∼ er. We term the former as a linear gradient, and the latteras radial.Note that we neglect any fluid and solid body inertia, and study themicroswimmers at zero Reynolds number. Next, we give details of the re-ciprocal theorem formulation.3.2.1 Reciprocal theoremThe velocity of a particle in complex fluids (or in fluids that show anydeviation from Newtonian fluids with uniform viscosity) is given byU = ηˆηRˆ−1FU · [−F + FT + FNN ] , (3.2)where U = [U Ω]⊤ is a six-dimensional vector comprising of rigid-bodytranslational and rotational velocities. The expression (3.2) is obtained usingthe reciprocal theorem of low Reynolds number flows [93] and its formulationas described in [43, 68].FT =ηηˆ∫∂BuS · (n · TˆU)dS (3.3)represents the Newtonian ‘thrust’ due to any surface deformation or activityuS of the particle. Here ∂B represents the surface of the particle. The non-Newtonian contribution or, pertinent to present work, the contribution dueto any deviation from uniform viscosity in Newtonian fluids,FNN = −∫VτNN : EˆU dV, (3.4)143.2. Theoretical formulationrepresents the extra force/torque on the particle due to the non-Newtoniandeviatoric stress τNN in the fluid volume V in which the particle is immersed;the total deviatoric stress in the fluid is τ = η0γ˙ + τNN . F = [F L]⊤represents the total hydrodynamic force and torque on the particle; in theabsence of any external force, F = 0 for a squirmer. The hat quantities areoperators from the resistance/mobility problem in a Newtonian fluid (withviscosity ηˆ)ˆ˙γ/2 = EˆU · Uˆ, (3.5)σˆ = TˆU · Uˆ, (3.6)Fˆ = −RˆFU · Uˆ, (3.7)obtained for the present case of spherical squirmers from the motion of asingle sphere in an unbounded and otherwise quiescent Newtonian fluid.3.2.2 Asymptotic analysisThe viscosity field in the present work is represented as η (x) = η0+εδη (x).To study the effect of small viscosity variations, quantified by ε, we expandflow quantities in a regular perturbation expansion of ε, e.g.,{u, p, τ} = {u0, p0, τ0}+ ε {u1, p1, τ1}+ ε2 {u2, p2, τ2}+ . . . , (3.8)where {u0, p0, τ0} are velocity, pressures and deviatoric stress solutions tothe Stokes equations with uniform viscosity η0. At the leading order,τ0 = η0γ˙0, (3.9)and at O (ε),τ1 = η0γ˙1 + δη (x) γ˙0, (3.10)and therefore, τNN,1 = δη (x) γ˙0. We consider corrections up to O(ε) inthis work. Note that the expansion is regular only for δη (x) ∼ O (1);for linear gradients of the form δη (x) ∼ x, the expansion is regular onlyfor x ∼ o (1/ε). The evaluation of the volume integral in equation (3.4),for effects of non-standard Newtonian rheology on the particle, then, inprinciple, requires singular perturbation methods. In the next section, wesee how the far field contribution from the viscosity gradients that we choosedoes not affect the answers obtained from the regular expansion of the formin equation (3.8).153.3. Results and discussion3.2.3 Remarks on the asymptotic analysisThe extra force and torque on a particle due viscosity gradients, up to O (ε),are given byFNN,1 =∫Vδη (x) γ˙0 : EˆU dV,LNN,1 =∫Vδη (x) γ˙0 : EˆΩ dV.(3.11)For a squirmer in linear or radial viscosity variations, δη (x) ∼ r, the farfield contribution, from distances r ∼ O (1/ε), to the extra force in (3.11)is O (r · r−3 · r−2 · r3) = O (ε), to torque is O (r · r−3 · r−3 · r3) = O (ε2),and therefore, can be neglected as we consider corrections to Newtonian(uniform viscosity) motion up to only O (ε).As the velocity field due to motion of a passive sphere decays slowercompared to that of a squirmer, in principle, one needs to use a singularperturbation approach to evaluate integrals in (3.11). However, for a passivesphere in linear viscosity profiles, the far-field contribution to the integralsat O (1) is identically zero (due to symmetry), and therefore, such systemsmay be studied using the regular perturbation scheme. However, for radialprofiles, the O (1) contribution is non-zero and one then needs a singularperturbation approach.3.3 Results and discussionBelow, we show results for the motion of squirmers in such small viscositygradients as discussed; our focus is on gradients that are externally imposedand do not depend on the squirmers’ positions. We begin by looking at apassive sphere.3.3.1 Passive sphereThe hydrodynamic force F and torque L on a rigid sphere of radius amoving with a velocity U and rotating with an angular velocity Ω in alinear viscosity field, e.g., δη (x) ∼ x, is given by, up to O (ε),F = −6piaη0U − 2piεa3Ω ×∇δη, (3.12)L = −8piη0a3Ω + 2piεa3U ×∇δη. (3.13)Note that the gradient in viscosity couples the force with the sphere’s angu-lar velocity, and the torque with the translational velocity; this is not the163.3. Results and discussioncase for a sphere in a Newtonian fluid with uniform viscosity. As discussedpreviously, we cannot study a passive sphere in radial gradients using aregular perturbation of the form in (3.8).When the gradients in viscosity are due to the particle itself, as in thestudy of Oppenheimer et al. [157] where the particle is hot compared to itssurroundings and the viscosity depends on temperature such that δη (x) =−aη0/r, the origin being the centre of the particle, we can reproduce theirresults using equations (3.11) to getF = −6piη0a(1− 5ε12)U , (3.14)L = −8piη0a3(1− 3ε4)Ω. (3.15)It should be noted that in this case the translational and rotational motionare not coupled—because of the symmetry of the gradient—and both thedrag and the torque are lower than those for the case of constant viscosityη0.These results for a passive sphere will be useful to contrast the resultsobtained for spherical squirmers.3.3.2 SquirmerThe exact expressions for translational and angular velocities of squirmerscan be obtained in linear viscosity profiles such as δη (x) ∼ x. These are,up to O (ε),U = 2B13 e+ ε3aB25η0(ee− I3)·∇δη, (3.16)Ω = εB1∇δηη0× e, (3.17)where e is the orientation of the squirmer. Note that the gradient in viscosityaffects both the angular and translational velocities of the squirmer; evenan axisymmetric squirmer can now rotate due to its own motion. In aNewtonian fluid with uniform viscosity, UN = (2B1/3) e and ΩN = 0.The rotation of the squirmer is in the opposite sense to that of a passivesphere dragged along the same direction in the same gradient. This can beunderstood by decomposing the swimming problem into a thrust problemand a drag problem [45]. In the thrust problem, the squirmer is held fixedand the thrust due to its surface velocity is calculated. In the drag problem,173.3. Results and discussionthe drag on a passive sphere translating with the velocity of the swimmeris calculated. Here, one finds that for a squirmer, the thrust contribution isopposite and dominating to the drag contribution. Therefore, the squirmerrotates in a direction opposite to that of a passive sphere. Also note thatof all squirming modes Bl, only the first two contribute to velocities in thelinear gradients considered here.In the case where the viscosity field due to the squirmer is radially in-creasing, (in the spirit of [157] where the viscosity field varies radially dueto the temperature of the particle), choosing δη(x) = −aη0/r where r ismeasured from the origin at the centre of the squirmer, the translationalvelocity of the squirmer is given byU = 2B13(1− ε12)e, (3.18)i.e., a squirmer swimming in thin shells of radially increasing viscositiesaround it swims slower than in a Newtonian fluid with uniform viscosity.The thrust and drag decomposition of the swimming problem shows thatthe viscosity gradient leads to decrease in both the thrust and the drag, butthe decrease in the thrust is more pronounced than the drag leading to aslower swimming [45]. The angular velocity of the squirmer is zero in thiscase because of the symmetry of the problem.The expressions in (3.16) and (3.17) allow studying the trajectories ofsquirmers in viscosity gradients. To do so, we first non-dimensionalise theequations; we scale lengths with the swimmer radius a, velocities with thefirst squirming mode B1, and stresses with η0B1/a. Equations (3.16) and(3.17) then becomeU = 23e+ ε3α5(ee− I3)·∇δη, (3.19)Ω = ε∇δη × e, (3.20)and henceforth, all quantities are dimensionless unless stated otherwise.From equations (3.19) and (3.20), we note that as the swimmer moves alongthe gradient i.e. e =∇δη/|∇δη|, it does not rotate (Ω = 0), and thereforemaintains its original orientation. Depending on its type of propulsion, andconsequently, on the sign of α, it can swim faster, slower or at the same speedas in a Newtonian fluid with uniform viscosity. Pusher swimmers (α < 0)swim slower, as their thrust generation is from the rear which, when movingalong the gradient, is in a fluid less viscous than that in the front. Pullers(α > 0), on the contrary, swim faster. Neutral swimmers (α = 0), which183.3. Results and discussion0 10 20 30 40x -60-50-40-30-20-100ypusherneutralpullerei∇ηFigure 3.1: The initial orientation of the swimmers einitial is along the pos-itive x-axis. The viscosity gradient is along the positive y-axis. After some-time all swimmers swim antiparallel to the viscosity gradient. Note that inthis orientation, pushers are the fastest, while pullers are the slowest. Thetrajectories are plotted for time t = 0 to t = 100.have the drag and thrust centres coinciding, swim with the Newtonian speed.The dynamics for pullers and pushers reverses when e = −∇δη/|∇δη|.When the gradient in viscosity is not aligned with the swimmer direc-tion, irrespective of the type of propulsion, squirmers show viscophobicity(negative viscotaxis). They rotate in the direction of lower viscosity. Wedemonstrate this in figure 3.1, where we consider motion of squirmers onlyin the plane of viscosity gradients. We consider α = 0 for neutral squirmers,α = 2 and α = −2 for pullers and pushers, respectively, and ε = 0.1. For∇δη (x) parallel to ey and the initial orientation of swimmers in the positivex-direction, as shown in figure 3.1, the final equilibrium orientation for allthe swimmers is antiparallel to the viscosity gradient; however, the trajec-tories chosen to attain the equilibrium orientation do depend on the type ofswimmer. In this equilibrium orientation, as discussed previously, pushersswim the fastest and pullers the slowest. Note that the viscophobicity asdemonstrated here can be understood using the thrust and drag decompo-sition [45]; on the decomposition one finds that the effect of the gradient onthe thrust is dominant to that on the drag.For the same values of α and ε, we also plot trajectories of squirmers mov-193.3. Results and discussion-20 0 20 40 60-60-40-20020 pusher-50 0 50 100x-100-50050100yneutral-300 -200 -100 0 100 200-400-2000200400pullerFigure 3.2: Planar trajectories of pushers (left), neutral swimmers (centre),and pullers (right). The initial position of the swimmers, (x = 1, y = 1),is marked by the red dot. The swimmers originally point in the positivex-axis. The swimmers are in a radial viscosity gradient: viscosity increasesradially outward from the point (0, 0), δη =√x2 + y2. Pushers find a stableorbit. The trajectory of neutral swimmers is bounded at long times. Pullersperform ‘unstable’ motion about the ‘origin’ of viscosity gradient. Notethe farthest the swimmers have travelled from this origin after equal times(t = 4000).ing in the plane of radially varying planar viscosity field, δη (x) =√x2 + y2,where the coordinates x and y are measured from an arbitrary point in spacewhich may be seen as a ‘viscosity sink’. It should be noted that we still useexpressions (3.19) and (3.20) to calculate the trajectories; we are in want ofexact expressions for radial gradients of such types. Although the expres-sions have been derived only for linear gradients, they are expected to holdfor radial gradients as well, for at the scale of the particle, little differenceshould exist between the two. The trajectories are plotted in figure 3.2.Note the qualitatively different behaviour of the three types of swimmers.The dynamics is discussed in more detail below.On trajectories of squirmers in radial gradientsThe two-dimensional motion of squirmers in the radial gradient discussedabove is easier to study with the orientation vector e represented as{cosϕ, sinϕ}, and the gradient viscosity vector for δη (x) = √x2 + y2,as ∇δη = {cos θ, sin θ}, where ϕ and θ are measured with the x-axis.Equations (3.19) and (3.20) then becomedψdt= 13ε sinψ −1r(23 sinψ +310εα sin 2ψ), (3.21)drdt= 23 cosψ +35εα(23 − sin2 ψ), (3.22)203.4. Conclusionwhere ψ = ϕ − θ and r = √x2 + y2. For steady state, ψ˙ = 0, r˙ = 0, andtherefore, when α ̸= 0, we havereq =1ε(2 + 95εα cosψeq), (3.23)ψeq = cos−1− 59εα1±√1 + 27ε2α225. (3.24)When α = 0, at equilibrium, we have ψeq =pi2 and req =2ε. With ϵ = 0.1,for α = ±2 (values for pusher and pullers), we obtain req = 20.21 andψeq = 1.5114. For neutral swimmers, α = 0, req = 20 and ψeq = pi/2 .We study linear stability of the equilibrium values by Taylor expandingequations (3.21) and (3.22) and obtainingdr′dt=(−23 sinψeq −35εα sin 2ψeq)ψ′ (3.25)dψ′dt= 13ε(ψ′ cosψeq)+ r′r2eq(23 sinψeq +310εα sin 2ψeq)− 1req(23ψ′ cosψeq +310εαψ′2 cos 2ψeq),(3.26)where dashed quantities represent small fluctuations from the equilibriumvalues. We find that for pushers, the equilibrium is a stable spiral, whilefor pullers, it is an unstable spiral. For neutral swimmers, the equilibriumis a centre. These characteristics are also observed in figure 3.2. For sucha simple gradient, we observe very different dynamics of the three types ofsquirmers.3.4 ConclusionWe observe that squirmers are, in general, viscophobic (unless they moveperfectly aligned with the gradient); they turn towards the less viscous re-gion. Using the squirmer model, we find that the dynamics of microswim-mers depend on their propulsion type; even a simple viscosity profile leadsto different dynamics for different swimmers. The dynamics of the swim-mers, owing to linearity of the problem, can be explained by decomposingthe swimming problem into a thrust problem and a drag problem; the effectsof the gradients on the thrust are seen to dictate the swimming response.We also show that the differences in the dynamics can be used to sort these213.4. Conclusionswimmers by smartly choosing the viscosity profile. As long as the swim-mers are not perfectly aligned along (parallel) the gradient direction, andany noise in the system will be helpful towards this end, we find that in lin-ear gradients, pushers will be the farthest from the point of release. In radialgradients, neutral and pusher swimmers will remain bounded / trapped nearthe viscosity ‘sink’. The puller swimmers can therefore be sorted out at far-ther distances.22Chapter 4Dynamics and rheology ofparticles in shear-thinningfluids†Particle motion in non-Newtonian fluids can be markedly different than inNewtonian fluids. Here we look at the change in dynamics for a few problemsinvolving rigid spherical particles in shear-thinning fluids in the absence ofinertia. We give analytical formulas for sedimenting spheres, obtained bymeans of the reciprocal theorem, and demonstrate quantitatively differencesin comparison to a Newtonian fluid. We also calculate the first correctionto the suspension viscosity, the Einstein viscosity, for a dilute suspension ofspheres in a weakly shear-thinning fluid.4.1 IntroductionParticles in fluids are ubiquitous in both natural and industrial processes.Blood, detergents, paints, aerated drinks, fibre-reinforced polymers, sewagesludges, and drilling muds are some examples where particles—rigid or dropsor bubbles—are present in a suspending fluid [9, 33]. The flow behaviourand rheological properties of such suspensions depend on parameters like theparticles’ shape, size and concentration, particle-particle interaction, particlesurface properties, fluid rheology and the type of flow. Even the simplest ofsuch suspensions – small, rigid non-Brownian particles in a Newtonian fluid– exhibits rich rheological properties like shear-thinning, shear-thickening,and normal stress differences which are characteristics of complex fluids[56, 191]. In many common examples like paints, foods, fracking fluids andbiological suspensions, the suspending fluid itself is non-Newtonian. The†This chapter has been previously published under the same title in Journal of Non-Newtonian Fluid Mechanics 262 (2018) 107–144 by Datt and Elfring. © 2018 ElsevierB.V.234.1. Introductionproperties of these suspensions is therefore expected to be even more complex[143, 155, 212].In order to understand the properties of these suspensions, it becomesimperative to first understand the interaction of a single particle with thesurrounding media and the mutual interaction between two such particles.In fact, it is known that the motion and orientational dynamics of particlescan be strongly affected by the rheology of the surrounding fluid medium[29, 49, 132, 133, 216]. For example, at zero Reynolds number, while thecenter-to-center distance between two equal spherical particles which aresedimenting along their line of centers through a quiescent fluid is fixedindefinitely at its initial value in a Newtonian fluid [192], this distance isfound to change in the presence of viscoelasticity of the fluid medium [174].The understanding of particle dynamics in complex fluids is also importantfor applications in particle manipulation in microfluidic devices (see recentreviews [47, 138]). Phenomena like cross-stream migration, in which rigidspheres in a pressure-driven tube flow of viscoelastic liquid migrate eithertowards or away from the wall in the absence of inertia, can be used forcell-trapping in biomedical applications [108].Towards a fundamental understanding of particle dynamics in non-Newtonian fluids, in this work, we theoretically study the dynamics ofrigid non-Browinian spherical particles in shear-thinning fluids in theabsence of any fluid or particle inertia. Unlike for viscoelastic fluids,where theoretical studies have been used to develop insights for manyexperimental observations [49, 132], similar studies have been relatively fewin shear-thinning fluids and most of these studies have focussed on using thepower-law model [16] to model the shear-thinning rheology [33]. However,as argued by Chhabra et al. [34], Chhabra & Uhlherr [35] and Chhabra[33], a fluid model with a zero-shear viscosity should be preferred to thepower-law model for slow flows around spheres. Here we use the Carreaumodel for shear-thinning fluids [16] (discussed in the subsequent section) tostudy the following problems motivated by some recent experiments:i) Two equal spherical particles sedimenting along their line of centresthrough a quiescent fluid. In Newtonian fluids, Stimson & Jeffery [192]showed that the initial distance of separation is maintained as the particlessediment.ii) Sedimentation of a spherical particle which is also rotating due tosome external field. In Newtonian fluids, the sedimenting velocity does notdepend on the rotation rate. The translational and rotational motion for asphere are decoupled in a Newtonian fluid [93].iii) Sedimentation of a spherical particle in a linear background flow. In244.2. Reciprocal TheoremNewtonian fluids, the sedimenting velocity of a sphere depends only on the(local) velocity of the background flow but is independent of the velocitygradient.iv) The influence of particles on the viscosity of a shear-thinning fluid.For a dilute suspension of neutrally buoyant particles in a Newtonian fluid,it was shown by Einstein [66] that the bulk shear viscosity of the suspensionincreases due to the presence of particles.In the following sections, we analyse these problems in shear-thinningfluids in detail, but before that we briefly discuss our theoretical approachand the rheology of shear-thinning fluids.4.2 Reciprocal TheoremWe are interested here in the motion of, or equivalently forces on, particlesin complex fluids. These integrated quantities can be evaluated withoutresolution of the associated flow field of the complex fluid by employing thereciprocal theorem. This approach was comprehensively reviewed by Leal[133], and we use here a generalized formalism developed in a number ofrecent papers for active particles [44, 67, 68, 69]. Following Elfring [68], themotion U or forces F of N particles in a complex flow may be given byU = ηˆηRˆ−1FU · [−F + FT + FNN ] , (4.1)where U = [U Ω], F = [F L] are 6N -dimensional vectors comprising trans-lation/rotation and hydrodynamic force/torque respectively on N particles.If the inertia of the particles is negligible (small Stokes numbers), as wewill assume here, then the hydrodynamic force must balance any externalor applied force (such as weight due to gravity) F = −Fext. The forceFT =ηηˆ∫∂B(uS − u∞)· (n · TˆU)dS, (4.2)is a Newtonian ‘thrust’ due to any surface deformation or activity of theparticles uS (although in all cases here we consider rigid passive particles,uS = 0) and ‘drag’ from any background flow u∞. Here ∂B represents thesurfaces of all the particles. The non-Newtonian contributionFNN = −∫Vτ ′NN : EˆU dV, (4.3)254.3. Shear-thinning fluidrepresents the extra force/torque on each particle due to a non-Newtoniandeviatoric disturbance stress τ ′NN = τNN − τ∞NN in the fluid volume V inwhich the particles are immersed.The formulas rely on operators from an N -body resistance/mobilityproblem in a Newtonian fluid (with viscosity ηˆ)ˆ˙γ ′/2 = EˆU · Uˆ ′, (4.4)σˆ′ = TˆU · Uˆ ′, (4.5)Fˆ ′ = −RˆFU · Uˆ ′. (4.6)where primes indicate disturbance quantities. The tensors Eˆ and Tˆ arefunctions of position in space that map the (arbitrary) motion of all Nparticles Uˆ to the fluid strain-rate and stress fields respectively, while theN -body rigid-body resistance tensorRˆFU =[RˆFU RˆFΩRˆLU RˆLΩ]. (4.7)We note that no specific Uˆ needs to be chosen in the rigid-body dual problemas only the linear operators EˆU , TˆU and RˆFU enter the picture. In many casesthere may be symmetries in the problem which simplify these operatorssubstantially, likewise we may know that the forces/torques are in someway simplified (collinear with gravity for instance) and hence need not evendetermine all components of the operators.Components of EˆU , TˆU and RˆFU for a single sphere that we use in thisstudy are provided in Appendix A.4.3 Shear-thinning fluidAs we outlined in the previous sections, the presence of a non-Newtonianstress τNN can significantly alter the motion of particles in flows. In thiswork, we consider the effects of shear-thinning fluids, which experience aloss in apparent viscosity η with increasing strain-rates γ˙; specifically, thedeviatoric stressτ = η(γ˙)γ˙, (4.8)where the viscosity is modelled using the Carreau model for generalisedNewtonian fluids [16]η (γ˙) = η∞ + (η0 − η∞)[1 + λ2t |γ˙|2](n−1)/2. (4.9)264.3. Shear-thinning fluidHere, η0 is the zero-shear-rate viscosity, η∞ is the infinite-shear rate viscos-ity, n is the power law index (n < 1 for shear-thinning fluids; smaller thevalue of n, more shear-thinning the fluid is) and λt is a time constant. Themagnitude of the strain-rate is given by |γ˙| = (Π)1/2 where Π = γ˙ij γ˙ij is thesecond invariant of the strain-rate tensor. Note that limγ˙→0 η (γ˙) = η0 andlimγ˙→∞ η (γ˙) = η∞, showing that at low and high strain rates the fluid be-haves like a Newtonian fluid with viscosity η0 and η∞, respectively. λt setsthe cross-over strain-rates at which non-Newtonian effects start to becomeimportant.In this work, we investigate strain-rates such that λt ≪ 1/γ˙c, where γ˙cis the characteristic strain-rate of the flow. In this case it is useful to writethe constitutive equation in the formτ = η0γ˙ + (η(γ˙)− η0) γ˙. (4.10)Although, we note that this rearrangement is not in any way restricted to lowstrain rates. Writing the equation as such, it is clear that the non-Newtoniancontribution τNN = (η(γ˙)− η0)γ˙.In dimensionless form, one may decouple the Newtonian and non-Newtonian contribution for a Carreau fluid asτ ∗ = γ˙∗ +{β − 1 + (1− β)[1 + Cu2|γ˙∗|2](n−1)/2}γ˙∗, (4.11)where stars (*) represent dimensionless flow quantities. The Carreau numberCu = γ˙cλt is the ratio of the characteristic strain rate in the flow γ˙c to thecrossover strain rate 1/λt. The viscosity ratio is given by β = η∞/η0 ∈ [0, 1].The characteristic length of the particle is chosen as the length scale in theproblems and as we consider only spherical particles, the length scale isa, the radius of the particles. η0γ˙c is the scale for stresses; the appropriatecharacteristic strain-rate, γ˙c, varies depending on the problem and therefore,is defined separately in each of the problems below.In this work, we consider the fluid behaviour to be weakly shear-thinning,in the sense that Cu≪ 1 [45], and therefore the viscosity is assumed to notdeviate substantially from the zero-shear viscosity η0. We then explore theleading-order weakly shear-thinning effects of the fluid rheology on particlemotion. To this end, we assume a regular perturbation expansion of allfields, e.g. u = u0 + Cu2u1 + . . ., and find the non-Newtonian stress to beτ ∗NN = −12Cu2 (1− n) (1− β) |γ˙∗0 |2γ˙∗0 +O(Cu4), (4.12)274.4. Sedimenting sphereswhere for shear-thinning fluids β < 1 and n < 1. We consider only theleading-order effects of shear-thinning viscosity and by using (4.12), we mayobtain the non-Newtonian force on particles at the expense of an integration(4.3) which then only requires the Newtonian flow field u0.4.4 Sedimenting spheresDue to their symmetry, a number of classic results involving motion ofspheres in Newtonian fluids can be predicted directly by employing thekinematic reversibility of the field equations and it is insightful to considerexamples where dynamics are altered (or not) by shear-thinning rheology.In this section, we explore one such case of two spheres sedimenting alongtheir line of centres but before that, it is instructive to first examine thesimple case of a single sphere moving through a shear-thinning fluid and tosee how the force-motion relationship is affected by the medium rheology.4.4.1 Single sphereThe drag force on a sphere of radius a, moving with a velocity U is givenby F = −6piaηU (dimensional), where η is the viscosity of the fluid. In ashear-thinning fluid with zero-shear rate viscosity η0 = η, the drag force isexpected to be less than the Newtonian value. This is because one expectsthe apparent viscosity around the sphere to decrease below η due to thestrain-rates ensuing from the motion of the sphere. Quantitatively, the dragforce in shear-thinning fluid can be evaluated using the reciprocal theorem.For a single sphere, we expect this force to be colinear with the velocity bysymmetry. Simplifying (4.3) we may writeF = −6piη0aU −∫VτNN : EˆU dV, (4.13)where EˆU (such that ˆ˙γ/2 = EˆU · Uˆ) is well known for a sphere translatingin Stokes flow [89]. The integral above may be easily evaluated to leadingorder as then the non-Newtonian stress depends only on the solution ofa translating sphere in a Newtonian fluid i.e. τNN [u0] (from(4.12)). Indimensionless form we find the drag in a shear-thinning fluid to beF ∗ = −6piU∗(1− 12 (1− β) (1− n)Cu2 9422275 |U∗|2). (4.14)If we (sensibly) take as the characteristic strain-rate γ˙c = |U |/a then U∗ = eis simply the unit vector in the direction of the motion (a convention we use284.4. Sedimenting spheresbelow). The term in the brackets is then an analytical calculation of thedrag correction factor (often given the symbol X in the literature [30, 35])valid to O(Cu2).We see that, as expected, the drag force on a sphere decreases in a shear-thinning fluid as compared to a Newtonian fluid. It should be noted thatthe reduction in drag below the Newtonian value is at odds with the con-clusions reached using the power-law fluid model which predict an increasein the drag force [33] because (as argued in that work) the power-law fluidmodel does not incorporate a zero shear-rate viscosity which is important inmodelling slow flows involving stagnation points and vanishingly small shearrates. In contrast, (4.14) predicts the reduction in drag observed in exper-imental results [35], and qualitatively agrees with a variational estimate byChhabra & Uhlherr [35], and numerical results from Bush & Phan-Thien[30], which both used the Carreau fluid model to characterise the fluid rhe-ology.It is straightforward to invert the drag force to obtain the velocity givena prescribed external force (for example weight due to gravity in sedimen-tation)U∗ = F∗ext6pi(1 + 12 (1− β) (1− n)Cu2 9422275|F ∗ext|2(6pi)2). (4.15)In this case an appropriate strain-rate scale is γ˙c = |Fext|/(η0a2) in whichcase F ∗ext would be a unit vector.4.4.2 Two spheresWe now consider sedimentation of two spheres of equal radii along the linejoining their centres. In a Newtonian fluid, one can use arguments of kine-matic reversibility and symmetry to find that the two spheres will sedimentwith equal velocities and will maintain their initial distance of separation[89]. Stimson & Jeffery [192] solved the hydrodynamically equivalent prob-lem of two spheres moving with a constant velocity along their line of centresand calculated the flow field and the forces on the spheres. When their radiiare equal, it was found that the forces on each of the two spheres are indeedequal but each less than on single sphere moving in a quiescent fluid withthe same velocity. Quantitatively, the force on either sphere can be writtenas F = −6piηaUλ (dimensional), where λ is a coefficient which depends onthe separation between the two spheres [192]. λ → 1 as the distance be-tween the two spheres approaches infinity, i.e. when the two spheres do not294.4. Sedimenting spheresinteract hydrodynamically with each other. It was also found that value ofλ decreases as the distance between the two spheres decreases. However, themethod of Stimson & Jeffery [192] could not be applied to the case of twospheres in contact with each other. This case was later solved by Cooley &O’Neill [38] who found λ = 0.645 (for the two sphere touching case). Theseresults and values of λ have also been observed experimentally [93].Non-Newtonian rheology can break kinematic reversibility and indeedviscoelasticity is known to change these results qualitatively. Where in New-tonian fluids two equal sedimenting spheres maintain their initial distanceof separation, it was found in the experiments of Riddle et al. [174] andanalyses of Brunn [28] and Ardekani et al. [4] that the two particles showeda tendency to aggregate in viscoelastic fluids. This effect of normal stresseson particle dynamics has been commented upon by Joseph & Feng [104].In shear-thinning fluids, it was observed in the experiments of Dauganet al. [46] that the two spheres would aggregate provided the initial distanceof their separation was smaller than some critical distance. Yu et al. [215], intheir experimental study, argued that in fact this tendency towards aggre-gation was due to thixotropy (memory of shear-thinning) and the corridorsof reduced viscosity in the wake of sedimenting particles lead to aggrega-tion [46, 105]. In the absence of memory, the two spheres would maintaintheir initial distance of separation [215]. Here, we theoretically study theequivalent problem of two equal spheres moving with a constant velocityalong their line of centres, as considered by Stimson & Jeffery [192], but ina shear-thinning fluid.We use the reciprocal theorem to calculate the forces on the two spheres.By (4.1) the force on the particles can be written generally asF = −ηηˆRˆFU · U + FNN . (4.16)For two equal spheres moving along their line of centres, by symmetry, weexpect all vectors in the problem to be collinear, which significantly simplifiesthe more general problem of the motion of two spheres. When the motion oftwo bodies is collinear (only translational motion is considered), it is useful todecompose the motion into a mean velocity U and relative velocity ∆U suchthat the velocity of each sphere may be written as U1 = U +∆U and U2 =U − ∆U . In this basis, the relevant resistance/mobility problems for thereciprocal theorem are i) two (equal) spheres translating with equal velocityalong the line joining their centres in a Newtonian fluid (corresponding toU as in this case ∆U = 0) with solution by Stimson & Jeffery [192] and ii)two (equal) spheres approaching each other with equal speed along the line304.4. Sedimenting spheresjoining their centres in a Newtonian fluid with solution due to Maude [141]and Brenner [23] (corresponding to ∆U as in this case U = 0). Note thatone requires two resistance/mobility problems for evaluating the force oneach of the two spheres. The resistance tensor for this problem is diagonalin the sense that in a Newtonian fluid mean translation leads only to amean force and likewise for relative force. The reciprocal theorem givenabove leads to expression for mean and relative force in a non-Newtowianfluid given byF = −η0ηˆRˆFU ·U −12∫VτNN : EˆU dV, (4.17)∆F = −12∫VτNN : Eˆ∆U dV, (4.18)where RˆFU is the (mean) translational resistance of the two spheres in aNewtonian fluid and EˆU and Eˆ∆U correspond to the strain-rate due to meanand relative motion respectively. Note that τNN is evaluated using thesolution of the problem in Newtonian fluids by Stimson & Jeffery [192] (from(4.12)).Upon evaluation of (4.18), for weakly non-linear shear-thinning fluidsone finds that there is no relative force∆F = 0, (4.19)meaning the forces on two equal spheres are equal in a shear-thinning fluidas in a Newtonian fluid. Although we obtain this result only for a weaklyshear-thinning fluid, we expect this to be the case for all generalized New-tonian fluids, regardless of the parameter regime. The reason is that thestress τNN maintains the symmetry of the Newtonian problem while theMaude-Brenner problem (and thus the operator Eˆ∆U ) displays a mirror-image symmetry and therefore the integral over the entire fluid volume mustbe zero. In fact, Brunn [29] briefly comments on this property of generalizedNewtonian fluids where results may come out to be similar to Newtonianfluids.As the force on the two spheres in a weakly shear-thinning fluid areequal, two sedimenting spheres do not show any tendency to aggregate ina shear-thinning fluid without memory, as was also found in the numericalwork of Yu et al. [215] discussed previously. We can further calculate theforce on each of the spheres, and compare it to the force on a single spherein a shear-thinning fluid from section 4.4.1. Since there is no difference in314.4. Sedimenting spheres1 10 100 1000 10000h / d345678F1Two spheresSingle sphere1 10 100 1000 100002.35h / d0.9840.9860.9880.990.992F / F0Two SpheresSingle Spherea) b)Figure 4.1: a) Variation of non-Newtonian drag F 1 as a function of thedistance of separation between the two spheres. b) Normalized drag as afunction of this distance. Note that the variation is non-monotonic and thatdrag reduction is minimized at h/d ≈ 2.35.drag force, the mean in (4.17) represents the hydrodynamic drag on eachsphere. The form of the force (dimensionless) isF∗ = F ∗0 +12Cu2 (1− β) (1− n)F 1e+O(Cu4), (4.20)where F ∗0 is the force on each sphere in a Newtonian fluid [192]. Herewe have non-dimensionalised lengths with sphere radius a and stresses withη0U/a where U is the magnitude of velocity of the spheres such that U∗ = e.Both F ∗0 and the coefficient F 1 obtained by evaluating the integral in (4.17)numerically, depend on the ratio of the centre distance between the spheresto their diameter, h/d, as shown in Figure 4.1a). Note that F 1 is positive,meaning the correction to drag force is in the direction of the motion e andso the drag in a shear-thinning fluid is less than in a Newtonian fluid. Tocontrast the results with the Newtonian fluids case, we also plot the ratioof the magnitude of the force in a weakly shear-thinning fluid (correct toO(Cu2)) to the drag in an equivalent Newtonian fluid for same configurationin Figure 4.1b). This is plotted in Figure 4.1b) for Cu2 = 0.1, β = 0.001and n = 0.25.We note that the ratio F/F0 is always less than 1 which shows that thedrag in a shear-thinning fluid is less than that in Newtonian fluid for thesame configuration. This is expected as the viscosity in a shear-thinningfluid decreases with strain-rate leading to less drag on each sphere whencompared to in Newtonian fluid. But what is perhaps surprising is the vari-ation of force with distance between the two spheres. We first note that asthis distance becomes large, the drag reduction on each sphere asymptotes324.5. Sedimentation of a rotating sphereto that on a single sphere in shear-thinning fluid. The drag reduction whenthe two spheres essentially do not interact hydrodynamically with each otheris greater than for any other distance of separation. Shear-thinning effectsare maximum in this configuration. When the spheres interact hydrody-namically the effective strain rates are reduced (due to screening) and thusthe drag reduction on each is always lower than for a single sphere. On theother end of the distances, we have the configuration when the two spheresare in contact with each other. This, interestingly, is not the configurationto observe minimum shear-thinning effects. In fact, the minimum shear-thinning effects or equivalently the maximum of the ratio F/F0 is observedwhen the clearance between the two spheres is of the order of their diameteri.e. when h/d ≈ 2.35. We believe that this nontrivial result may be due tothe complex flow field around the spheres, which also includes ring vortices,and its effect on the dissipation rates [50].4.5 Sedimentation of a rotating sphereWe now consider the case of a sphere which translates as well as rotates in ashear-thinning fluid. The calculations are inspired by the recent experimen-tal work of Godínez et al. [82] who study the hydrodynamically equivalentproblem of sedimentation of a rotating sphere in a power-law fluid. By im-posing a controlled rotation on a sedimenting sphere, Godínez et al. [82]measured the increase in the sedimentation velocity, which could then beused to predict the values of power law indices of the fluids. They consideredrotation of the sphere only about the sedimenting axis. Here we considerthe problem more generally.Translation and rotation of a sphere in a Newtonian fluid are decoupledand owing to the linearity of the Stokes flow, one may superimpose thesolution of translation alone and rotation alone to get the solution of atranslating-rotating sphere in a Newtonian fluid. In other words a spherethat rotates in a Newtonian fluid will sediment at the same rate as when itdoes not rotate. This decoupling of translation and rotation is not expectedto hold in a non-linear fluid. We explore this for a weakly shear-thinningfluid, again using the reciprocal theorem.According to the reciprocal theorem, as before, we haveF = −ηηˆRˆFU · U −∫VτNN : EˆU dV. (4.21)We non-dimensionalise length with the sphere radius, a, stresses by η0U/awhere U is the magnitude of velocity of the sphere and hence U∗ = e. We334.5. Sedimentation of a rotating sphereconsider a general angular velocity which in dimensionless form, Ω∗, is notnecessarily a unit vector. Then from (4.21) we getF ∗ = −6pie ·{I − 12Cu2 (1− n) (1− β)(9422275I +552385[2 |Ω∗|2 I −Ω∗Ω∗])}.(4.22)In the absence of any rotation Ω∗ = 0, the force corresponds with (4.14) asexpected. With nonzero rotation, the drag force is further reduced due to theadditional strain-rate caused by rotation. When the rotation is aligned withthe translation,Ω ∝ U , for example when the axis of rotation is aligned withgravity for a sedimenting sphere as in the experiments of Godínez et al. [82],the drag force remains collinear with U . When the rotation is not alignedwith translation, a lateral force may arise due to the term in the directionof the axis of rotation ∝ (e · Ω∗)Ω∗. When the rotation is orthogonal totranslation there is no lateral drift and the change in the drag force due torotation is twice that of when the rotation is aligned with translation and sowould maximize sedimentation velocity for a given rotation rate. Conversely,in the mobility problem for a given external force F ∗ext, a rotating spherewill sediment with a translational velocity given byU∗ = F∗ext6pi ·{I + 12Cu2 (1− n) (1− β)(9422275|F ∗ext|2(6pi)2I + 552385[2 |Ω∗|2 I −Ω∗Ω∗])},(4.23)where F ∗ext is a unit vector if the strain rate scale γ˙c is chosen as|Fext|/(η0a2). On comparison with the experimental results, it is notedthat Godínez et al. [82] find a power law dependence of the sedimentingvelocity on the rotation rate, |U | ∝ |Ω|(1−n), across a range of rotationrates such that strain-rate around the sphere is predominantly due torotation and not due to translation, our analytical result in equation(4.23), valid for small Carreau numbers, draws a similar picture, namely,increasing the rotation rate and decreasing the power-law index, n, increasethe sedimentation velocity of the sphere.We also calculate the hydrodynamic torque on the particleL∗ = −8piΩ∗ ·{I − 12Cu2 (1− n) (1− β)(245 |Ω∗|2 I + 414385 [2I − ee])}.(4.24)The first term in the shear-thinning correction is due to the particle rotationalone as there is a reduction in the torque due to the shear-thinning causedby the rotation. The second term in the correction is due to both the344.6. Sphere under an external force in a linear flow of shear-thinning fluidtranslation and rotation of the sphere and may generate a torque which isnot aligned with the direction of rotation. Clearly, in a shear-thinning fluidtranslational and rotational dynamics are coupled.4.6 Sphere under an external force in a linearflow of shear-thinning fluidWe now consider the dynamics of a spherical particle driven by an externalforce in an unbounded linear-flow. In a Newtonian fluid, we know that thesedimenting velocity of a sphere is not altered by the velocity gradient insimple shear flow. However, this may not be the case in non-linear fluids. Infact, in viscoelastic fluids, it is found that the terminal velocity of a spheredecreases when the applied shear-flow is perpendicular to gravity in whatis called a cross-shear-flow [27, 49, 96]. Gheissary & van den Brule [80]used sedimentation of a sphere in cross-shear flow to predict the rheologicalproperties of different shear-thinning fluids. The cross-shear flow is a modelsystem used for transport of particles in hydraulically-induced fractures [8].Einarsson & Mehlig [64] recently extended the analyses in viscoelastic fluidsto the case when gravity (or another external force) and the vorticity direc-tion of the applied flow are not aligned. Here, we perform a similar analysisfor a shear-thinning fluid.Using the reciprocal theorem we calculate the velocities (both rotationaland angular) of the particle asU = ηˆηRˆ−1FU ·[Fext − ηηˆ∫∂Bu∞ · (n · TˆU)dS −∫Vτ ′NN : EˆU dV], (4.25)where Fext = [Fext 0]T . Here, Fext is an arbitrary external force actingon the particle. The particle is immersed in a 2D linear flow given byu∞∗ = A∞∗ · x∗ (dimensionless), in a Cartesian basis we may writeA∞∗ = 1 + λ 1− λ 0− (1− λ) − (1 + λ) 00 0 0 (4.26)where we have scaled length with the radius of the sphere and stresses withηoγ˙c, where γ˙c is characteristic of the applied strain-rate such that we haveA∞∗ in the form above. It is useful to also decompose u∞∗ = 12 γ˙∞∗ · x∗ +Ω∞∗ × x∗ into symmetric and antisymmetric parts associated with strain-rate and rotation-rate respectively. Note that λ = −1 corresponds to purelyrotational flow, λ = 0 is shear flow and λ = 1 extensional flow [119].354.7. Suspension of spheres in a shear-thinning fluidOn evaluating (4.25) we get the translational velocity of the particleU∗ ={I + 12Cu2 (1− β) (1− n)[9422275|F ∗ext|2(6pi)2I + 695539|γ˙∞∗|2I + 100377007γ˙∞∗ · γ˙∞∗]}· F∗ext6pi.(4.27)The first two terms on the right hand side correspond to the velocity due toan external force in an otherwise quiescent shear-thinning fluid as in (4.15).The remaining terms demonstrates the coupling between the backgroundflow field and external force. When the force is perpendicular to the planeof applied flow, we see that the velocity of the particle is further increasedabove its quiescent fluid value due to the thinning of the fluid by the externalflow field. However, interestingly, for any general direction of the externalforce, the velocity of the particle may not be in the direction of the forcing.This is due to the lack of symmetry of the background flow field in onedirection. It is also worth noting that for a purely rotational flow, one doesnot see a shear-thinning effect arising from the background flow.Using (4.25) we also evaluate the angular velocity of the sphere, whichis given byΩ∗ = Ω∞∗ + 12Cu2 (1− β) (1− n) 31894004[F ∗ext6pi × γ˙∞∗ · F∗ext6pi]. (4.28)Note that in the absence of any forcing the sphere rotates with just thebackground angular velocity just like in a Newtonian fluid where the angularvelocity of the sphere in a background flow is independent of viscosity. Thiswas also found in the numerical simulations of D’Avino et al. [48]. Also, ifthe external force is along any of the principal directions of strain, or thebackground flow is purely rotational, the angular velocity of the sphere willbe just due to the rotational component of the background flow. However,for an arbitrary direction of the external force, the angular velocity may bedifferent than that imposed by background flow field.4.7 Suspension of spheres in a shear-thinningfluidSuspensions of particles in shear-thinning fluids are encountered in a widerange of chemical, biochemical and material processing industries, and assuch there has been considerable interest in studying the flow properties ofsuch suspensions [32, 90, 114, 130, 161, 199]. Most of these studies consider364.7. Suspension of spheres in a shear-thinning fluidparticles in power-law fluids, in other words, it is assumed that the strain-rates are large enough so that the fluid rheology is captured by a power lawmodel. Here, we complement these studies by quantifying the first effectsof the non-Newtonian rheology of the suspending fluid in the realm of smallstrain-rates.We calculate the average stress in a dilute suspension of neutrally buoy-ant rigid spheres in a weakly shear-thinning fluid, subject to a linear back-ground flowu∞∗ = A∞∗ · x∗ (4.29)as discussed in the previous section. The average stress in a suspension ofrigid spheres in a weakly shear-thinning fluid, correct to O (Cu2), is evalu-ated as⟨σ̂∗⟩ = ⟨γ˙∗⟩ − 12Cu2 (1− β) (1− n) ⟨|γ˙∗|2γ˙∗⟩+ ⟨σ̂∗p⟩, (4.30)where σp is the additional stress within the suspended particles, and theaverage quantities (denoted with angular brackets) are obtained by takingan ensemble average over all possible configurations of the particles [10,116, 173]. We use the wide hat symbol ̂ to refer to the symmetric anddeviatoric component of a second-order tensor and note that the isotropicterms in the average stress do not contribute to suspension rheology [65, 173].In a homogeneous and dilute suspension of particles, we know⟨σ̂∗p⟩ = nŜ∗ (4.31)where n is the particle number density equal to ϕ/Vp, where ϕ ≪ 1 is theparticle volume fraction, Vp is the volume of a single particle, and S is theparticle stresslet [10, 116] defined asS =∫∂B12 [n · σr + rn · σ] dS. (4.32)In order to calculate the average stress in the suspension, we start byevaluating the particle stresslet using the reciprocal theorem [65, 68, 128].Elfring [68] derives an expression for the stresslet in a weakly non-Newtonianfluid which is (in dimensional form)S = S∞ − ηηˆ∫∂Bu∞ ·(n · TˆE)dS −∫Vτ ′NN : EˆE dV. (4.33)374.7. Suspension of spheres in a shear-thinning fluidClearly, evaluation of the stresslet S up to O (Cu2) using equation (4.32)would require calculating the first correction to the flow field in the shear-thinning fluid; however, equation (4.33), obtained using the reciprocal the-orem, bypasses this calculation and the first correction to the stresslet isobtained by using the Newtonian flow field.We first evaluate the stresslet S∞ due to the stress σ∞ from the back-ground flow. In the absence of particles, the background stress σ∞ is simplyσ∞∗ = γ∞∗ − 12Cu2 (1− n) (1− β) |γ˙∞∗|2γ˙∞∗ + O(Cu4), (4.34)where γ˙∞∗ = A∞∗ +A∞∗⊤ is the applied strain rate. Using the definitionof the stresslet (4.32), we therefore have, up to O (Cu2),S∞∗ = 4pi3 γ˙∞∗ − 12Cu2 (1− β) (1− n) 4pi3 |γ˙∞∗|2γ˙∞∗. (4.35)Evaluating the integral terms in (4.33), we obtainS∗ − S∞∗ = 2piγ˙∞∗ − 12Cu2 (1− β) (1− n) 68469pi17017 |γ˙∞∗|2γ˙∞∗, (4.36)and therefore altogether haveS∗ = 10pi3 γ˙∞∗ − 12Cu2 (1− β) (1− n)(273475pi51051)|γ˙∞∗|2γ˙∞∗. (4.37)The first term on the right hand side is the stresslet in a Newtonian fluid.From the equation above, it can be seen that the total stresslet in a shear-thinning fluid is less than that in Newtonian fluid.We now proceed to calculate the average stress in equation (4.30). Wenote that the mean Newtonian viscous stress, ⟨γ˙∗⟩, is equal to the bulkapplied strain-rate γ˙∞∗ [89, 116]. The second term on the right hand sidein equation (4.30) can be evaluated by first performing a formal ensembleaverage based on the ergodic hypothesis [173] on ⟨|γ˙∗|2γ˙∗⟩. Writing thestrain-rate in terms of the mean and fluctuating components γ˙∗ij = ⟨γ˙∗ij⟩+γ˙′∗ij ,we obtain⟨|γ˙∗|2⟩ = ⟨γ˙∗ij γ˙∗ij⟩ = ⟨γ˙∗ij⟩⟨γ˙∗ij⟩+ ⟨γ˙′∗ij γ˙′∗ij ⟩, (4.38)where dashed quantities are fluctuating values. Here, we have used the fact⟨γ˙′ij⟨γ˙ij⟩⟩= 0 and as such, obtain |γ˙∗|2′ = 2γ˙′∗ij ⟨γ˙∗ij⟩+γ˙′∗ij γ˙′∗ij−⟨γ˙′∗ij γ˙′∗ij ⟩. Using384.8. Conclusionthese, the ensemble average:⟨|γ˙|2∗γ˙∗⟩ = ⟨|γ˙|2∗⟩ ⟨γ˙∗⟩+ ⟨|γ˙|2′∗γ˙ ′∗⟩ (4.39)=(⟨γ˙∗ij⟩ ⟨γ˙∗ij⟩+ ⟨γ˙′∗ij γ˙′∗ij ⟩)⟨γ˙∗⟩+ 2⟨γ˙′∗ij ⟨γ˙∗ij⟩γ˙′∗⟩+ ⟨γ˙′∗ij γ˙′∗ij γ˙′∗⟩.(4.40)Performing this ensemble averaging step has been shown [173] to removeterms that in a volume average give rise to divergent integrals for dilutesuspensions in second-order fluids [87, 95]. We now replace ensemble averageby volume average in (4.40), and evaluate the quantities both inside thesolid spheres and in the fluid volume, noting that inside the solid particlesγ˙′∗ = −γ˙∞∗, since the total strain-rate inside the particle is zero [116].Following Koch & Subramanian [116], evaluation of (4.40) gives⟨|γ˙∗0 |2γ˙∗0⟩ =(1 + 125ϕ28)|γ˙∞∗|2γ˙∞∗. (4.41)Summing all the contributions to the average stress in equation (4.30),we have, finally,⟨σ̂∗⟩ =[1 + 2.5ϕ− 12Cu2 (1− β) (1− n) (1 + bϕ) |γ˙∞∗|2]γ˙∞∗. (4.42)where b = 288675/34034. The term inside the square bracket gives the effec-tive viscosity of the suspension. The presence of particles thickens the fluidat the leading order (Einstein viscosity) where as at O (Cu2) it decreasesthe effective viscosity due to enhanced thinning of the fluid. We also notethat decreasing n linearly reduces the total correction to fluid viscosity fromthe Einstein correction as discussed by Tanner et al. [199] for dilute sus-pensions in power law fluids. The presence of particles in a shear-thinningfluid could lead to interesting rheological behaviour when the thickeningand thinning effects of particles compete at the same order. Also, the formof above expression suggests that a dilute suspension of rigid spheres in aCarreau fluid will behave as a Carreau fluid. In fact, our results agree withrecent results by Domurath et al. [58] who use a numerical homogenizationtechnique to obtain the effective viscosity of a dilute suspension in a Bird-Carreau model and find that the effective viscosity too can be modelledusing a Bird-Carreau model with modified values of the parameters.4.8 ConclusionIn this work, we considered a few problems involving spheres in shear-thinning fluids at zero Reynolds number. Using the reciprocal theorem, we394.8. Conclusionanalytically demonstrated how shear-thinning rheology may lead to qualita-tive changes in the particle dynamics compared to Newtonian fluids. Specif-ically, we showed that the translational and rotational dynamics of a sphereare coupled in shear-thinning fluids which can lead to interesting dynamicsin problems involving sedimentation of rotating spheres (a setup which maybe used as a rheometer) and sedimentation in a background flow field. Wealso showed that for two equal spheres sedimenting along the line joiningtheir centres, the symmetry arguments used in Newtonian fluids will pre-dict the observed result in a generalised Newtonian fluid. Although thesetwo spheres will sediment maintaining their initial distance of separation,the variation of the shear-thinning effects with initial separation distanceis non-monotonic. Finally, we considered a dilute suspension of spheresin a weakly shear-thinning fluid and showed that the resulting suspensionwill also be a weakly shear-thinning fluid with a viscosity that varies due tocompeting effects arising from the presence of particles: the particles thickenthe fluid (the Einstein viscosity correction) but also increase effective strain-rates thereby enhancing shear-thinning. At higher strain-rates, outside thescope of our weakly non-linear assumption, it would be interesting to in-vestigate strain rates at which the thinning effect supersedes the thickeningone.40Chapter 5Squirming throughshear-thinning fluids ‡Many microorganisms find themselves immersed in fluids displaying non-Newtonian rheological properties such as viscoelasticity and shear-thinningviscosity. The effects of viscoelasticity on swimming at low Reynolds num-bers have already received considerable attention, but much less is knownabout swimming in shear-thinning fluids. A general understanding of thefundamental question of how shear-thinning rheology influences swimmingstill remains elusive. To probe this question further, we study a sphericalsquirmer in a shear-thinning fluid using a combination of asymptotic anal-ysis and numerical simulations. Shear-thinning rheology is found to affecta squirming swimmer in nontrivial and surprising ways; we predict andshow instances of both faster and slower swimming depending on the surfaceactuation of the squirmer. We also illustrate that while a drag and thrustdecomposition can provide insights into swimming in Newtonian fluids, ex-tending this intuition to problems in complex media can prove problematic.5.1 IntroductionSelf-propulsion at small length scales is widely observed in biology; com-mon examples include spermatozoa reaching the ovum during reproduction,microorganisms escaping predators and microbes foraging for food [21, 72].While swimming at low Reynolds numbers is well studied for Newtonian flu-ids [129], an understanding of the effects of complex (non-Newtonian) fluidson locomotion is still developing. Many biological fluids, such as blood orrespiratory and cervical mucus, display complex rheological properties in-cluding viscoelasticity and shear-thinning viscosity [120, 142]. A viscoelas-tic fluid retains a memory of its flow history, whereas the viscosity of a‡This chapter has been previously published under the same title in Journal of FluidMechanics Rapids, 784 (2015) R1 by Datt et al. © 2015 Cambridge University Press.415.1. Introductionshear-thinning fluid decreases with the shear rate. While it is importantto elucidate how non-Newtonian fluid rheology influences propulsion at lowReynolds numbers because microorganisms swim through biological fluidspossessing these properties, an improved understanding may also guide thedesign of artificial microswimmers [172] and novel microsystems [140] ex-ploiting these nonlinear fluid properties.Recent research has begun to shed light on the effects of viscoelasticity(see the reviews by Sznitman & Arratia [195] and Elfring & Lauga [70]),but much less is known about swimming in shear-thinning fluids at lowReynolds numbers. Dasgupta et al. [42] measured a decreased swimmingspeed of a waving sheet in a shear-thinning viscoelastic fluid relative to aNewtonian fluid. In contrast, an asymptotic study of a sheet driven bysmall-amplitude waves showed that the swimming speed of a waving sheetremains unchanged in an inelastic shear-thinning fluid compared to that ina Newtonian fluid [205]. A recent experiment by Gagnon et al. [77] on thelocomotion of the nematode Caenorhabditis elegans has also suggested thatshear-thinning viscosity does not modify the nematode’s beating kinematicsor swimming speed. In addition, numerical studies [148, 149] examined avariety of two-dimensional swimmers and showed that faster or slower swim-ming in shear-thinning fluids can occur depending on the class of swimmerand its swimming gait. The results were understood in terms of the fluidviscosity distribution surrounding the thrust and drag elements of the swim-mer. By estimating separately the propulsive thrust and drag force on theswimmer, Qiu et al. [172] obtained a scaling relation predicting the swim-ming velocity of a single-hinge swimmer (a microscallop), which is enabledto move at low Reynolds numbers by shear-thinning rheology.The question that emerges from recent literature is when (and why) aswimmer goes faster or slower in a shear-thinning fluid [126]. To addressthis question we study a canonical idealized model swimmer, the squirmer,in a shear-thinning fluid described by the Carreau-Yasuda model using acombination of asymptotic analysis and numerical simulations.We predictand show instances of both faster and slower swimming depending on thesurface actuation of the squirmer. We also explore separately the effects ofshear-thinning on the propulsive thrust generated by the squirmer and thedrag force it experiences, and demonstrate that extension of these findingsto swimming in non-Newtonian fluids can prove problematic.425.2. Theoretical framework5.2 Theoretical frameworkThe hydrodynamics of spherical bodies propelling themselves with surfacedistortions, otherwise known as squirmers, was first studied by Lighthill[137] and Blake [17]. We follow this approach and model a squirmer withprescribed time-independent tangential surface distortions. The resultingslip velocity around the squirmer is decomposed into a series of Legen-dre polynomials of the form uθ(r = a, θ) =∑∞l=1BlVl (θ), where Vl(θ) =−(2/l(l + 1))P 1l (cos θ) with P 1l being the associated Legendre function ofthe first kind and θ the polar angle measured with the axis of symmetry.The coefficients Bl are related to Stokes flow singularity solutions. In aNewtonian fluid, theB1 mode (a source dipole) is the only mode contributingto the swimming velocity, and the B2 mode (the stresslet) is the slowestdecaying spatial mode and thus dominates the far field velocity generatedby squirmers. Therefore, often only the first two modes, B1 and B2, ofthe expansion are considered [59, 101, 213]. The ratio of the two modes,α = B2/B1, characterises the type of swimmer in a Newtonian fluid: α > 0describes a puller, which generates impetus from its front end (e.g. thealga Chlamydomonas), whereas α < 0 represents a pusher, which generatespropulsion from its rear part (e.g. the bacterium Escherichia coli), and theα = 0 case corresponds to a neutral squirmer which induces a potentialvelocity field. In a Newtonian fluid, the swimming speed of a squirmerUN = 2B1/3 [17, 137], which is independent of the fluid viscosity becausedrag and thrust change equally with viscosity. Any modes other than B1only modify the surrounding flow structure but do not contribute to theswimming speed of a squirmer. This simple picture, however, does notapply to squirming in a shear-thinning fluid, as we discuss later, where allmodes can potentially contribute to the swimming velocity, and adding anyother modes to B1 will nontrivially affect the locomotion of the squirmer.5.2.1 Shear-thinning rheology: the Carreau-Yasuda modelShear-thinning fluids experience a loss in apparent viscosity with appliedstrain rates, a property that results from changes in the fluid microstructure.As the rate of strain exceeds the rate of structural relaxation, one observesmicrostructural ordering in the fluid [26]. Here, we capture the change inapparent viscosity due to this ordering using the Carreau-Yasuda model forgeneralised Newtonian fluids [16]. The variation of viscosity with applied435.2. Theoretical frameworkstrain rate is given byη = η∞ + (η0 − η∞)[1 + λ2t |γ˙|2]n−12 , (5.1)where η0 and η∞ are the zero- and infinite-shear rate viscosities respectively.The power law index n characterises the degree of shear-thinning (n < 1) andthe relaxation time λt sets the crossover strain rate at which non-Newtonianbehaviour starts to become significant. The magnitude of the strain ratetensor is given by |γ˙| = (Π/2)1/2, where Π = γ˙ij γ˙ij is the second-invariantof the tensor. As an example, measured values for human cervical mucuscan be fitted by the Carreau-Yasuda model with values η0 = 145.7 Pa s,η∞ = 0 Pa s, λt = 631.04 s and n = 0.27 [99, 205].We non-dimensionalise the flow quantities taking the first mode, B1,of the surface actuation as the scale for velocity and the radius, a, of thesquirmer as the characteristic length scale. The strain rates are scaled withω = B1/a and the stresses by η0ω, such that the constitutive equation takesthe dimensionless formτ ∗ ={β + (1− β)[1 + Cu2|γ˙∗|2]n−12}γ˙∗, (5.2)where τ is the deviatoric stress tensor, and dimensionless quantities aredenoted by stars (*). The Carreau number Cu = ωλt is the ratio of thecharacteristic strain rate, defined by the surface actuation ω, to the crossoverstrain rate, defined by the fluid relaxation 1/λt. The viscosity ratio is givenby β = η∞/η0 ∈ [0, 1].It is evident from (5.2) that when the actuation rate ω is much smalleror much larger than the fluid relaxation rate 1/λt, i.e. when Cu → 0 orCu→∞, the shear-thinning fluid reduces to a Newtonian fluid of constantviscosity η0 (dimensionless viscosity 1) or η∞ (dimensionless viscosity β) re-spectively. Recalling that for a given surface actuation the swimming speedof a squirmer in the Newtonian regime is independent of the fluid viscosity,we therefore expect the swimming speed of a squirmer in a shear-thinningfluid to converge to its Newtonian value in the limits Cu→ 0 or Cu→∞.Non-monotonic variation of the swimming speed with Cu is expected for anyswimmer with prescribed kinematics and has been observed by Montenegro-Johnson et al. [148] for some two-dimensional model swimmers. In this studywe employ both asymptotic analysis and numerical simulations to investi-gate these effects of shear-thinning rheology on swimming at low Reynoldsnumbers.445.2. Theoretical framework5.2.2 Asymptotic analysisThe deviatoric stress tensor τ ∗ in (5.2) is a non-linear function of the strainrate tensor γ˙∗. Assuming only a weak nonlinearity, we may uncouple theNewtonian and non-Newtonian contributions, writingτ ∗ = γ˙∗ + εA∗, (5.3)with ε≪ 1 as a dimensionless measure of the deviation from the Newtoniancase (ε = 0).We observe that in the limits Cu = 0 or β = 1, (5.2) reduces to aNewtonian constitutive equation. Thus, one may expect weakly nonlinearbehaviour when the fluid relaxation rate is much faster than the surfaceactuation rate (ε = Cu2 ≪ 1), or when the zero-shear-rate viscosity is veryclose to the infinite-shear-rate viscosity (ε = 1− β ≪ 1).Henceforth we shall work in dimensionless quantities and therefore dropthe stars (*) for convenience.Expansion in Carreau numberExpanding all fields in regular perturbation series in ε = Cu2, we obtain,order by order, the constitutive equationsτ0 = γ˙0,τ1 = γ˙1 +(n− 1)2 (1− β) |γ˙0|2γ˙0, (5.4)hence A = (n−1)2 (1− β) |γ˙0|2γ˙0 to leading order in (5.3). It should be notedthat the first correction to the Newtonian behaviour is linear in n, whichpoints to a linear dependence of the swimming speed on n upon using (5.8),elucidating the trend suggested by the two-dimensional numerical findingsin Montenegro-Johnson et al. [149]. We also remark that this expansion isvalid only when Cu2|γ˙|2 is o(1) and is therefore not uniformly valid acrossall values of strain rates.Expansion in viscosity ratioExpanding in perturbation series with ε = 1 − β gives us, order by order,the constitutive equationsτ0 = γ˙0, (5.5)τ1 = γ˙1 +{−1 +(1 + Cu2|γ˙0|2)n−12}γ˙0, (5.6)455.2. Theoretical frameworkwhere in this limit A ={−1 + (1 + Cu2|γ˙0|2)n−12 } γ˙0 to leading order in(5.3). It should be noted that this asymptotic expansion is uniformly validfor all strain rates or Carreau numbers, which permits a full-range study ofthe non-monotonic swimming behaviour.5.2.3 The reciprocal theoremStone & Samuel [193] demonstrated the use of the Lorentz reciprocal the-orem in low-Reynolds-number hydrodynamics [93] to obtain the swimmingvelocity of a squirmer for a given prescribed surface actuation uS with-out calculation of the unknown flow field, provided one can solve the re-sistance/mobility problem for the swimmer shape (with surface S). Lauga[122, 125] then developed integral theorems extending this method for usewith complex fluids. We use these methods in the subsequent calculationsto obtain the swimming velocity of a squirmer in a shear-thinning fluid;the methodology adopted below closely follows the formulation in Elfring &Lauga [70].We represent the velocity field and the associated total stress tensor fora force- and torque-free swimmer with u and σ respectively. We considerthe corresponding resistance problem in a Newtonian fluid to simplify thecalculation of the swimming velocity. The resistance problem (denoted witha hat) involves the rigid-body motion with translational velocity Uˆ androtational velocity Ωˆ, and the corresponding velocity field and associatedstress tensor are represented by uˆ and σˆ respectively. Due to the linearityof the Stokes equation, we may write uˆ = Lˆ · Uˆ, σˆ = Tˆ · Uˆ and Fˆ = −Rˆ · Uˆ.Here, for compactness, both the translational and the rotational componentsof velocity are contained in Uˆ and, similarly, the corresponding matricescontain both the translational and rotational terms. In weakly nonlinearcomplex fluids, the swimming velocities U = [U Ω]⊤ are given byU = Rˆ−1 ·[∫SuS · (n · Tˆ ) dS − ε∫VA :∇Lˆ dV]. (5.7)The integral over the volume of fluid V external to S in the equation mea-sures the change in swimming dynamics due to the non-Newtonian behaviourof the fluid. For a spherical squirmer with axisymmetrical tangential surfacedistortions, there is no rotational motion and the translational velocity isgiven simply byU = − 14pi∫SuSdS − ε8pi∫VA :(1 + 16∇2)∇G dV, (5.8)465.3. Results and discussionwhere G = 1r(I + rrr2)is the Oseen tensor (or Stokeslet). The first termon the right-hand side is the result of swimming in a Newtonian fluid [193],and the last term in the equation contains the weakly nonlinear effect, whichcan be evaluated analytically in some special cases and can be computed ingeneral by numerical quadrature readily.5.2.4 Numerical solutionThe numerical simulations of the momentum equations at zero Reynoldsnumber with the Carreau-Yasuda constitutive relation (5.1) are imple-mented in the finite element method software COMSOL. We use a squarecomputational domain of size 500a × 500a, discretized by approximately30000–50000 Taylor-Hood (P2 − P1) triangular elements. The mesh isrefined near the squirmer in order to properly capture the spatial variationof the viscosity. Since slowly decaying flow fields are expected at lowReynolds numbers, a large domain size is important to guarantee accuracy.The simulations are performed in a reference frame moving with theswimmer and the far-field (inlet) velocity is varied to obtain a computedzero force on the squirmer. In addition to comparing with the asymptoticanalysis in this work, we have validated our implementation against theanalytical results for a three-dimensional squirmer in a Newtonian fluid[17, 137] and a two-dimensional counterpart in a shear-thinning fluid [149].5.3 Results and discussionAs a first step we investigate the effect of shear-thinning rheology uponswimming speed by considering the small-Cu regime and use (5.8) to derivean analytical formula for the leading-order swimming speed U of a two-modesquirmer (with B1 and B2 modes)UUN= 1 + Cu2 (1− β) (n− 1)2 C1[1 + C2α2], (5.9)where C1 = 0.49 and C2 = 2.25 are numerical constants, and UN is theNewtonian swimming speed. In a shear-thinning fluid we have n < 1 andβ < 1 , and hence we find that this two-mode squirmer can only swimslower than in a Newtonian fluid (U/UN < 1) in the small-Cu regime. Thetwo-dimensional numerical simulations in Montenegro-Johnson et al. [149]reported that a neutral squirmer (α = 0) swims slower in a shear-thinningfluid, which is consistent with our analytical results for a three-dimensional475.3. Results and discussionFigure 5.1: In (a) we show results from numerical simulations for a neu-tral squirmer (α = 0, l), puller (α = 5, s) and pusher (α = −5, t)for ε = 0.99 and n = 0.25. (b) The non-monotonic variation in velocityis well captured by asymptotics for a neutral squirmer (solid line), and apusher/puller (dashed line) at ε = 0.1 and n = 0.25.squirmer (5.9) in the small-Cu regime, but we also find that the same con-clusion of a decreased swimming speed holds for pushers (α < 0) and pullers(α > 0) as well. In contrast to the case of swimming in a viscoelastic fluid,where the pusher and puller attain different velocities given the same mag-nitude of α [217], (5.9) reveals that in a shear-thinning fluid a pusher andpuller have the same swimming velocity because the function for swimmingspeed is even in α; this asymptotic result is verified by numerical simulationsto hold for different ranges of Cu and β (as shown by the overlapping of theupper and lower triangles in figure 5.1).To further characterise the variation of swimming speed, over the fullrange of Cu, we consider the asymptotic limit ϵ = 1 − β ≪ 1, aided bynumerical simulations for larger values of ϵ. Biological fluids often havea small viscosity ratio β and hence ε = 1 − β is typically close to 1. Infigure 1a, we present the numerical results for a neutral squirmer, pusher,and puller in the biological limit using the values ε = 0.99 and n = 0.25 toemulate human cervical mucus [99, 205]. We demonstrate in the upper inset(neutral squirmer) and lower inset (pusher and puller) in figure 1a that thenumerical solutions for the swimming speed ratio converge to the asymptoticsolutions (solid line in the upper inset; dashed line in the lower inset) whenε→ 0. In figure 1b, the results are presented at ε = 0.1 and we note that allqualitative features of the impact of a shear-thinning fluid in the biologicallimit (ε ≈ 1, figure 1a) on the swimming speed are well captured by theasymptotic analysis (when ε≪ 1, figure 1b) and as expected, the numerical485.3. Results and discussionsimulations (symbols) agree very well with the asymptotic theory (lines)when ε is small (figure 1b).The non-monontonic variation of the swimming speed with Cu may beexpected based on the asymptotic behaviour of the constitutive relation dis-cussed at the end of section 5.2.1. To understand the variation more quan-titatively, recall the form of A in (5.6) and observe from the integral expres-sion for swimming velocity (5.8) that at low strain rates, the non-Newtoniancontribution A ∼ 12Cu2(n − 1)|γ˙0|2γ˙0 vanishes as Cu → 0. At high strainrates, A ∼ −γ˙0 + (Cu|γ˙0|)n−1 γ˙0; the first term, −γ˙0, vanishes under theintegration in (5.8) [70], and the remaining term gives a non-Newtonian con-tribution that vanishes as Cu→∞ because for a shear-thinning fluid n < 1.The swimming speed therefore displays a non-monotonic variation with Cu,and since the speed decreases when Cu is small as shown by (5.9), a mini-mum swimming speed may be expected to occur at intermediate values ofCu (the ‘power-law’ regime of the model), where the non-Newtonian effectis most significant. However, for a given swimming gait, if the actuationrate of the swimmer is small enough or large enough, the shear thinningfluid may appear to have no effect at all on the swimming speed.To understand the reduction in swimming speed, inspired by the qual-itative descriptions given in Montenegro-Johnson et al. [149], we look intothe thrust and the drag of the swimming problem separately.5.3.1 Drag and thrustWe separate the swimming problem into a drag problem (a sphere under-going rigid body translation U inducing hydrodynamic drag) and a thrustproblem (a sphere held fixed undergoing only tangential surface distortionsthereby generating thrust). The superposition of these two sub-problemsgives the entire swimming problem in a Newtonian fluid in the Stokes regime;this is obviously not the case in a shear-thinning fluid due to its nonlinearconstitutive equation. However, by looking at the thrust and the drag prob-lems, one may gain insight into the more complex non-Newtonian swimmingproblem [149, 172].We derive the expressions for drag and thrust in a shear-thinning fluidagain via the reciprocal theorem approach (section 5.2.3) by utilizing thesolution to the resistance problem in a Newtonian fluid. The drag force ona sphere moving with a velocity U in weakly shear-thinning fluid is given byFD = −6piU− 34ε∫V AD :(1 + 16∇2)∇G dV , where AD is formed by the so-lution to the Newtonian drag problem. Similarly, the thrust force generatedby a sphere held stationary with surface actuation uS in a weakly shear-495.3. Results and discussionFigure 5.2: In (a), the symbol ∆ denotes the difference between drag, orthrust, in a shear-thinning fluid and a Newtonian fluid: the dash-dot curverepresents the drag reduction in a shear-thinning fluid compared with aNewtonian fluid, while the solid and dashed curves represent loss in thrustfor squirmers with α = 0 and α = ±5 respectively (n = 0.25, ε = 0.1). In(b) we show that the difference between drag and thrust is positive both fora neutral squirmer (solid) and a pusher/puller (dashed) in a shear-thinningfluid. All quantities are dimensionless.thinning fluid is given by FT = −32∫S uS dS− 34ε∫V AT :(1 + 16∇2)∇G dV ,where AT is formed by the solution to the Newtonian thrust problem.One could expect a drag reduction when a rigid sphere is pulled with aconstant velocity through a shear-thinning fluid since the fluid viscosity isreduced by the fluid straining motion. However, it is interesting to see infigure 5.2a that the thrust reduction caused by the shear-thinning rheologyis larger than the drag reduction for a large range of Cu. This more severereduction in thrust than drag then suggests slower swimming speeds com-pared with the Newtonian case, which correctly predicts the trend found bydetailed calculations (figure 5.1). In addition, for very small or large valuesof Cu, the difference between the magnitudes of drag and thrust (FD −FT )vanishes as shown in figure 5.2b, respecting the limits where the swimmingspeed should recover the Newtonian value (figure 5.1).Although conceptually intuitive, the drag and thrust decomposition isnot complete as it neglects the contribution of non-linear products in thenon-Newtonian stress to the full swimming problem, namely A ̸= AT +AD,due to the non-linearity in the constitutive equation. We will give a counter-example in section 5.3.2 below to illustrate a scenario when these intuitivearguments fail.505.3. Results and discussion5.3.2 Addition of other squirming modesThe results from the detailed asymptotic and numerical analysis as well asthe intuitive model for a two-mode squirmer seem to suggest that the shear-thinning rheology acts to hinder the locomotion. This raises the simplequestion of whether this conclusion still holds if other modes of surfaceactuation are present. The picture is clear for a Newtonian fluid: only theB1 mode contributes to swimming and the addition of other modes does notalter the swimming speed. However, can the shear-thinning rheology renderother modes, typically not considered in the Newtonian analysis, effective forpropulsion? We address these questions using the asymptotic and numericaltools developed in the previous sections.We first note that the B3 mode alone leads to locomotion in a shear-thinning fluid in stark contrast to a Newtonian fluid as only the B1 modehas a non-zero surface average (see (5.8)). Indeed any odd mode alone maylead to locomotion in a shear-thinning fluid (even modes alone do not swimby symmetry). We also find quite distinctive behaviour when the B3 modeis combined with other modes. In the Cu ≪ 1 regime, we can derive ananalytical expression allowing us to predict the values of α and ζ = B3/B1for faster or slower swimming. To quadratic order in Cu, we findUUN= 1 + Cu2 (1− β) (n− 1)2 C1[1 + C2 (1 + C3ζ)α2 + C4(C5ζ2 + C6ζ − 1)ζ],(5.10)where the additional numerical constants are given by C3 = 0.51, C4 = 0.70,C5 = 0.18, and C6 = 1.66. Again the swimming speed is even in α and werecover (5.9) when ζ = 0 as expected. From (5.10) we can predict that fasterswimming (U/UN > 1) will occur if∂2U∂Cu2(α, ζ)∣∣∣∣∣Cu=0> 0, (5.11)in other words when the term in the square brackets in (5.10) is negative. Infigure 5.3a we plot the level set curve below which faster swimming occurs inthe small Cu regime; we find that this can only occur when ζ is negative forany α. For example, when α = 0 we must have ζ < −10.11, while α = ±5,ζ < −2.22 leads to faster swimming.In figure 5.3b we show the variation of swimming speed for two swimmerswith α and ζ chosen below the level set curve (the upper solid and dashedlines) and two swimmers with α and ζ chosen above the level set curve(the lower solid and dashed lines). We note that the swimming speeds of515.3. Results and discussionFigure 5.3: In (a) we show the level set curve below which faster swimmingoccurs for specific values of α and ζ = B3/B1 when Cu≪ 1. In (b) we showthe swimming speed of a neutral squirmer (solid line) and a pusher/puller(dashed line) at values of ζ chosen below the curve in (a) so that the swim-ming speed is larger than the Newtonian value at small Cu (the upper solidline: α = 0, ζ = −15 ; the upper dashed line: α = ±5, ζ = −4 ); conversely,with values of ζ above the curve in (a) the swimmers always swim slowerthan in a Newtonian fluid as shown (lower solid line: α = 0, ζ = 15; lowerdashed lines: α = ±5, ζ = 4). Here ε = 0.1, n = 0.25.the faster swimmers in the small-Cu regime experience a subsequent fallbelow the Newtonian value and then a rise above it as Cu increases, beforeasymptoting to the Newtonian swimming speed at high Cu. This indicatesthat microorganisms (with a given swimming gait) can swim both faster andslower than in a Newtonian fluid depending on the actuation rate of thatgait. In contrast, the two swimmers that swim slower in a Newtonian fluidin the small-Cu regime remain slower for larger Cu with a non-monotonicvariation similar to that observed previously (see figure 5.1). These resultsalso hold qualitatively for large values of ϵ.We emphasize that the thrust and drag reduction model is unable toexplain the faster swimming speed with the addition of a B3 mode becausein these cases the thrust still decreases more than the drag over a widerange of Carreau numbers, if they are considered separately. This serves asa counter-example demonstrating how analyzing drag and thrust separatelymay not adequately describe swimming in complex media. This fact pointsspecifically to the interaction between thrust and drag fields, due to the non-linearity in the constitutive equation, as the cause of faster than Newtonianswimming.525.4. Conclusion5.4 ConclusionWe show in this work that shear-thinning rheology affects a squirmer with aprescribed swimming gait in nontrivial and surprising ways; we predict, ana-lytically, instances of both faster and slower swimming than in a Newtonianfluid depending on the details of the prescribed boundary conditions. Indeedwe demonstrate that even with the same squirming modes a squirmer canswim faster or slower depending on its rate of actuation. In general, theseresults point to the importance of both the spatial and the temporal detailsof the swimming gait of a microorganism and ultimately the difficulty inpredicting the resulting effect of the non-Newtonian fluid a priori. In lightof this, an important next step would be to incorporate models of internalforce generation for biological swimmers and determine how the fluid rhe-ology affects the resultant gait itself in concert with propulsion. Finally,we remark that the drag and thrust decomposition of the swimming prob-lem is indeed effective in Newtonian fluids and may also be insightful incomplex fluids in some instances, but one should use caution when extend-ing the results to non-Newtonian swimming as the inherent non-linearity ofthe problem can be significant enough for a Newtonian-like decompositionto yield qualitatively flawed predictions as illustrated by the example weprovide.53Chapter 6An active particle in acomplex fluid §In this work, we study active particles with prescribed surface velocitiesin non-Newtonian fluids. We employ the reciprocal theorem to obtain thevelocity of an active spherical particle with an arbitrary axisymmetric slipvelocity in an otherwise quiescent second-order fluid. We then determinehow the motion of a diffusiophoretic Janus particle is affected by complexfluid rheology, namely viscoelasticity and shear-thinning viscosity, comparedto a Newtonian fluid, assuming a fixed slip velocity. We find that a Janusparticle may go faster or slower in a viscoelastic fluid, but is always slowerin a shear-thinning fluid as compared to a Newtonian fluid.6.1 IntroductionActive particles are self-driven units which can convert stored or ambient freeenergy into systematic motion [139, 180]. These particles are found on lengthscales from subcellular to oceanic, and range from aquatic, terrestrial andaerial flocks to colloidal particles propelled through fluid by catalytic activityat their surfaces. The interactions of active particles with the medium theyare found in, and amongst themselves, give rise to fascinating collectivebehaviour and beautiful pattern formation [139]. Active particles in fluidmedia can be either living, like swimming microorganisms [129], or synthetic,like crystals of light-activated colloidal surfers [162], swimming droplets [202]and chemically self-propelled nano-motors [107]. For sufficiently small sizesof active particles, inertial forces are negligible compared to viscous forces,and one may assume the fluid to be under an instantaneous equilibrium offorces [170].Several microorganisms propel themselves using small surface distortionsas in the coordinated beating of cilia on Opalina and Paramecium [184]. As§This chapter has been previously published under the same title in Journal of FluidMechanics, 823, (2017) 675–688 by Datt et al. © 2017 Cambridge University Press.546.1. Introductionsuch, these swimmers are often modelled as spheres with a prescribed surfaceslip velocity [165]; the slip velocity serves as a coarse-grained description ofany deformation or dynamics on the particle body that leads to its motion[17, 137]. Likewise, a chemically active colloidal particle with asymmetriccatalytic properties generates a non-uniform distribution of reaction prod-ucts and hence, also a flow within a thin ‘inner’ region near the particle’ssurface [2]. The surface flow and the resultant diffusiophoretic motion mayalso be modelled by prescribing an apparent slip velocity on the particlesurface [106]. The motion of these particles, arising due to a surface slipvelocity is, by now, well-understood for particles that move in Newtonianfluids at low Reynolds numbers [22, 71]. In general, the propulsive forcegenerated by the surface slip velocity balances the hydrodynamic drag forcedue to the rigid body motion of the particle. For simple bodies, the swim-ming velocity is given directly by the surface average of the prescribed slipvelocity [67] and because of this simplification, detailed models of the surfaceslip velocity for living and synthetic active particles are often unnecessary.In contrast, an understanding of dynamics of active particles in non-Newtonian fluids is still developing [164]. Unlike in Newtonian fluids, theconstitutive equation for stress is nonlinear in non-Newtonian fluids and asa result a straightforward linear decomposition of the flow field into dragand thrust components fails [45]. Consequently, a surface average of theslip velocity does not yield the velocity of the particle, and so a detaileddescription of the surface slip velocity may be significant in complex fluids.Despite this, many recent studies consider, as a point of comparison withNewtonian fluids, the two-mode swimmer model [52, 135, 149, 217], althoughrecently it was shown that neglected details of the surface slip velocity canhave a qualitative effect on the motion of the particle in a shear-thinningfluid [45].In this work, we analyse the motion of an active particle in a weaklynonlinear complex fluid with a general axisymmetric slip velocity by meansof the reciprocal theorem [125, 193]. This allows us to consider a completerange of prescribed motions on the particle surface and to determine whatdetails matter and why. We note that the swimming gait (apparent sur-face slip velocity) of the swimmer may itself be affected in complex fluidsas compared to Newtonian fluids, due to, for example, constraints on powerfor biological swimmers or changes in solute diffusivity for diffusiophoreticparticles. Here, however, we consider swimmers with the same swimminggait as in Newtonian fluids. As an example, we consider the slip velocityof self-diffusiophoretic Janus particles and discuss the effects of viscoelastic-ity and shear-thinning rheology on the particles’ propulsion velocity. These556.2. Modelling active particlesactive colloidal particles, at times, may swim through polymer suspensions[31], and an understanding of their dynamics in complex fluids may lead tointeresting applications in biological and chemical engineering [169]. Recentstudies on the effects of rheology on the motion of Janus particles [86, 157]have shown that the particle translational and rotational dynamics are cou-pled in media with viscoelasticity or local viscosity variations. Further, mo-tivated by recent works on the dynamics of active particles in backgroundflow of non-Newtonian fluids [3, 51, 140], we generalise the reciprocal theo-rem formulation [70, 125, 128] to include a background flow in the spirit ofprevious classical work on passive particles in weakly nonlinear flows [133].6.2 Modelling active particlesBiological microswimmers possess variety of different geometries and swim-ming modes; many, like ciliates (Opalina) and multicellular colonies of flag-ellates (Volvox), are approximately spherical in shape and propel due to thebeating of closely packed cilia on their surface [184]. These swimmers, inan idealised model, are mathematically represented as spheres with smallamplitude radial and tangential motions of elements of the surface. Theoriginal model (now known as the squirmer model), by Lighthill [137] andBlake [17], considered only axisymmetric surface distortions so the swim-mers could swim only along their axis of symmetry. Recently, Pak & Lauga[159] extended the model to arbitrary surface deformations allowing three-dimensional translational and rotational swimming kinematics of the swim-mer.Synthetic active particles too can be conceived in many shapes with avariety of propulsion mechanisms [207]. Self-phoretic particles, in particular,are colloids which are able to generate local gradients through the catalyticphysiochemical properties on their surface [84, 85, 145]. The short-rangeinteraction between the surface of the swimmer and the self-generated outerfield gradient (solute concentration, temperature or electric field) locallycreates fluid motion in the vicinity of particle boundary that leads to particlepropulsion due to phoresis [2]. When the interaction layer is thin comparedto the particle size, phoretic effects can be represented by the generation ofslip velocities on the particle surface [106, 145].In this work, we focus on spherical phoretic particles [85, 145], with an566.2. Modelling active particlesaxisymmetric slip velocity expressed here asuS (θ, t) =∞∑p=1αp (t)Kp (cos θ) eˆθ (6.1)withKp (cos θ) =(2p+ 1)p (p+ 1)P′p (cos θ) sin θ, (6.2)where eˆθ is a unit vector in the direction of increasing polar angle θ in spher-ical coordinates and Pp is the pth Legendre polynomial [144]. The flow fielddue to the swimmer in Newtonian fluids is completely characterised and de-termined by the intensities of the ‘squirming’ modes, αp [17]. Of particularsignificance are the first two modes: α1, which fixes the swimming veloc-ity [137], and α2, which defines the strength of the force dipole generatedby the swimmer Σ = 10piα2 [145]. Consequently, for analyses of collectivebehaviour [54, 218], or transport of nutrients [100, 144], in Newtonian flu-ids, active particles are very often modelled with a truncated slip velocityexpansion which retains only the first two terms. We consider here onlysteady slip velocities on the particle surface, which is often appropriate forphoretic particles; however, in general, especially for models of biologicalorganisms where the surface motion arises from a cyclical deformation, theslip velocities may depend on time t [166]. This time dependence of the sur-face actuation is then particularly important for fluids which possess historydependence, like polymer solutions, especially when the time scale of surfaceactuation is of the same order as the fluid relaxation time [69].Self-diffusiophoretic particles propel due to asymmetric surface chemicalreactions [2, 20, 84] which cause an induced imbalance of osmotic effectsin a thin interaction layer on the particle surface. The resulting flow inthis thin layer, the apparent slip velocity, is proportional to the local so-lute concentration gradient and the specifics of solute–surface interactions(phoretic mobility). Under the assumption that diffusion is fast enough sothat the chemical reaction at the surface is controlled by the far-field so-lute concentration (fixed-flux formulation, Dämkohler number = 0) and onneglecting the distortion of solute distribution due to flow resulting fromphoretic effects (Péclet number = 0), one obtains the squirming modes in(6.1)αp =pAp2p+ 1MD, (6.3)576.3. Swimming in a background flow of a weakly non-Newtonian fluidAfdAbFigure 6.1: Self-phoretic particle with two compartments of different activ-ity, Af and Ab. We consider particles with a constant uniform mobility overthe surface. When θd = pi/2, the particle has compartments of equal cover,which we call a symmetric Janus particle.where the surface activity A (θ) = ∑ApPp (cos θ) (and positive values denoteabsorption of solute), the phoretic mobilityM is assumed to be constant overthe surface and D is the solute diffusivity (see Michelin & Lauga [145] fordetails).We consider Janus-type particles with a discontinuous change in activitybetween two distinct compartments of the surface activity, A(θ) = Af forθ < θd while A(θ) = Ab for θ > θd as illustrated in figure 6.1. Here, we takethe rear compartment to be inert, Ab = 0, in which case the coefficients aregiven by [145]A0 =Af2 (1− cos θd) , An =Af2 [Pn−1(cos θd)− Pn+1(cos θd)] (n ≥ 1),(6.4)which then set the squirming modes and the entire flow field for Janusparticles in Newtonian fluids.6.3 Swimming in a background flow of a weaklynon-Newtonian fluidConsider a general active particle (or swimmer) B with surface ∂B immersedin a background flow u∞ of an incompressible and weakly nonlinear complexfluid. The velocity on the swimmer surface ∂B isu (x ∈ ∂B) = U +Ω × x+ uS , (6.5)586.3. Swimming in a background flow of a weakly non-Newtonian fluidwhere U is the translational velocity of the particle, Ω is the rotationalvelocity and uS is the prescribed deformation velocity on its surface (theswimming gait).The rheology of the non-Newtonian fluid is assumed to be only weaklynonlinear [70, 125], and thus, a constitutive equation of the formτ = ηγ˙ + εA[u], (6.6)where τ is the deviatoric stress, η is the viscosity and γ˙ the strain-rate tensorsuch that ηγ˙ gives the Newtonian contribution. A[u] is a symmetric tensorand a nonlinear functional of u and ε is a small dimensionless parametercharacterising the deviation from Newtonian behaviour, for example, smallDeborah number in case of viscoelastic fluids or small Carreau number forshear-thinning fluids.We consider the flow field to be inertialess and in mechanical equilibriumwith ∇ · σ = 0, where σ is the stress tensor corresponding to the velocityfield u. We define disturbance fields u′ = u− u∞ and σ′ = σ − σ∞ whereu∞ and σ∞ correspond to the velocity and stress fields of the backgroundflow in the absence of the particle. Due to the nonlinearity of constitutiveequation (6.6), u′ and σ′, in general, do not represent velocity and stressfields of the same problem (except when ε = 0).Stone & Samuel [193] demonstrated a shortcut to obtain the swimmingvelocity of an arbitrary swimmer in a Newtonian fluid with a given pre-scribed surface actuation uS without calculation of its unknown flow fieldusing the Lorentz reciprocal theorem in low Reynolds number hydrodynam-ics [93], provided one can solve the rigid-body resistance/mobility problemfor a body of the same shape. Using this approach Lauga [122, 125] thendeveloped integral theorems to determine the swimming velocity in complexfluids. We use these methods below, following the formulation in [69, 70],to obtain the swimming velocity of a swimmer in a weakly non-Newtonianfluid but include the possibility of a non-zero background flow for generality.For the resistance problem (denoted with a hat), we consider rigid-bodymotion with translational velocity Uˆ and rotational velocity Ωˆ, through aNewtonian fluid with corresponding velocity field uˆ and associated stresstensor σˆ = ηˆ ˆ˙γ. As both flows (due to the swimmer and due to rigid-bodymotion) are in mechanical equilibrium, we haveuˆ · (∇ · σ′) = u′ · (∇ · σˆ) = 0. (6.7)Integrating over the volume of fluid, V, exterior to B and applying thedivergence theorem while enforcing the incompressibility of the flows, we596.3. Swimming in a background flow of a weakly non-Newtonian fluidget∫∂Vn·σ′·uˆ dS+∫Vτ ′ :∇uˆ dV =∫∂Vn·σˆ·u′ dS+∫Vτˆ :∇u′ dV = 0, (6.8)where we have defined τ ′ = ηγ˙ ′ + εA′ and A′ = A [u]−A [u∞]. The surface∂V that bounds the fluid volume V is composed of the body surface, ∂B, andan outer surface (fluid or solid, possibly at infinity). Here, n is the normalto the surface, ∂V, pointing into V.Provided the fields, u′ and σ′, decay appropriately in the far field, wemay neglect the outer surface of ∂V (we shall show this is the case forweakly viscoelastic linear background flows in a subsequent work). For flowsbounded by no-slip walls these terms will be identically zero. Upon substi-tution of the boundary conditions on ∂B for each field and enforcing that thenet hydrodynamic force, F = ∫∂B n·σ dS , and torque, L = ∫∂B x×(n·σ) dS,are both zero on a free swimmer in the absence of inertia, the left-hand sideof (6.8) simplifies toη∫Vγ˙ ′ :∇uˆ dV + ε∫VA′ :∇uˆ dV = 0. (6.9)while the right-hand side of (6.8) simplifies toFˆ ·U + Lˆ ·Ω +∫∂Bn · σˆ ·(uS − u∞)dS − ε ηˆη∫VA′ :∇uˆ dV = 0, (6.10)where we have utilised the fact that ˆ˙γ : ∇u′ = γ˙ ′ : ∇uˆ. We will here usesix-dimensional vectors for compactness, U = [U Ω]⊤ and Fˆ = [Fˆ Lˆ]⊤,and from the linearity of the Stokes equation, write uˆ = Lˆ · Uˆ, σˆ = Tˆ · Uˆand Fˆ = −Rˆ · Uˆ, where Rˆ is symmetric. Finally, upon combining (6.9) with(6.10) we obtainU = Rˆ−1 ·[∫∂B(uS − u∞)·(n · Tˆ)dS − ε ηˆη∫VA′ :∇Lˆ dV], (6.11)which gives us a relation for the propulsion velocity of a swimmer in thebackground flow of a weakly non-Newtonian fluid. The correction to theNewtonian swimming speed, due to the tensor A′, typically depends uponthe unknown field u but, upon expanding perturbatively in ε, the correctiondepends only on the Newtonian solution to leading order.For a spherical particle of radius a, the translational velocity is givensimply byU = − 14pia2∫S(uS − u∞)dS − ε 18piη∫VA′ :(1 + a26 ∇2)∇G dV(6.12)606.4. Janus particle in non-Newtonian fluidswhere G = (I + rr/r2) /r is the Oseen tensor (or Stokeslet). As expected,when ε = 0, one obtains the result for a swimmer in a background flow ofNewtonian fluid [67].6.4 Janus particle in non-Newtonian fluidsAs examples of an active particle in a complex fluid, we study a Janus par-ticle in a weakly viscoelastic fluid and in a weakly shear-thinning fluid butassume the same surface slip velocity as in the Newtonian fluid (given by(6.3). We note that we expect the non-Newtonian rheology will also affectthe slip velocity for phoretic particles but focus here only on kinematic differ-ences for a fixed swimming gait. Viscoelasticity and shear-thinning rheologyare two important non-Newtonian properties [16] and also the characteristicsof many biological fluids [120, 142] wherein these artificial swimmers havepotential applications [158]. As discussed in §6.2, we assume the diffusionof the solute to be fast enough so that the effects of Péclet and Damköhlernumber can be neglected and we shall consider the particle in an unboundedand otherwise quiescent background (u∞ = 0). We first analyse the Janusparticle in a weakly viscoelastic fluid.6.4.1 Viscoelasticity: second-order fluidViscoelastic fluids exhibit both viscous and elastic responses to forces. Suchfluids possess a memory, and stresses in them depend on the flow history.For flows which are both slow and slowly varying, viscoelasticity may bemodelled without any memory of the past stresses as a second-order fluid[150],τ = ηγ˙ − Ψ12∇γ˙ +Ψ2γ˙ · γ˙. (6.13)Here, η is the total viscosity of the solution and Ψ1 and Ψ2 are the first andsecond normal stress-difference coefficients, respectively. The first normalstress difference is generally positive in viscoelastic flows i.e Ψ1 > 0. Thetriangle denotes the upper-convected derivative∇γ˙= ∂γ˙∂t+ u ·∇γ˙ − (∇u)⊤ · γ˙ − γ˙ ·∇u. (6.14)In order to study the effect of fluid rheology on the particle, we first non-dimensionalise the equations by scaling lengths with the particle radius,a; velocities with the first swimming mode α1, which without any loss of616.4. Janus particle in non-Newtonian fluidsgenerality is assumed to be positive, and stresses with ηω, where ω = α1/ais the scale of strain rate. The resulting dimensionless constitutive equationisτ ∗ = γ˙∗ −De(∇γ˙∗ + bγ˙∗ · γ˙∗), (6.15)with De = ωΨ1/2η, the Deborah number, which is the ratio of the relaxationtime scale of the fluid to the characteristic timescale of the flow and b =−2Ψ2/Ψ1 ≥ 0. Henceforth, we work in dimensionless quantities and dropthe stars (*) for the sake of convenience. For small De (weakly viscoelasticlimit), we expand the flow quantities in a regular perturbation expansion inDe [52, 70, 121] to get, at the leading order,τ0 = γ˙0, (6.16)and at O (De)τ1 = γ˙1 + A, (6.17)with A = −(∇γ˙0 +bγ˙0 · γ˙0). The angular velocity of a spherical swimmer iszero due to axisymmetry while its translational velocity, correct to O (De),is given by (6.12) where now ϵ = De.The flow field for a swimmer with prescribed surface velocity (6.1) in aquiescent Newtonian fluid is given by [101, see]u0 =− 12r3e+32r3e · rrrr+∞∑p=2( 1rp+2− 1rp)(p+ 12)ΘpPp(e · rr)rr+∞∑p=2(p2rp+2 −(p2 − 1) 1rp)(p+ 12)ΘpWp(e · rr)(e · rrrr− e),(6.18)where e is the swimming direction and r is the position vectorwith r = |r| from the centre of the sphere. Θp = αp/α1 andWp (x) = 2/ (n (n+ 1))P ′p (x). Using the Newtonian velocity field,one can calculate the strain-rate field around the swimmer, γ˙0, and thusobtain the expression for A. Substituting the expression for A in (6.12) andusing the orthogonal properties of Legendre polynomials, one obtains, aftersome lengthy but straightforward calculations,U/UN = 1 +De (b− 1)∞∑p=1CpΘpΘp+1, (6.19)626.4. Janus particle in non-Newtonian fluidswhereCp =6p(p+ 1)2(p+ 2) . (6.20)Recall that UN = α1 is the (dimensional) swimming speed in Newtonianfluids. Frequently, the slip velocity description is truncated at two modesi.e. Θp = 0 ∀ p > 2, and depending on whether Θ2 < 0, Θ2 = 0 or Θ2 > 0the swimmer is identified as a pusher, neutral or puller swimmer, respec-tively, in Newtonian fluids [71]. However, swimmers like starfish larvae[81] and Janus particles possess significant values of higher modes. Whenconsidering such swimmers in non-Newtonian fluids, one should be carefulwhile truncating the series because unlike in Newtonian fluids, swimmingspeeds may be qualitatively affected by higher modes [45]. Indeed, as canbe noted from (6.19), setting the modes α1 = 1, α2 = 1 and α3 = 2 (with ap-propriate units) produces qualitatively different swimming behaviour thanα1 = 1, α2 = 1 and α3 = −2 when just the first three modes are considered.Therefore, the expression (6.19), while giving the contribution of all spec-tral modes in the slip velocity expansion to the swimming velocity, helps topredict when it may be reasonable to neglect higher modes and use a simple‘two-mode’ description to obtain the swimming speed.We consider the case of a symmetric Janus particle, where preciselyone half is chemically active and the other inert, θd = pi/2. The spectralcoefficients for activity in this case are zero for even modes (from (6.4)),and consequently Θ2p = 0. Hence, from (6.19), one finds that a symmetricJanus particle (with a constant uniform surface mobility) swims only at itsNewtonian speed – a result also true for a two-mode neutral swimmer [52]but here obtained without any restriction on the number of modes beingconsidered. Interestingly, one could obtain this result by observing that thenon-Newtonian contribution in (6.12) is a volume integral of the contractionof an even tensor A (under x → −x) and an odd kernel and thereforevanishes. Similarly, looking at the power consumption of a squirmer, P ,correct to the first order [52]2P =∫Vγ˙0 : γ˙0 dV +De∫VA : γ˙0 dV, (6.21)one finds once again that for a symmetric Janus particle the non-Newtoniancontribution gives a null result. Thus, a symmetric Janus particle in asecond-order fluid swims and expends power as if in an equivalent Newto-nian fluid (De = 0), correct to the first order in De, for the same surfaceslip velocity as in the Newtonian fluid. We note that the non-Newtonian636.4. Janus particle in non-Newtonian fluidsrheology will affect the solution of the ‘inner’ region for phoretic particles[145]. Additional non-Newtonian stresses arise on the particle surface, andeven the solute diffusivity may change due to viscosity variations. For athin interaction layer, neglecting effects of Péclet and Damköhler number,the slip velocity will change at O (De) similarly to the case of electrophoresisconsidered by Khair et al. [111]. Here, however, our emphasis is on studyingthe changes in the propulsion velocity from its Newtonian value for a given(but arbitrary) slip velocity on the particle surface.A similar result was obtained by Leal [131] for axisymmetric passiveparticles with fore–aft symmetry in a second-order fluid, where such particlestranslate, to the first approximation, at the same rate as in an equivalentNewtonian fluid. On comparison with present results, one may expect evennon-spherical active particles with fore–aft symmetry in second-order fluidsto behave as if in equivalent Newtonian fluids.When the two halves of the Janus particle are not exactly equal, i.e.θd ̸= pi/2, then the even spectral modes of the activity, A2p, are no longerequal to zero and hence Θ2p ̸= 0. Consequently, the non-Newtonian contri-bution to the swimming velocity may now be non-zero, and can be easilycalculated for any level of active surface coverage, θd. We find that whenθd > pi/2, the particle swims faster than in a Newtonian fluid and whilefor θd < pi/2 it swims slower, provided b < 1 (see [37] and [53] for a recentdiscussion on permissible values of b). Interestingly, one can qualitativelypredict this result by considering the two-mode description, by observingthat Θ2 = 2 cos θd. The former particle behaves as a pusher, Θ2 < 0, andthus swims faster, where as the latter is a puller, Θ2 > 0, and therefore swimsslower than in a Newtonian fluid (from (6.19)), as also reported for two-modeswimmers by De Corato et al. [52]. Quantitatively, the viscoelastic contri-bution decays for higher modes as Cp ∼ 1/p2 and a two-mode descriptiongives the viscoelastic contribution with a relative error of less than 0.1 for| cos θd| ≤ 0.35; however, the approximation grows worse upon increasingthe fore–aft asymmetry of the particle and a three-mode description is bet-ter for | cos θd| ≥ 1/√5. This is shown in figure 6.2, where we plot thescaled first-order velocity, U (M)1 /UN = (b− 1)∑Mp=1CpΘpΘp+1 from (6.19)for different coverage areas of activity with varying number of modes. Notethat as θd approaches 0 or pi, the Newtonian velocity UN → 0 and U (M)1 /UNdiverges.The asymptotic results for a small De expansion are seen to be validfor only very small values of De (≈ 0.02 for two-mode swimmers with O (1)modes [52]). This may be understood by noting that squirming modes of646.4. Janus particle in non-Newtonian fluids0 0.5 1 1.5 2 2.5 3θd00.51(U1(∞) - U1(M) )/ U1(∞)M = 4M = 3M = 2M = 10 0.5 1 1.5 2 2.5 3θd-505U 1(M)  / U NFigure 6.2: Variation of the scaled first-order swimming velocity U (M)1 /UNwith θd obtained for the firstM+1 modes (dashed lines), and b = 0.2. U (∞)1corresponds to the convergence value (M = 99) and is depicted by the solidline. Inset plot shows the relative error.656.4. Janus particle in non-Newtonian fluidsmagnitude O (1) result in strain rates of magnitude O(10) on the surfaceof the particle in a Newtonian fluid and, therefore, O(102) values of thenon-Newtonian contribution A, which thereby renders the Deborah numberexpansion accurate for only very small values of De. Numerical results usingthe Giesekus model, at higher values of De, find all swimmers – pusher,puller and neutral – swimming slower and expending less power than inan equivalent Newtonian fluid [217]; one may also expect results obtainedusing the second-order fluid model to deviate from those obtained with theGiesekus model, at moderate Deborah numbers, due to the saturation ofpolymer elongation in the latter and the associated differences in extensionalrheology. In the experimental study of Janus particles in viscoelastic fluidsby [86], the Deborah (Weissenberg) numbers were quite small, and hencein a regime where one may then expect the second-order model to, at leastqualitatively, predict the viscoelastic fluid behaviour [132].6.4.2 Shear-thinning rheology: Carreau modelShear-thinning fluids experience a loss of apparent viscosity with appliedstrain rate. The Carreau model [16] and its perturbation to the form in(6.6) has recently been covered by Datt et al. [45]. We consider the per-turbation of the flow quantities in the viscosity ratio, ε = 1 − β whereβ ∈ [0, 1] is the ratio of infinite shear-rate viscosity to zero shear-rate vis-cosity, as this expansion is uniformly valid for all strain rates and obtainA ={−1 + (1 + Cu2|γ˙0|2)(n−1)/2} γ˙0. Here, Cu, the Carreau number is theratio of the characteristic strain rate in the flow, to the cross-over strainrate in the fluid and n characterises the degree of shear thinning (n < 1).With this form of A, it is difficult to obtain an analytical expression for thepropulsion velocity similar to that obtained for the viscoelastic case (6.19).However, one can numerically calculate the propulsion velocity with highermodes and then compare the results with just the first two modes. This isdone in figure 6.3 for n = 0.25, where we plot U (M)1 /UN for two values ofµ ≡ cos θd.We find that irrespective of the position of θd, the Janus particleswims slower in a shear-thinning fluid than in a Newtonian fluid. Thenon-monotonic variation of the first-order swimming speed with Cu in figure6.3 is similar to as found by Datt et al. [45] for any two-mode squirmer.Though the two-mode description qualitatively predicts the results: all –neutral, pusher and puller – swimmers swim slower, with pusher and pullersswimming at the same velocity [45], it is apparent from figure 6.3 thathigher modes may significantly alter the results. Additionally, we note that666.4. Janus particle in non-Newtonian fluids0.01 0.1 1 10 100Cu-0.3-0.25-0.2-0.15-0.1-0.050U 1(M) / U N µ  = 0 µ  = ± 0.9Figure 6.3: Variation of the scaled first-order swimming velocity U (M)1 /UN(obtained forM+1 modes) with Cu for two values of µ ≡ cos θd. Solid linescorrespond to M = 30, and M = 28 for µ = 0 (symmetric) and µ = ±0.9respectively (additional modes lead to negligible differences). Dashed-linescorrespond to the swimming velocity with just the first two modes.676.5. Conclusion and future workthe values of Θ2 and Θ3 for any Janus particle lie in the range where Dattet al. [45] predict a smaller swimming velocity than in Newtonian fluids.6.5 Conclusion and future workIn this work, we studied active particles with prescribed surface velocities innon-Newtonian fluids. Using the reciprocal theorem, we derived a generalform of the propulsion velocity of an active particle in a weakly nonlinearbackground flow. Using this formalism, we calculated the swimming speedfor an active particle with a general, axisymmetric slip velocity in an other-wise quiescent second-order fluid extending results previously obtained fora two-mode description. We then considered the motion of diffusiophoreticJanus particles in weakly viscoelastic and shear-thinning fluids. We showedthat a Janus particle with two equal halves, in a weakly viscoelastic fluid,will swim at the same speed as in a Newtonian fluid due to its fore–aft sym-metry (provided the surface slip velocity remains unchanged). When thissymmetry is broken the particle may swim faster or slower than in a New-tonian fluid and this may be predicted by considering the Janus particle asa pusher or puller based on the two-mode squirmer description. Conversely,in a weakly shear-thinning fluid, a Janus particle always swims slower thanin a Newtonian fluid.While analysing Janus particles, we neglected any changes to the slipvelocity due to fluid rheology as well any dynamics due to the distortion ofthe solute concentration field of phoretic particles because of the velocityfield. The latter may not be true for large proteins or molecules, when thediffusion constant is small and the Péclet number becomes significant. Thiscoupling of the velocity and concentration field leads to interesting dynamicsin Newtonian fluids [145, 146] and is an avenue for further inquiry in non-Newtonian fluids. We also expect the fluid rheology to affect the slip velocityof the particle: the gait of a biological microswimmer may be modified bynon-Newtonian stresses, likewise the slip velocity of a diffusiophoretic Janusparticle. For a complete understanding of the dynamics of active particlesin complex fluids, one should also consider such changes to the gait itself.PostscriptConsider a squirmer in a 2D linear shear flow, u∞ = Γ · x, of a second-order fluid (A =∇γ˙0 +bγ˙0 · γ˙0, where b is the ratio of the two normal stresscoefficients). We find that the rotational dynamics of the squirmer at first68Postscriptorder, O (De), is given, in dimensionless form (with the scales used in thecurrent chapter), byω1 = −56Θ2e× (E · e) (6.22)where E = Γ + Γ⊤, Θ2 = α2/α1 and e is the orientation of the squirmer.Note that at the leading order (in the Newtonian fluid), the squirmer willrotate with just the rotational velocity of the background flow, as the ax-isymmetric distribution of the slip velocity on its surface does not lead toany self-rotation.The result in (6.22) shows that of all the modes, only the second modecontributes to the rotational velocity of the swimmer. The expression’sresemblance to Jeffery’s equations [25, 103] suggests that the slip velocitydistribution may render the spherical swimmer appear spheroidal to thefluid flow. We are currently working on the problem, dealing with howto rigorously prove the expression (6.22) (presently, the absence of highermodes in the expression has been tested numerically).Note that De Corato & D’Avino [51] have already addressed the dynam-ics of a three mode squirmer in a sheared viscoelastic fluid. The novelty weadd is through the form (6.22) with its similarity to Jeffery’s equations, andthe hypothesis that the result holds for a general n-mode squirmer.69Chapter 7A note on higher-orderperturbative corrections tosquirming speed in weaklyviscoelastic fluids¶Many microorganisms swim in fluids with complex rheological properties.Although much is now understood about motion of these swimmers in New-tonian fluids, the understanding is still developing in non-Newtonian fluids—this understanding is crucial for various biomimetic and biomedical appli-cations. Here we study a common model for microswimmers, the squirmermodel, in two common viscoelastic fluid models, the Giesekus fluid model andfluids of differential type (grade three), at zero Reynolds number. Throughthis article we address a recent commentary that discussed suitable values ofparameters in these models and pointed at higher-order viscoelastic effectson squirming motion.7.1 IntroductionWith ideas of minimally invasive surgery, targeted drug delivery, and otherbiomimetic applications [79, 154, 208], an understanding of motion of mi-croswimmers in complex fluids has become imperative. Subsequently, manyrecent articles have focussed on motion of microswimmers in complex flu-ids (see reviews [70, 195]). While biological fluids demonstrate many non-Newtonian fluid properties [204], one common property is viscoelasticity[120, 201]. We consider this property in this article.Viscoelastic fluids show both viscous and elastic properties, and retainmemory of their flow history [16]. Recent experimental studies on biologi-¶This chapter has been submitted for publication under the same title by Datt andElfring.707.1. Introductioncal swimmers [134, 171, 181] have addressed how an organism may changeits swimming stroke as it “senses” the viscoelasticity of the fluid medium.Elastic stresses in the fluid can also directly contribute to changes in theswimming speed given a swimming stroke (see, for e.g., [121]). The presentwork is a theoretical study of swimmers in viscoelastic fluids. A model ofmicroswimmers conducive to theoretical treatment is the squirmer model[137]. The model, developed by Lighthill [137] and Blake [17], consists ofa rigid body that generates thrust due to the presence of (apparent) slipvelocities on its surface. It has been used to understand various single andcollective behaviours of microswimmers in Newtonian fluids [165]. In vis-coelastic fluids, Zhu et al. [217] studied the motion of squirmers using numer-ical simulations and found that all squirmers—pushers, pullers and neutralswimmers—swim slower than in a Newtonian fluid for a wide range of valuesof the Weissenberg number (measure of viscoelasticity in the fluid). Later,De Corato et al. [52] using a theoretical approach (and the squirmer model),showed that in fact for very small values of the Deborah (Weissenberg) num-ber not considered in the work of Zhu et al. [217] pusher swimmers swimfaster, puller swimmers slower and neutral swimmers at the same speed asin a Newtonian fluid. We note that in these studies, as will be the case inthe present study, the swimming speeds in viscoelastic and Newtonian fluidsare compared for the same swimming stroke.The work of De Corato et al. [52] used the second-order fluid model tostudy weakly viscoelastic effects on squirming motion. The use of the secondorder fluid model with parametric values as chosen by De Corato et al. [52]was critiqued by Christov & Jordan [37] who argued that the parametricvalues be chosen in accordance with thermodynamic constraints and rec-ommended the use of other viscoelastic models which “better elucidate thetransient effects of fluid viscoelasticity on a squirmer”. De Corato et al. [53]then showed that in fact using the Giesekus model to study weakly viscoelas-tic effects, to O (De), gives results identical to those previously obtained bythem using the second-order fluid model. The motivation for this work inlarge part is due to this discussion; here we study the squirming motion tohigher orders in Deborah number both in the Giesekus fluid and in fluidsof differential type. We find that unlike in a second-order fluid that obeysthermodynamic constraints, weak viscoelastic contributions to the squirm-ing speed are non-zero in a fluid of grade three (third-order fluid) obeyingthermodynamic constraints. These contributions are qualitatively differentto those obtained due to viscoelasticity as modelled by the Giesekus fluid.In the following, we briefly discuss the squirmer model and the second-order fluid model with the points of contention, and then present our results.717.2. Theoretical framework7.2 Theoretical framework7.2.1 The squirmer modelThe spherical squirmer model consists of a sphere with prescribed axisymm-teric surface velocities (surface velocities may be thought of as originatingfrom surface distortions in biological microswimmers like Opalina) whichgenerate thrust forces to propel the swimmer [17, 137]. We consider onlytangential surface velocities on the swimmer (the swimmer maintains itsshape) so that the surface velocity uS = uSθ eθ, where uSθ can be expressedasuSθ = Σ∞l=1BlVl (θ) , (7.1)using Vl (θ) = − (2/(l(l + 1)))P 1l (cos θ); P 1l (cos θ) are associated Legendrepolynomials of the first kind, and θ is the polar angle measured from the axisof symmetry [17]. The coefficients Bl are generally referred to as squirmingmodes. In Newtonian fluids, the swimming speed of the squirmer is dueto just the first mode, UN = 2/3B1, and the second mode B2 gives thestresslet due to the squirmer [101]. As velocities due to higher modes decayfaster than the first two modes (in fact, B2 gives the slowest decaying spatialcontribution to the flow field), and since higher modes do not contribute tothe swimming speed, in Newtonian fluids, often only the first two modesare considered, i.e., Bn = 0 for n > 3. For the purpose of this study, inaccordance with the bulk of literature in the field [165], we too consideronly the first two modes. At this point we feel it is important to notethat in general considering only the first two modes in complex fluids maybe problematic as shown in the recent works of Datt et al. [44, 45]. Theinterested reader may refer to the description of non-axisymmetric squirmingmodes in Newtonian fluids by Pak & Lauga [159].When the ratio β = B2/B1 is negative, the squirmer generates thrustfrom its rear end, like the bacterium E. coli.; when β > 0 the thrust is gen-erated from the front end, as in the “breaststroking” algae Chlamydomonas.When β = 0, the thrust and drag centres coincide, and flow field aroundthe swimmer is due to a potential dipole. The three types of squirmers arecalled pushers, pullers, and neutral swimmers, respectively [165].7.2.2 The second-order fluid modelThe deviatoric stress in an incompressible second-order fluid is given byτ = ηA1 + α1A2 + α2A21, (7.2)727.2. Theoretical frameworkwhereA1 ≡ L+ L⊤,An ≡ DAn−1Dt + L⊤An−1 + An−1L,(7.3)with L⊤ = ∇u, and D/Dt denoting the material derivative [61, 198]. Hereη is the shear viscosity and α1 and α2 are material moduli. The secondorder fluid model has been used to study the first effects of viscoelasticityon the motion of both passive and active particles (see for e.g., [4, 29, 160]).However, there has been much discussion on the permissible values of α1and α2 in the model. Dunn & Fosdick [62] have shown that considering(7.2) as exact, the model is consistent with thermodynamics whenη ≥ 0, (7.4)α1 ≥ 0, (7.5)α1 + α2 = 0. (7.6)However, often these constraints, citing experimental investigations (in-correctly, according to Dunn & Rajagopal [61]), are not strictly adheredto. In particular, α1, which corresponds to the first normal stress differencecoefficient, is generally taken to be negative [61].7.2.3 The reciprocal theoremThe reciprocal theorem of low Reynolds number hydrodynamics [93] canbe used to calculate the first effects of the fluid rheology on the swimmingspeed of microswimmers [125]. The details of the reciprocal theorem for thespecific case of squirmers in viscoelastic fluids may be found, among others,in the works of Lauga [122], De Corato et al. [52] and Datt et al. [44].Consider a weakly non-linear fluid of the form [125]τ = ηγ˙ + εΣ [u] , (7.7)where τ is the deviatoric stress, η is the shear viscosity, and γ˙ is the strainrate tensor so that the first term on the right hand side in (7.7) gives theNewtonian contribution. Here ε is the small parameter that quantifies thedeviation from the Newtonian behaviour and Σ gives the non-Newtoniancontribution. The translational velocity of a squirmer of radius a in such afluid is, obtained by using the reciprocal theorem,737.3. Results and discussionU = − 14pia2∫SuS dS − ε 18piη∫SΣ :(1 + a26 ∇2)∇G dV, (7.8)where G = (1/r) (I + rr/r2) is the Oseen tensor, and S denotes the surfaceof the swimmer, and V , the fluid volume [44].7.3 Results and discussionDe Corato et al. [52] studied the motion of a squirmer in a second-orderfluid. Considering only small deviations from Newtonian behaviour, theyexpanded all flow quantities in the small parameter ε = De, where Debo-rah number De = −α1B1/ηa is a measure of the relaxation time scale ofthe fluid to the characteristic time scale of the flow (note that for steadysurface slip velocity squirmers, the Deborah and Weissenberg numbers areequivalent [168]). De Corato et al. [52] assumed α1 < 0, in contradictionwith the thermodynamic stability criterion as pointed out by Christov &Jordan [37]. The thermodynamic constraint α1 + α2 = 0 was also relaxed.De Corato et al. [52] found that the perturbation calculations predicted thatpushers swim faster, pullers slower and neutral swimmers at the same speedas in Newtonian fluids, provided that the swimming gait remains unchangedbetween the viscoelastic and Newtonian fluids. Their numerical simulationsin a Giesekus fluid found the analytical results to hold up to De ≈ 0.02 [52].It was commented that the deviation of the results due to theoretical calcu-lations from those due to numerical simulations at larger De was because ofhigher order viscoelastic effects that were neglected in the analytical resultsfor which only O (De) corrections were analysed [52].The critique of the work of De Corato et al. [52] by Christov & Jordan[37] was focussed on the former not respecting the thermodynamic con-straints of the second-order fluid model. In particular, Christov & Jordan[37] remarked that since α1+α2 should be equal to zero, most corrections toflow quantities (but pressure) including the swimming speed of the squirmerwill be zero, since all these corrections are proportional to the sum α1 +α2.Citing [62], Christov & Jordan [37] also pointed out that for α1 < 0 a steadysolution to the problem should not be expected. Finally, Christov & Jor-dan [37] suggested calculating corrections to the swimming motion with thethermodynamic constraints (meaning going to higher powers in De for anynon-zero contributions) or using a different viscoelastic model, such as theupper-convected Maxwell model.747.3. Results and discussionDe Corato et al. [53] showed that even with using a more involved modellike the Giesekus fluid model (which reduces to the upper-convected Maxwellmodel for a choice of a model parameter), one obtains equations identicalto the second-order fluid in the limit of small De at O (De). Further, for itspermissible values, the Giesekus fluid gives identical results to those fromthe second order fluid as used by De Corato et al. [52]. In fact, they main-tain that the second order fluid model should be seen as an approximation tomore complex viscoelastic models in slow and nearly steady flows (and there-fore (7.2) not be seen as exact). Perhaps, in order to avoid any confusion,one may restrict the use of the term “second-order fluid model” only when itis treated as an exact model obeying the thermodynamic constraints; wherea slow and nearly steady flow approximation is used one can start witha more involved model and reduce it to simpler constitutive equations ateach order in the perturbation series in De. Below we use this terminologyand study the squirmer in a Giesekus fluid and in fluids of grade n (thesecond-order fluid is a fluid of grade two) and calculate the corrections tothe swimming speed in these fluids up to higher orders in De.7.3.1 Giesekus fluidThe polymeric stress in an incompressible Giesekus fluid is given as [150]τp + λ∇τp +αmληpτp · τp = ηpγ˙, (7.9)where the mobility factor αm must take values between 0 and 1/2 [150, 217].The total deviatoric stress in the fluid is τ = τs + τp where τs = ηsγ˙ is thecontribution from the Newtonian solvent. The total viscosity in the fluidη = ηs + ηp. Here we consider the case when ζ = ηs/η = 0; when ζ = 0 andαm = 0, (7.9) reduces to the upper-convected Maxwell fluid model [150].We non-dimensionalise equations by scaling lengths by the squirmer ra-dius a, velocities with the first squirming mode B1, and stresses with ηB1/a,and obtain the dimensionless constitutive equationτ ∗ +De∇τ ∗ +αmDeτ ∗ · τ ∗ = γ˙∗, (7.10)where the Deborah number De = λB1/a. Henceforth, we drop the starsfor convenience. We expand all flow quantities in a regular perturbationexpansion in De, and using standard methods to calculate the flow fields in757.3. Results and discussionStokes flow [93] obtain the swimming speed of the squirmer, up to O (De3),U = 23 +215β (−1 + αm)De+β2(−20568− 98136αm + 65266α2m)+ 84 (−193 + 176αm (−3 + 2αm))45045 De2+ β482431950(170 (3005646 + αm (6190100 + 3αm (−10014053 + 4815243αm)))+ β2(224764987 + αm (1298121442 + 3αm (−1659132865 + 875113652αm))))De3.(7.11)At this point, examining equation (7.11) for specific values of β and αmbecomes instructive; we choose β = −1 for pushers, 0 for neutral squirmers,and 1 for puller type squirmers and αm = 0.2. These values correspond tothe values used in the work of De Corato et al. [52]. From (7.11) we find,for pushers,UUN= 1 + 0.16De− 2.05De2 − 2.62De3, (7.12)for pullers,UUN= 1− 0.16De− 2.05De2 + 2.62De3, (7.13)and for neutral squirmers,UUN= 1− 0.80De2. (7.14)The swimming speeds in (7.12), (7.13), and (7.14) are plotted in figure 7.1along with their respective Padé approximant P 12 (De) [13]. When correc-tions up to only O (De) are considered, we note that pushers swim faster,pullers slower and neutral swimmers at the same speed as in a Newtonianfluid; this is shown in the work of De Corato et al. [52]. With terms up toO (De3), we note that all the squirmers swim slower than in a Newtonianfluid (except for very small values of De) as found in the numerical workof Zhu et al. [217]. Clearly, the inclusion of higher order terms changes thetheoretical predictions significantly.One may calculate the higher order terms in the expansion to predictresults for larger values of De. This is done by Housiadas & Tanner [97],up to O (De8), for steady sedimentation of a passive sphere in a viscoelas-tic fluid. Housiadas & Tanner [97] also quantify when the results from theseries should not be considered (using positive definiteness of the conforma-tion tensor). Sauzade et al. [179] and Elfring & Lauga [70] also performed a767.3. Results and discussion0 0.2 0.4 0.6 0.8 1De-101U / UN β = −1 β = 0 β = +10 0.05 0.1 0.150.9511.05Figure 7.1: Swimming speeds in the Giesekus fluid as a function of De.The solid lines include corrections up to O (De3). The dashed lines arePadé approximations to the series for the speeds in the text. The dottedlines include only O (De) corrections. The addition of the higher ordermodes decreases the speeds of the squirmers. As seen here, all squirmers atlarge values of De swim slower than in a Newtonian fluid, as found in thenumerical work of Zhu et al. [217].777.3. Results and discussionhigher-order perturbation analysis, using techniques to improve the conver-gence properties of the series, for the swimming speed of a two-dimensionalswimming sheet where the small parameter was the amplitude of the waveson the sheet. We have not pursued these endeavours here, for the motivationfor this study was to see the differences between the different viscoelasticmodels considering only the first few terms.The results in the foregoing were obtained using the Giesekus model forviscoelasticity. They would remain qualitatively the same if one were to usethe upper-convected Maxwell model. But what happens to a squirmer in afluid of grade n, when the fluid is “regarded as a fluid in its own right, notnecessarily an approximation to any other one” [203] ?7.3.2 A fluid of grade threeConsider an incompressible fluid of grade three [75]:τ = ηA1 + α1A2 + α2A21 + β1A3 + β2 [A1A2 + A2A1] + β3(trA21)A1, (7.15)where η, α1, α2, β1, β2, and β3 are material moduli. The equation is dimen-sional. Thermodynamics stipulates [75] thatη ≥ 0 α1 ≥ 0, |α1 + α2| ≤√24ηβ3,β1 = 0 β2 = 0 β3 ≥ 0.(7.16)We scale flow quantities as before, and consequently, equation (7.15)with (7.16), in its dimensionless form, becomesτ = γ˙ +De[∆γ˙ +Qγ˙ · γ˙]+De2 [tr (γ˙ · γ˙)Pγ˙] , (7.17)whereDe = α1B1/ηa, Q = α2/α1 and P = β3η/α21.∆γ˙ is the lower convectedderivative of γ˙ [150], denoted by A2 in equation (7.15). We expand allflow quantities in a regular perturbation expansion of De and calculate thepropulsion speed up to O (De2), which in dimensionless form comes to beU = 23−215β (1 +Q)De− 2β2 (1 +Q) (161 + 559Q)− 48 (616 + 1383β2)P45045 De2.(7.18)Note that when P = 0, we obtain a fluid of grade two, where 1+Q = 0 (7.6),and consequently, no contribution to the swimming speeds of the squirmers787.3. Results and discussion0 0.2 0.4 0.6 0.8 1De0123456U / UN β = −1 β = 0 β = +1Figure 7.2: Swimming speeds in fluids of grade three. Solid lines: P = 3/2,Q = −7. Dashed lines: P = 3/2, Q = 5. Solid and dashed lines for β = 0overlap. Depending on the values of Q for a given P, either of the puller orpusher can swim faster or slower than in a Newtonian fluid at small De.we consider. This is in contradiction to the results obtained through theweak De expansion in a Giesekus fluid to O (De) where pushers and pullersswim faster and slower, respectively, than in a Newtonian fluid. This wasdiscussed in the exchange between Christov & Jordan [37] and De Coratoet al. [53] described previously.To observe the effects of a fluid of grade three, we choose P = 3/2 (anarbitrary choice in as much as the physics of the problem is concerned).From equation (7.16), we know that −7 ≤ Q ≤ 5. We plot the swimmingspeeds for two cases: P = 3/2, Q = −7 and P = 3/2, Q = 5 in figure 7.2.From figure 7.2 and equation (7.18), we see that depending on the valueof Q, either of the puller or the pusher can swim faster than in a Newtonianfluid at O (De). The higher order correction, O (De2), gives a positivecontribution to the swimming speed.In contrast to the results from the Giesekus fluid, the parameters in afluid of grade three allow for a wider range of possibilities—either of pullersor pushers can swim faster or slower at small values of the Deborah number.Here, we demonstrate this using the parameter Q for a given P. About797.4. Conclusionthis range of possibilities, perhaps it is useful to recall the observation fromTruesdell [203] that “it is possible that two fluids of grade 3 could behavejust alike in every viscometric test yet react altogether differently to sometest of a different kind”. At higher De, all squirmers swim faster in fluids ofgrade three than in a Newtonian fluid, when in Giesekus fluids they wouldswim slower.7.4 ConclusionWe calculated the higher order corrections to the swimming speeds in twoviscoelastic fluids: the Giesekus fluid and the fluid of grade three. Thehigher order corrections significantly add to the results at O (De); even atrelatively small values of De, the corrections lead to qualitatively differentspeeds. This again raises the question about the range of values of De atwhich the expansion can predict results (also see [44]). Importantly, weobserve that the two fluids, the Giesekus fluid and the fluid of grade three,predict qualitatively different swimming speeds for the squirmers. Clearly,the answer to what viscoelastic model to use depends on what all we wishto model—in this, we are guided by experiments.80Chapter 8Two-sphere swimmers inviscoelastic fluids**We examine swimmers comprising of two rigid spheres which oscillate peri-odically along their axis of symmetry, considering when the spheres oscillateboth in phase and in anti-phase, and study the effects of fluid viscoelasticityon the swimmers’ motion. These swimmers display reciprocal motion inNewtonian fluids and consequently, no net swimming is achieved over onecycle in such fluids. Conversely, in viscoelastic fluids, we find that the effectof viscoelasticity acts to propel the swimmers forward in the direction of thesmaller sphere when the two spheres are of different sizes. Finally, we com-pare the motion of rigid spheres oscillating in viscoelastic fluids with elasticspheres in Newtonian fluids where we find similar results.8.1 IntroductionRecent review articles on swimming at small length scales [11, 71, 88, 129,154, 177] point to the immense interest in recent years on understanding thetopic that has wide ranging applications from biomedical engineering [208]to autonomous de-pollution of water and soil [78]. Several theoretical modelsfor understanding swimming at low Reynolds number in Newtonian fluidshave been developed, such as the swimming sheet [200], and the squirmer[137]. The swimming techniques used in these two seminal models, whichwere drawn from observing biological swimmers, demonstrate effective waysto circumvent the scallop theorem, which stipulates that a reciprocal swim-ming gait cannot lead to net motion at low Reynolds numbers in Newtonianfluids [170]. Beyond the swimming sheet and the squirmer, other theoreticalmodels have been proposed; many aiming simplicity. Purcell in his famous1976 talk “Life at low Reynolds number” proposed the “simplest animal”**This chapter has been previously published under the same title in Physical ReviewFluids, 3, 123301 (2018) by Datt, Nasouri, and Elfring. © 2018 American Physical Society.818.1. Introductionthat could swim: a planar three-linked swimmer, which could move by alter-nately moving its front and rear segments [12, 170]. The Najafi-Golestanianswimmer [151] propels forward using its collinear assembly of three equalspheres, connected with thin rods which vary in lengths as the spheres oscil-late in a non time-reversible way [1, 73, 83]. Avron et al. [6] proposed anothermodel, more efficient than the three-sphere model, where the swimmer con-sists of just a pair of spherical bladders which exchange their volumes whilealso varying their distance of separation. These models have been instru-mental in understanding swimming at low Reynolds number and thereforein designing optimal swimmers in Newtonian fluids [63, 144, 147, 197].In many instances, microswimmers swim in fluids which are not New-tonian and show complex rheological properties [164]. Among others, oneexample is of a mammalian sperm in the female reproductive tract [72]where cervical mucus displays viscoelasticity and shear-thinning viscosity[120]. Consequently, several model swimmers studied in Newtonian fluidshave also been studied in non-Newtonian fluids for a comparison of theirswimming dynamics [39, 44, 45, 76, 94, 121, 149]. The change in the swim-mer’s dynamics—whether a change in its propulsion velocity for a fixedswimming gait or a change in the gait itself for either a fixed actuation forceor fixed energy consumption—is found to be swimmer dependent [70] and ingeneral, we see that it is fraught with peril to generalize results obtained forone swimmer to others [45, 69]. Perhaps more interestingly, and closer tothe present work, are strategies that do not lead to swimming in Newtonianfluids but can be useful in complex fluids. Lauga [122] first showed this fora squirmer with a surface velocity distribution that does not lead to any netmotion over one cycle in a Newtonian fluid, but does so in a viscoelasticfluid. Keim et al. [109] then demonstrated experimentally this elasticity en-abled locomotion for a rigid assembly of two connected spheres undergoingrotational oscillations about an axis perpendicular to their mutual axis ofsymmetry. Böhme & Müller [19] observed the same for axisymmetric swim-mers performing reciprocal torsional oscillations. Pak et al. [160] modelled asnowman swimmer, which has two unequal spheres that rotate about theircommon axis, that can swim only in complex fluids. Indeed it is knownthat the scallop theorem does not hold in complex fluids [124]; fluid inertia,nearby surfaces, elasticity of the swimmer body, or interaction with otherswimmers are some other reasons why a reciprocal gait for a swimmer maylead to net motion [124]. In truth, the motivation for this work came fromthe interesting experimental and computational works of Klotsa et al. [115]and Dombrowski et al. [57] who show that an assembly of two rigid collinearspheres with a single degree of freedom can swim in the presence of inertia,828.1. Introductionand can in fact also reverse its direction at higher Reynolds number. Felder-hof [74] then theoretically studied the effect of inertia on the motion of suchcollinear swimmers.In this work, we consider two different two-sphere ‘swimmers’. The firstis simply an assembly of two spheres connected as a rigid body that isoscillated by some external force aligned along the axis of symmetry of thetwo spheres. Strictly speaking, this is not a swimmer because the motion ofthe body arises as a consequence of the external force; however, we will seethat by imposing a sinusoidally varying force (with zero mean value) we canachieve a rectified ‘swimming’ motion in a complex fluid. This is similar tothe two-sphere system developed by Pak et al. [160] that achieved net motionunder an imposed torque exerted by an external (magnetic) field, althoughimposing an oscillatory force is perhaps easier to accomplish experimentally.The second swimmer is a two-sphere assembly where the swimming gait isprescribed as the sinusoidal variation of the distance between the two sphereswith no imposed external force. This is similar to the Najafi-Golestanianswimmer [151] except that here instead of three spheres we have only twoand therefore only a single degree of freedom.We emphasize that neither of these swimmers can achieve any net mo-tion over a complete cycle in a Newtonian fluid at zero Reynolds number,irrespective of the radii of the spheres. This is due to the reciprocal forcing ofthe first swimmer and the reciprocal prescribed swimming gait of the second[170]. In contrast, we will show that in a viscoelastic fluid, both swimmersmove in the direction of the smaller sphere when the spheres are of unequalradii and nowhere if the spheres are identical. This motion is a nonlinearviscoelastic response elicited from the deformation of the microstructure ofthe fluid and is therefore absent in Newtonian fluids. In light of this, a two-sphere assembly in a viscoelastic fluid may also be used as a micro-rheometeras previously demonstrated in the works of Khair & Squires [112] and Paket al. [160], but an assembly of two rigidly connected spheres oscillating in afluid is perhaps the simplest such example of a nonlinear micro-rheometer.Here we use the method of perturbation expansion to study the two-sphereswimmers in an Oldroyd-B fluid which for small extension rates is a rea-sonable approximation of polymeric fluids [121]. To conclude this work, wecompare our results with another two-sphere swimmer wherein the spheresthemselves deform elastically in a Newtonian fluid—a comparison of two-sphere swimmers in the presence of elasticity, either of the fluid or the solid.838.2. Swimmer in a viscoelastic fluidd0B1aB2a↵ekFigure 8.1: Schematic of the two-sphere swimmer. The spheres, labeled B1and B2, have radii aα and a, respectively (α > 1). The spheres are (onaverage) a distance d0 apart and e∥ is the unit vector pointing from B1 toB2.8.2 Swimmer in a viscoelastic fluid8.2.1 Two-sphere swimmersIn order to describe the motion of a swimming object, we decompose thecontributions of the velocity of the body,v(x ∈ ∂B) = U +Ω × r + vS , (8.1)where U and Ω are the rigid-body translation and rotation, and the swim-ming gait is denoted by vS . Here the body B, with boundary ∂B, is com-posed of two spheres of radius a and αa, labeled B2 and B1 respectively(B = B1 ∪ B2). Without lack of generality, we assume α ≥ 1. The distancebetween the two spheres is d, which is along e∥ (from large to small sphere)as shown in figure 8.1.When the two spheres are connected as a rigid body, the distance be-tween the two sphere centres is a fixed constant d = d0; there is no swimminggait vS = 0, but an oscillatory external force is applied on the body,Fext = F cos(ωt)e∥. (8.2)This may be imposed by applying an oscillating external magnetic field ifthe spheres are magnetic, or if the spheres are not density matched with thefluid, simply by oscillating the medium (although in that case there wouldbe a mean force on the spheres as well). We will refer to this as an in-phaseswimmer because the two spheres move in unison (see figure 8.2a).In contrast to the first swimmer, the distance d between the spheres ofthe second swimmer varies sinusoidally according tod = d0 + 2δ sin(ωt), (8.3)848.2. Swimmer in a viscoelastic fluid(b)(a)B1B1 B2B2B1B1 B2B1B1B1B1B2B2B2B2B2B2B2B1B1ekekFigure 8.2: Schematic showing one complete cycle for the two swimmers:(a) The in-phase swimmer maintains the distance between the spheres as itmoves forward. (b) In the anti-phase swimmer, the spheres converge anddiverge. The steps in grey show the transition from one half cycle to thenext. The red dot marks the position of the swimmer.as equal and opposite velocities are imposed on the two spheresvS(x ∈ ∂B1) = δω cos(ωt)e∥, (8.4)vS(x ∈ ∂B2) = −δω cos(ωt)e∥. (8.5)Here d0 is the average distance, δ is the amplitude of oscillation and ω is thefrequency. We refer to this swimmer as the anti-phase swimmer (see figure8.2b).For the sake of comparison between the two swimmers, we set the mag-nitude of the force F in (8.2) such that to leading order, the magnitude ofthe velocity of the induced oscillations would also be δω for the in-phaseswimmer (see Appendix B for further details).8.2.2 Theory for swimming in complex fluidsThe motion U of an arbitrary swimmer (or active particle) in a non-Newtonian fluid, with deviatoric stressτ = ηγ˙ + τNN , (8.6)858.2. Swimmer in a viscoelastic fluidwhere τNN is the additional non-Newtonian stress, at zero Reynolds numberis given byU = ηˆηRˆ−1FU · [Fext + FT + FNN ] , (8.7)where U = [U Ω] is six-dimensional vector comprising rigid-body transla-tional and rotational velocities, respectively (we use bold sans serif fontsfor six-dimensional vectors and tensors and bold serif for three dimensionalones) [43, 68]. The six-dimensional vector Fext = [Fext Lext] contains anyexternal force and torque acting on the swimmer. The termFT =ηηˆ∫∂B(vS − v∞) · (n · TˆU)dS, (8.8)is a Newtonian ‘thrust’ due to any surface deformation vS of the swimmerin a background flow v∞. Here, we consider an otherwise quiescent fluid sothat v∞ = 0. The non-Newtonian contributionFNN = −∫VτNN : EˆU dV, (8.9)represents the extra force/torque on each particle due to a non-Newtoniandeviatoric stress τNN in the fluid volume V in which the particles are im-mersed.These formulae rely on operators from a resistance/mobility problem ina Newtonian fluid (with viscosity ηˆ)ˆ˙γ = 2EˆU · Uˆ, (8.10)σˆ = TˆU · Uˆ, (8.11)Fˆ = −RˆFU · Uˆ. (8.12)The tensors EˆU and TˆU are functions of position in space that map therigid-body motion Uˆ of the swimmer to the fluid strain-rate and stress fields,respectively, while the rigid-body resistance tensorRˆFU =[RˆFU RˆFΩRˆLU RˆLΩ]. (8.13)Both problems considered here are axisymmetric, with the forcing and thegait aligned with the axis of symmetry of the swimmer. In this case, theresistance matrix RˆFU is diagonal and only translational motion occurs,simplifying matters substantially.868.2. Swimmer in a viscoelastic fluidWe consider here only the time-averaged or (post-transient) mean veloc-ity of the swimmer,U = ηˆηRˆ−1FU · [Fext + FT + FNN ], (8.14)where the overline represents a time-averaged quantity. The in-phase swim-mer does not change shape therefore the resistance is constant and FT = 0because vS = 0; furthermore, the prescribed force is periodic with zeromean, Fext = 0. In contrast, the anti-phase swimmer has no external forc-ing Fext = 0, but undergoes a reciprocal shape change and so, while theresistance is not constant, we know that Rˆ−1FU · FT = 0 by the scallop theo-rem [102]. We see then that, for both swimmers, the net motion is only dueto the non-Newtonian contribution from the rheology of the fluid mediumU = ηˆηRˆ−1FU · FNN . (8.15)By the symmetry of the problem, any net motion must be in the directionof the axis of symmetry e∥ i.e. U = Ue∥ withU = − ηˆηRˆFU∥∫VτNN : EˆU∥ dV, (8.16)where RˆFU∥ = e∥ · RˆFU · e∥ is the scalar resistance to translational motionof the two-sphere assembly in the direction of the axis of symmetry, whereasEˆU∥ = EˆU · e∥ is a second order tensor equal to the strain-rate field due torigid-body translation (with unit speed) in the direction e∥. RˆFU∥ and EˆU∥are obtained by way of the Stimson-Jeffery solution of two spheres movingwith equal velocities along their axis of symmetry in a Newtonian fluid [192].Finally, we note that although the geometry of the anti-phase swimmer is notconstant, we solve the problem asymptotically for small deformations abouta mean geometry such that RˆFU∥ , EˆU∥ , and the boundary of the volumeintegral in (8.16) are constant, which allows us to pass the time-averageoperator onto the non-Newtonian stress alone [69, 125].8.2.3 Constitutive equationWe are interested here in the effects of nonlinear viscoelasticity that enablethe net motion of the swimmers. Until this point, we have only assumedthat the stress in the fluid may be separated into a Newtonian and non-Newtonian contribution. The deviatoric stress τNN in a viscoelastic fluid878.2. Swimmer in a viscoelastic fluidtypically follows a nonlinear evolution equation. For simplicity, we use theOldroyd-B constitutive equation [16] but other constitutive relationships canbe easily used within this formalism. Oldroyd-B is a single relaxation timeviscoelastic (Boger fluid) fluid that is governed by∇τNN=ηNNλγ˙ − 1λτNN , (8.17)where λ is the relaxation time of the fluid and ηNN is an additional viscositydue to the (polymeric) microstructure. The upper convected derivative isdefined∇A= ∂A/∂t+ v ·∇A−((∇v)⊤ ·A+A ·∇v)where v is the fluidvelocity field.The problems we consider here are periodic (with period τ = 2pi/ω) and,neglecting any transient evolution from an initial condition, we may simplifymatters by assuming that all functions may be written as Fourier series, forexample, the velocity field v = ∑p v(p)epiωt. Following this for the stress,we have [69]τ(p)NN = (η∗(p)− η)γ˙(p) +N (p) (8.18)where the tensor N (p) represents the contribution of the nonlinear terms toeach mode and the complex viscosityη∗(p) = 1 + piDeβ1 + piDe η0. (8.19)The Deborah number, De = λω, characterizes the relative rate of actuationof the spheres to the relaxation of the fluid. The viscosity ratio β = η/η0is the relative viscosity of the Newtonian part of the fluid (solvent) whereη0 = η+ ηNN represents the (total) zero-shear-rate viscosity of the fluid. Inparticular, by substituting (8.18) into (8.16) one may show thatU = − ηˆη0RˆFU∥∫VN : EˆU∥ dV, (8.20)where N = N (0), and we see that linear viscoelasticity does not lead tonet motion of these swimmers because by definition N (p) = 0 for linearlyviscoelastic fluids (see Appendix B for further details).8.2.4 Small amplitude expansionWe assume that the oscillation amplitudes are much smaller than all otherlength scales, δ ≪ a, d0, and define dimensionless quantities ϵ = δ/a ≪ 1888.2. Swimmer in a viscoelastic fluidand ∆ = d0/a. In addition, we define a dimensionless clearance between thespheres, ∆c = ∆ − (1 + α). We solve for the flow by employing a regularperturbation expansion in small deformations ϵ to all flow quantities{v, τ , p, . . .} = ϵ {v1, τ1, p1, . . .}+ ϵ2 {v2, τ2, p2, . . .}+ . . . . (8.21)The swimming speed is then given byU = −ϵ2 ηˆη0RˆFU∥∫VN2 : EˆU∥ dV +O(ϵ4). (8.22)Because the tensor N represents the nonlinear terms in the viscoelasticconstitutive equation, there are no terms linear in ϵ. The quadratic termdepends only on the leading order flow field, N2[v1, τ1], which is a solu-tion to a linearly viscoelastic flow that has exactly the same flow field as aNewtonian flow with equivalent prescribed velocity boundary conditions.When the spheres move together as a rigid body (the in-phase swimmer),the solution for v1 is easily obtained using the solution for two spheresmoving with equal velocities along the line joining their centers by Stimson& Jeffery [192]. Similarly when the spheres approach one another (anti-phaseswimmer), the solution for v1 is available due to the work of Maude [141]for two spheres approaching each other in a Newtonian fluid (see [188] forsome corrected errors). Thus knowing the O (ϵ) fields, we may evaluate thetensor N2, which for an Oldroyd-B fluid is given byN2 = −12Re{De (1− β)(1 + iDe)[v(−1)1 ·∇γ˙(1)1 −(∇v(−1)1)⊤· γ˙(1)1 − γ˙(1)1 ·∇v(−1)1]}.(8.23)Finally, we obtain the leading order motion for either swimmer by evaluating(8.22) to findU = δω δa(De (1− β)1 +De2)U , (8.24)where the dimensionless quantity U is evaluated using numerical integrationof an analytical expression.Note that under this small-amplitude expansion |γ˙| ∼ ϵω and conse-quently Weissenberg numbers, Wi = |γ˙|λ ∼ ϵDe, are asymptotically smallerthan Deborah numbers. Thus, provided ϵ is made small enough, these re-sults are valid for arbitrary values of Deborah number even for fluids suchas Oldroyd-B fluid that are unphysical for order one Weissenberg numbers(see also [121, 125]).898.2. 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eenzt87/3j8z8f/fvyfJx/tTD9W/3f+0Gl8nhz8F46lP1E=</latexit>Figure 8.3: The swimming speed coefficient U is plotted with variation in theclearance ∆c between the two spheres for different size ratios α. The squaresymbols (connected by dashed lines) represent the anti-phase swimmer andthe circles (connected by solid lines) represent the in-phase swimmer. Allquantities are dimensionless.8.2.5 Results and discussionWe find that the two-sphere assembly can swim in a viscoelastic fluid atfinite Deborah numbers, provided the two spheres are of different sizes. Thedifference in the sphere sizes leads to the fore-aft asymmetry required forswimming. We see from (8.24) that the swimming speed is maximized whenDe = 1. In the limit when the actuation is much slower than the relaxationof the fluid, De→ 0, or much faster, De→∞, there is no swimming U = 0,indeed the term in the brackets of (8.24), which governs this behavior, issimply the dimensionless elastic modulus of the fluid [69]. We report thevalues of U for the two swimmers for a few configurations in figure 8.3. Bothswimmers swim with the smaller sphere as the head. At small separations,the anti-phase swimmer is an order of magnitude faster; however, at largeseparations this difference in magnitude fades away.The direction of the motion of these swimmers can be largely predictedby studying a single sphere oscillating in a viscoelastic fluid. The viscoelasticsteady streaming flow that results from this motion draws fluid in towardsthe center of the sphere along the axis of oscillation [18]. Larger spheresgenerate stronger viscoelastic flows for a given velocity but the relation-908.2. Swimmer in a viscoelastic fluidship is sublinear in radius and so one would expect that when two unequalspheres interact, because of the relative resistances, the net effect of the in-teracting viscoelastic streaming flows would be to push the assembly in thedirection of the smaller sphere. This is essentially a ‘far-field’ superpositionargument, where there is no difference between in-phase and anti-phase os-cillations, and one should take great care when applying this logic to closelyinteracting spheres in a nonlinear non-Newtonian fluid; however, this pre-diction qualitatively agrees with our exact two-body problem solutions. Wealso note that Keim et al. [109] find that a similar two-sphere assembly un-dergoing rotational oscillations instead moves towards the larger sphere, butin that case the spheres are moving perpendicular to their axis of symmetryand so we expect the viscoelastic steady streaming flow to be reversed alongthat axis.Examining more closely first the in-phase swimmer, a rigid body of suchshape moving in a weakly viscoelastic fluid (e.g. a second-order fluid underslow flows [16]) will experience a net viscoelastic force pointing towards thesmaller sphere and so the total drag on the body when the larger sphere leadsincreases while it decreases when the smaller sphere is at the front. Leal [131]has also shown that for sedimenting slender bodies, when the trailing endis sharp and the leading edge is blunt the drag increases in a second-orderfluid. In light of this, when the two-sphere body oscillates periodically in aviscoelastic fluid, one expects the net viscoelastic contribution to the forceon the body over one cycle to point towards the smaller sphere. The speed ofthe swimming depends on the strength of this viscoelastic contribution andthe hydrodynamic resistance to the steady translation of the body. As canbe seen from figure 8.3, such a swimmer has an optimum in the swimmingvelocity at a certain separation for a given ratio of the sphere sizes.For the anti-phase swimmer, the viscoelastic force seems to depend onthe strength of squeeze flow between the two spheres which increases as theseparation between the spheres decreases. Combined with the low hydrody-namic resistance of the assembly when the spheres are close, swimming ismonotonically faster with smaller separations (for a given size ratio). Whenthe spheres are far apart, the strength of the squeeze flow decreases and thetwo types of swimmers swim with speeds of the same order.Clearly, a size ratio of 1 will not lead to swimming. One also expectsa very large size ratio to be equally inefficient due to a decrease in the netfore–aft asymmetry over a complete cycle. This non-monotonicity with sizeratio is also observed at small distances in figure 8.3, although at very largedistances, when the interaction between the spheres has much decreased,higher size ratio leads to better swimming. However, this may not be the918.3. Swimmer with elastic spheresregime one would focus on for optimal swimming.We also note that the effect of viscoelasticity on the swimmers is foundto be opposite to the effect of inertia as described in the analytical workof Felderhof [74]. There, the two-sphere swimmer moves with the largersphere as the head, as might be expected given that weakly inertial steady-streaming flow can push fluid out from an oscillating sphere along the axisof oscillation [18, 175, 187]. However, recent numerical work by Dombrowskiet al. [57] reports that the smaller sphere leads at small Reynolds numberonly to switch to larger sphere leading at higher Reynolds number. We donot observe such switching of swimming direction with the Deborah numberin our analysis which is valid for small oscillation amplitudes. We alsonote that although the results presented here are for an Oldroyd-B fluid,one may perform a small-amplitude analysis with other viscoelastic fluidmodels like the second-order fluid, Giesekus fluid, and FENE-P [16] andfind qualitatively similar results. Any quantitative differences that occur aredue to parameter values (concerning for example, the presence or absence ofsecond normal stress differences) particular to the model or slightly differentdefinitions of the Deborah number [121].In the next section, we study a two-sphere swimmer with elastic spheresin a Newtonian fluid and demonstrate that the direction of propulsion is thesame as this two-(rigid)-sphere swimmer in viscoelastic fluid.8.3 Swimmer with elastic spheresWe now compare the two-sphere swimmers in a viscoelastic fluid with swim-mers with elastic spheres in a Newtonian fluid. This calculation closely fol-lows the work of Nasouri et al. [153] who studied a two-sphere swimmer withone rigid and other elastic sphere in a Newtonian fluid. Here, similar to theprevious section, we consider model swimmers that consist of two spheres ofradii a and αa, but this time we relax the rigidity constraint by assumingthat the spheres are isotropic, incompressible neo-Hookean solids.To study the behavior of this system, one must first understand thedeformation of a single elastic sphere in Stokes flow. Neglecting intertia,momentum balance for the elastic solid yields∇ · σs + f(t) = 0, (8.25)where σs is the stress due to elastic deformation and f is the applied bodyforce density on the sphere. For an isotropic, incompressible neo-Hookeansolid, this stress field can be expressed using the displacement vector u as928.3. Swimmer with elastic spheres[91, 156]σs = −psI +G(D ·DT − I), (8.26)where G is the shear modulus and D = I +∇u is the deformation gradienttensor. The Lagrange multiplier ps enforces the incompressibility of the solidthroughdet (D) = 1, (8.27)where ‘det’ is the determinant. The traction across the solid-fluid interfacemust be continuous so thatσs · n = σ · n, (8.28)where n is the normal vector to the deformed sphere and σ is the stress fieldin the fluid domain which can be determined by solving the Stokes equationsover the deformed boundary.If we scale lengths with a, velocities with δω, forces with G/a, time witha/δω, stress in the solid domain with G and stress in the fluid domain withηδω/a, from equation (8.28) a dimensionless parameter ε = ηδω/aG thennaturally arises as the ratio of viscous forces to elastic forces. Here we focuson the case wherein the sphere is only weakly elastic; elastic forces are muchlarger than viscous forces and so ε≪ 1. Since the motion is axisymmetric,one can show that the elastic sphere reaches equilibrium with a relaxationtime scale of τrelax ∼ O(aε/δω). Thus, under the assumption of ε ≪ 1, wecan assume that elastic deformations are quasi-static: the sphere deformsinstantly and we then have rigid-body motion [98].Similar to the viscoelastic case, for the in-phase swimmer, for the sakeof comparison, we set the magnitude of the applied external force to beF = δωRFU∥ so that to leading order the speed of oscillation is δω. For theanti-phase swimmer, we define the gait according to (8.4) and (8.5) but inthis case the velocity is prescribed on the deformed boundaries.We now return to our two-sphere swimmer, with both spheres beingweakly elastic. In a Newtonian fluid, the dynamics of the motion of thebody is given byU = R−1FU · [FT + Fext] . (8.29)The thrust force may be generically decomposed into the thrust generatedby each sphere FT = FT1 +FT2 . Because the spheres are deforming, we will938.3. Swimmer with elastic spheresassume that the spheres are well separated, and compute the hydrodynamicthrust generated by each sphere with hydrodynamic interactions solved toleading order using a far-field approximation, ∆≫ 1.For individual spheres, (8.8) reduces to Faxén’s first law for each sphereFT1 = −R1 ·(vS1 −F1 [v∞2 ]), (8.30)FT2 = −R2 ·(vS2 −F2 [v∞1 ]), (8.31)where R1 and R2 are the resistance tensors for each sphere and, F1 andF2 are the respective Faxén operators. Here, v∞1 is the background flowfield induced by sphere B1, and vice versa for v∞2 . Recalling that spheresare only weakly elastic (since ε≪ 1), the spheres only slightly deviate fromtheir spherical shape so that the hydrodynamic resistance and Faxén’s lawsare unchanged from an undeformed sphere to leading order [24, 113]. Thenet thrust generated by the swimmer at the leading order is therebyFT = 6piηa(−αvS1 + αv∞2,1 − vS2 + v∞1,2), (8.32)where v∞2,1 indicates the background flow from sphere 2 evaluated at thecenter of sphere 1 (and vice versa). For the externally forced (in-phase)swimmer, the gait is zero vS1 = vS2 = 0. For the anti-phase swimmer, theimposed gait is periodic and given that we are interested in only the meanmotion, averaging over a period τ = 2pi/ω, for both swimmers, leads toFT = 6piηa(αv∞2,1 + v∞1,2). (8.33)We see clearly, in this far-field result, that the thrust is dictated purely bythe elastic steady streaming flow generated by each sphere acting on theother.By solving equations (8.25) to (8.28) asymptotically, one can determinethe flow field around an oscillating elastic sphere. This flow field, upon aver-aging, will give the steady streaming flows v∞1 and v∞2 (see [153] for technicaldetails). By prescribing an external force of magnitude F = δωRFU∥ for thein-phase swimmer, the magnitude of the deformation and thus the magni-tude of the steady-streaming flows is equal for both swimmers. We note,in particular, that the elastic steady-streaming flow of each sphere drawsfluid inward along the axis of symmetry in much the same way as the vis-coelastic steady streaming flow. Here, we find that v∞2,1 · e∥ ∝ δωϵ3/∆2 andv∞1,2 · e∥ ∝ −δωϵ3/α∆2. The net thrust is thenFT =7497934048piηd0δωα(1− 1α2)ϵ3∆3e∥. (8.34)948.4. ConclusionBoth oscillating elastic spheres generate steady-streaming flows but the mag-nitude of each flow is inversely proportional to the radius while the resistanceof each sphere is linearly proportional to the radius and so the net thrustforce is in the direction of the smaller sphere (α ≥ 1).With a hydrodynamic resistance of RFU∥ = 6piηa(1 + α), and using thefact that the average external force is zero,Fext = 0, (8.35)(in the case of the anti-phase swimmer the prescribed force itself is zero),we obtain the time-averaged velocityU = 2499368096δω(1− 1α)ϵ3∆2e∥. (8.36)The swimming motion is always in the direction of the smaller sphere, similarto the rigid swimmer in the viscoelastic fluid (the swimmer swims with thesmaller sphere as the head). Furthermore, since we solved this problemassuming the spheres are well separated using far-field approximations ofthe flow, the speed of the swimmer is ultimately independent of whether thespheres oscillate in-phase or anti-phase.8.4 ConclusionWe studied the effects of elasticity on the motion of two-sphere swimmerswhere the two spheres oscillate in-line. When the two spheres are rigid andthe fluid viscoelastic, we find that the swimmers swim with the smaller-sphere as the head. However, the swimming speed is dependent on the typeof swimmer: anti-phase swimmers, in general, swim faster than the in-phaseswimmers. We also find that when the spheres themselves are elastic and thefluid Newtonian, the swimmer again moves in the direction of the smallersphere.We note that the effects of elasticity on the swimmer are found to beopposite of the effect of inertia described in the theoretical work of Felderhof[74] who showed that the two-sphere swimmer moves with the larger sphereas the head, but we do not observe a reversal of the swimming directionas a function of the Deborah number, analogous to what is observed uponincreasing Reynolds number in the numerical work of Dombrowski et al.[57].95Chapter 9Conclusion and outlookThe motivation for the work in this thesis was to understand how some sim-ple motions of particles are affected by the rheological properties of the fluid.We focussed on small particles that can swim, and restricted our attentionto two rheological properties, shear-thinning viscosity and viscoelasticity,chosen for their apparent ubiquity in non-Newtonian fluids. We followed atheoretical approach, often drawing from recent experimental and numericalworks. In this section, we briefly recapitulate and discuss the findings, thelimitations, and the future of this work.9.1 The findingsWe started the thesis by looking at the dynamics of a squirmer in a Newto-nian fluid with externally imposed gradients in the fluid viscosity in chapter3. The work was motivated by the recent study of Liebchen et al. [136]explaining the physical mechanism of viscotaxis. We showed that viscositygradients change the motion of a squirmer drastically in comparison to itsmotion in a Newtonian fluid of uniform viscosity. Specifically, we find thatthe squirmers are in general viscophobic, although the details of their dy-namics are dependent on whether they swim as pushers, pullers, or neutralswimmers. The differences in these details among the swimmers can be usedto sort them based on their swimming style.Next we discuss the effects of shear-thinning viscosity on the motionof passive particles in Chapter 4. By analysing the motion of a sphere in abackground flow field due to an external force, or somewhat equivalently themotion of a rotating and sedimenting sphere (a set-up proposed as a rheome-ter [82]), we showed how translational and rotational motion can couple evenfor particle shapes for which no such coupling is present in Newtonian flu-ids. We also showed that for two equal spheres that sediment along the linejoining their centres, while the principle of kinematic reversibility holds andspheres maintain their initial distance of separation, the reduction in theirdrag is a non-monotonic function of their distance of separation. As for thecase of a dilute suspension of rigid spheres in shear-thinning fluids, we found969.1. The findingsthe corresponding Einstein viscosity, reflecting how the presence of a par-ticle both adds and subtracts from the shear-viscosity; which contributiondominates depends on the applied shear-rate.The effects of shear-thinning viscosity on the swimming speed ofa squirmer were studied in Chapter 5. Prior to our work, there wererelatively few studies analysing the motion of swimmers in shear-thinningfluids. We showed that a squirmer can swim both slower and faster ina shear-thinning fluid as compared to a Newtonian fluid for the sameswimming stroke. The faster or slower swimming, in general, cannot beexplained by a simple thrust and drag decomposition of the swimmingproblem—an approach which previously had been used for developingintuition in shear-thinning fluids. We also showed that squirming modesthat do not contribute to swimming in Newtonian fluids can contribute toswimming in shear-thinning fluids, and that, in general, squirming throughshear-thinning fluids possesses some rich physics.While in shear-thinning fluids we showed that higher-order squirmingmodes, and not just the first mode as in Newtonian fluids, can contributeto swimming, we showed exactly how these modes couple and contributeto the swimming speed in a weakly viscoelastic fluid. In Chapter 6, wegive the expression for the swimming speed of a general, n-mode, squirmerin a second-order fluid. As an example of squirmers in which the highermodes are known, we chose diffusiophoretic Janus particles and studied theirdynamics both in second-order and shear-thinning fluids. In the particularcase of Janus particles, we quantified the error in swimming speeds if onewere to use only the first two swimming modes to calculate the effects of thetwo non-Newtonian properties: viscoelasticity, and shear-thinning viscosity.Whereas in Chapter 6 we studied only the first effects of viscoelasticitythrough a second-order fluid on the squirmer motion, we considered higher-order effects using two different viscoelastic models—the Giesekus modeland a fluid of differential type (grade three)—in Chapter 7. Importantly, wefind that the swimming speeds as predicted by the expansion in Deborahnumber for the two fluid models are qualitatively different. The work raisesquestions regarding the adequacy of the Deborah number expansion, andthe choice of viscoelastic model.In Chapter 8, we studied an assembly of two spheres connected with amassless, hydrodynamically non-existent, rod for effects of viscoelasticity.We found that on oscillating the two spheres along the rod, the entire as-sembly moves forward, provided that the two spheres are of unequal sizes.We contrasted this analysis with a similar assembly of two elastic spheresin a Newtonian fluid. We found that in both cases the assembly moves in979.2. The limitationsthe direction of the smaller sphere. Note that, constrained by the scalloptheorem, such an assembly of rigid spheres cannot swim in a Newtonianfluid; that this simple design swims in a viscoelastic fluid is possible onlydue to the complexity of the fluid.The topics in this thesis, for most part, concerned the newly emergingfield of active matter, which has been attracting researches from the moreconventional fields in physics, chemistry and biology. We believe the workin this thesis offers some fundamental contributions to this rapidly grow-ing field, be it in sorting and designing active particles, or more generallythrough understanding their motions in fluids where they commonly are, orwill be, found.9.2 The limitationsClearly, many particles are not spherical, and modelling swimmers assquirmers may not be accurate on occasions. Moreover, many non-Newtonian fluids have more complex rheological properties than justshear-thinning viscosity and viscoelasticity. While these points are impor-tant, suffice to say we believed disregarding them was essential for anyimmediate theoretical progress. Hence, we do not delve into these pointshere.More importantly, we ask what when no parameter is small in our anal-ysis? As an example, consider the two-sphere swimmer assembly discussedin Chapter 8. Will the direction of the swimmer motion be reversed whenthe amplitude of oscillations of the spheres is not small? Asymptotic resultsin this work by themselves cannot answer such questions. Even how smallis small enough for the analytical results to hold can be known, generally,only through experiments and numerics.Further, in the thesis, the quantities of interest were swimming speeds,sedimenting velocities, and drag forces on particles—integral quantities thatcould be calculated relatively easily using the generalized reciprocal theo-rem. The shortcut of the reciprocal theorem meant not travelling the morelaborious path of calculating the flow field corrections first, and then cal-culating such quantities as mentioned, but perhaps also a want of a deeperphysical intuition developed through analysing flow fields.In our opinion, we still need more of all—theoretical, numerical, andexperimental—research for a complete understanding of the problems wehave studied in this work; theoretical work, being more tractable of thethree, in part due to its self-declared assumptions, has often struck first in989.3. The futurethis battlefield.9.3 The futureMany microorganisms swim in complex fluids. We know more about mi-croswimmers in Newtonian fluids than in complex fluids. From a prob-lem solving perspective, following Einstein’s suggestion ‘that since the basicequations of physics are nonlinear, all of mathematical physics will have tobe done over again’ [183] one may just start solving swimming problemsthat have been understood in Newtonian fluids in complex (non-linear) flu-ids. In fact, not a bad advice when comes to understanding nature, owingto the ubiquity of swimming phenomena in biology where many fluids showcomplex rheological behaviour not restricted to just shear-thinning viscos-ity and viscoelasticity, but also involving thixotropy and microstructuralanisotropy.Immediate areas to venture into include understanding the interactionsbetween swimmers in complex fluids, for microswimmers, e.g., sperm cells,swim together in large numbers, and understanding how viscoelasticity of thefluid and inertia together affect the motion of a swimmer—microswimmersdo accelerate when escaping predators (for effects of inertia alone, see [110,210] and references therein).Dynamics of model microswimmers can also be analysed in fluids thatare active themselves; such is done in the latest work of Soni et al. [186],who answer if an active fluid can do work on a Taylor sheet swimmer, andtherefore make it swim faster than in a passive fluid for the same stroke.The same stroke assumption that we have also used in our works, shouldbe relaxed too in order to capture how the strokes themselves are affectedby the fluid’s rheological properties. This can be realised by prescribingthrust forces on the swimmers rather than swimming strokes, as Curtis &Gaffney [39] have done for the Najafi-Golestanian three-sphere swimmer inan Oldroyd-B fluid, and thereby showing the qualitative different resultsachieved through prescribing forces and prescribing kinematics.One problem, we would like to see addressed in the near future is thestochastic dynamics of active particles in complex fluids. While large macro-scopic swimmers can sustain their swimming direction for long times, mi-croswimmers experience orientation decorrelation because of their smallsizes [117, 178]. At the size of microswimmers, thermal fluctuations can‘lead to a Brownian loss of the swimming direction, resulting in a transi-tion from short-time ballistic dynamics to effective long-time diffusion’ [178].999.3. The futureMany bacteria also experience what is called tumbling. In tumbling, ‘a bac-terium typically runs in a directed sense for a certain time interval beforechanging its orientation abruptly, and by a large amount. It then runs inthe new direction, possibly correlated to the old one’ [118].The stochastic dynamics play an important role at the length scale ofmicroswimmers [11, 92]. Even reciprocal swimmers, which on average do notswim, have ‘enhanced diffusivities, possibly by orders of magnitude, abovetheir normal Brownian diffusion’ [123]. As many microswimmers are foundin complex fluids, Patteson et al. [163] experimentally investigated the runand tumble dynamics of E. Coli bacteria in polymeric solutions. 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Additional details on related expressions for both passive andactive particles can be found in the recent work of Nasouri & Elfring [152].For a single sphere of radius a in a Newtonian fluid with viscosity ηˆRˆFU = 6piηˆaI, (A.1)RˆLΩ = 8piηˆa3I, (A.2)RˆLU = 0, (A.3)RˆFΩ = 0. (A.4)Additionally,n · TˆU = −3ηˆ2a [I 2Θ] , (A.5)where Θij = ϵijkxk and n is the unit normal to the surface.The entities corresponding to ˆ˙γ/2 = EˆU · Uˆ[EˆU ]ijk =3axk4r3(δij − 3xixjr2)+ 3a34r5[xk(−δij + 5xixjr2)− xiδjk − xjδik](A.6)[EˆΩ]ijk = −3a32r5 (xixlϵljk + xjxlϵlik) . (A.7)are detailed in [157]. Relevant to the stresslet calculation, we have[EˆE ]klij =ηˆa52r5 (δikδjl + δjkδil) +5ηˆa34r5 (δilxjxk + δkixjxl + δjlxixk + δkjxixl)− 5ηˆa52r7 (δklxixj + δjlxixk + δilxjxk + δjkxixl + δikxjxl)+ 5ηˆ(7a52r9 −5a32r7)xixjxkxl +5ηˆa32r5 δklxixj ,(A.8)119Appendix A. Some expressions for motion of spheres[TˆE ]klij =ηˆa5r5(δikδjl + δjkδil) +5ηˆa32r5 (δilxjxk + δkixjxl + δjlxixk + δkjxixl)− 5ηˆa5r7(δklxixj + δjlxixk + δilxjxk + δjkxixl + δikxjxl)+ 5ηˆ(7a5r9− 5a3r7)xixjxkxl,(A.9)which can be found in the supplementary information of [128].For the problem involving two spheres sedimenting along their commonaxis, the stream functions for the two auxiliary cases in Newtonian fluids,two spheres moving with the same velocity and two spheres approachingeach other with equal speed, were reported by Stimson & Jeffery [192] andBrenner [23], respectively. The stream functions are expressed in the formof infinite series solutions. To ensure convergence, we considered aroundthe first 30 to 40 terms of the series. The stream functions are used tocalculate the strain-rate tensors corresponding to EˆU (Stimson-Jeffery) andEˆ∆U (Brenner, Maude). The stress tensor τNN for the case considered in themanuscript is evaluated using strain-rates from the Stimson-Jeffery solution.For the limiting case of two spheres touching each other and sedimenting,we use the solution of the problem in Newtonian fluids by Cooley & O’Neill[38] to evaluate both τNN and EˆU . The stream function for this case isexpressed in form of a definite integral from zero to infinity (equation 3.4 inthe reference). A non-infinite value of the upper limit of the integral has tobe chosen for evaluation; we find that convergence of the solution is achievedat a value of around 15.120Appendix BLinear viscoelasticityEquation (8.7) delineates a relationship between forces and velocities andwith (8.18) gives, for each Fourier mode,ηˆη∗(p)RˆFU ·U(p) = F (p)ext+η∗(p)ηˆ∫∂BvS(p)·(n·TˆU)dS−∫VN (p) : EˆU dV. (B.1)For a rigid-body motion under periodic external forcing, vS = 0. Assum-ing that the magnitude of the forcing is small so that nonlinear viscoelasticterms are negligible to leading order, we obtain a (complex) linear viscoelas-tic relationship between force and velocity for each mode,R∗(p)FU · U(p) = F (p)ext , (B.2)where the complex resistance R∗FU =η∗ηˆRˆFU .In our problem, there is only a single force mode 2F (1) = F (the othermodes are zero, see (8.2)). Setting the magnitude of the velocity to be|U | = δω then leads to a force with magnitude F = δω|η∗(1)|RˆFU∥/ηˆ. Usingthe complex viscosity of Oldroyd-B (see (8.19)) we obtain that taking F =δωη01+βDe21+De2 RˆFU∥/ηˆ leads to a velocity U = δω cos(ωt + ϕ)e∥ to leadingorder.121

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