Modelling the Fluid Mechanics of Cilia and Flagella in...

EPJ manuscript No.(will be inserted by the editor)

Modelling the Fluid Mechanics of Cilia and Flagella inReproduction and Development

Thomas D. Montenegro-Johnson13, Andrew A. Smith13, David J. Smith123, Daniel Loghin1, and John R. Blake13

1 School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK2 School of Engineering & Centre for Scientific Computing, University of Warwick, Coventry, CV4 7AL, UK3 Centre for Human Reproductive Science, Birmingham Women’s NHS Foundation Trust, Edgbaston, Birmingham,

B15 2TG, UK

Received: 29th August 2012

Abstract. Cilia and flagella are actively bending slender organelles, performing functions such as motility,feeding and embryonic symmetry breaking. We review the mechanics of viscous-dominated microscaleflow, including time-reversal symmetry, drag anisotropy of slender bodies, and wall effects. We focus onthe fundamental force singularity, higher order multipoles, and the method of images, providing physicalinsight and forming a basis for computational approaches. Two biological problems are then consideredin more detail: (1) left-right symmetry breaking flow in the node, a microscopic structure in developingvertebrate embryos, and (2) motility of microswimmers through non-Newtonian fluids. Our model of theembryonic node reveals how particle transport associated with morphogenesis is modulated by the gradualemergence of cilium posterior tilt. Our model of swimming makes use of force distributions within a body-conforming finite element framework, allowing the solution of nonlinear inertialess Carreau flow. We findthat a three-sphere model swimmer and a model sperm are similarly affected by shear-thinning; in bothcases swimming due to a prescribed beat is enhanced by shear-thinning, with optimal Deborah numberaround 0.8. The sperm exhibits an almost perfect linear relationship between velocity and the logarithmof the ratio of zero to infinite shear viscosity, with shear-thickening hindering cell progress.

1 Introduction

The active locomotion of cells and transport of fluids onmicroscopic scales has been a benchmark problem in ap-plied mathematics for the past 60 years, since Taylor [1]demonstrated that a two-dimensional sheet could swimutilising only viscous forces. The field had been an activearea of research for zoologists for some time [2–5], thougha leap forward in understanding was made when experi-mentalists and theoreticians began to collaborate. It wasin this spirit of collaboration, fostered by Taylor and Gray,that Hancock [6] first developed slender body theory, apowerful method based upon modelling slender swimmersby distributions of force singularities, which in turn hasled to the development of the singularity methods used inthe present study.

Microscale fluid propulsion is usually achieved in na-ture through the beating of cilia and flagella. These areslender, hair-like organelles that perform a range of func-tions from locomotion to sensory reception. In reproduc-tion, flagella propel sperm cells, allowing sperm and eggto meet, and cilia transfer the fertilised embryo from theampulla to the uterus. Then, in the early stages of verte-brate embryonic development, cilia are responsible for theproduction of a directional fluid flow which breaks left-right symmetry in vertebrates [7]. This occurs in a fluid

filled cavity that appears on the embryo shortly after fer-tilisation, called the node.

In eukaryotic cells, cilia and flagella induce active bend-ing along their length via a remarkable, evolutionarily-conserved internal structure known as the axoneme, whichwas discovered with the advent of electron microscopy [8–10]. The axoneme comprises 9 inextensible outer micro-tubule doublets, as shown in fig. 1, and passive linkingelements which stiffen the assembly. The combination ofrelative, localised microtubule sliding, their inextensibilityand the restraining effects of linking structures, generatesbending. This is the ‘sliding filament theory’ first demon-strated by Satir [11]. For human sperm, as with mostmotile cilia and flagella, a central pair of microtubulesruns along the length of the axoneme. This configurationis referred to as the “9+2” axoneme. Nodal cilia, however,lack this central pair. These “9+0” cilia were thought tobe immotile until the relatively recent work of Nonaka etal. [7], which showed that they ‘whirled’ with a near rigid-body motion, quite distinct from the extensively studiedbeat patterns of 9+2 cilia.

In Newtonian fluids, such as sea water, microscopiccilia and flagella must execute a beat pattern that is nottime reversible [12] in order to generate a net fluid flow.This is due to the lack of time dependence in the gov-

2 T.D. Montenegro-Johnson et al.: Fluid mechanics of cilia and flagella in reproduction and development

Outer dynein armOuter fiber(doublet)

Membrane

Centralfiber

Radial linkhead

Radiallink

Centralsheath

Nexinbridge

Innerdynein arm

1

2

3

4

56

7

8

9 AB

Fig. 1. A schematic cross-section of the “9+2” axoneme, re-drawn from Fawcett [15].

erning fluid equations, which are discussed in sect. 2. Forsperm flagella, time reversibility is broken by propagatinga bending wave along the length of the flagellum. How-ever, nodal cilia are incapable of executing sophisticatedwaveforms, and so instead break symmetry by tilting theiraxis of rotation in a given direction. Hydrodynamic inter-action with the cell wall then leads to an effective-recoverystroke asymmetry.

In this paper, we first summarise the fluid mechanicsthat govern the flow generated by cilia and flagella. Wediscuss the flow solution generated by a singular force,and the relationship between this fundamental solutionand the physics governing the flow induced by the motionof slender bodies. We show how the effects of a no-slipwall may be incorporated through the inclusion of ‘imagesystems’ involving higher-order flow singularities, such aspoint stresses and torques. These are demonstrated graph-ically via singularity diagrams. We then discuss compu-tational techniques that have arisen from modelling theaction of cilia and flagella on the fluid by distributions ofthese flow singularities.

A combination of these computational techniques isused to model cilia-driven flow in the embryonic node inmice (fig. 2). The node is modelled at various stages ofdevelopment, and the effects of posterior cilium tilt onthe generation of directional flow is examined.

For many biological flow problems, for instance theswimming of sperm through cervical mucus, the assump-tion of Newtonian fluid rheology, as discussed in sect. 2,is invalid. In such cases, the fluid dynamics is governedby complicated nonlinear equations. We apply a finite ele-ment method developed by Montenegro-Johnson et al. [13]to an artificial swimmer described by Najafi and Golesta-nian [14], and examine the effects of nonlinear fluid rhe-ology on the swimmer’s progression. We then apply thesame techniques to model a two-dimensional analogue ofhuman sperm.

(a) (b)

(c) Reichert’s membrane

Nodal flow Return flow

R L

Microvillum;NVP extrusion

NVPNode floor

Cilium NVP breakup;cell signalling

Fig. 2. (a) Electron micrograph of a mouse embryo at 7.5days post fertilisation, indicating the anterior-posterior andleft-right directions (VN, ventral node; NP, notochordal plate;FG, foregut). Bar 100µm, reprinted from Hirokawa et al. [16]with permission from Elsevier, © 2006. (b) Cilia covering thenodal pit cells, reprinted from Nonaka et al. [17]. (c) Schematicfigure showing a simplified model for nodal flow. Cilia rotate,with axis of rotation tilted towards the posterior, creating aright-to-left flow above the cilia tips. This is balanced by a re-turn flow, due to the overlying Reichert’s membrane. The flowis believed to transport Nodal Vesicular Parcels (NVPs) whichbreak up at the left of the node, delivering morphogen pro-teins to initiate asymmetric development, reprinted with kindpermission from Springer Science+Business Media, originallypublished by Springer and Journal of Engineering Mathemat-ics [18, fig. 1(f)] © Springer Science+Business Media B. V.2010.

2 Fluid mechanics of cilia and flagella

By considering the forces acting on an arbitrary volume offluid and applying the laws of conservation of momentumand mass, we may derive the Cauchy equations

ρ(∂u∂t

+ (u ⋅ ∇)u) = ∇ ⋅σ +F , (1a)

∂ρ

∂t+∇ ⋅ (ρu) = 0, (1b)

which govern the motion of the fluid. Here, F is the bodyforce acting on the fluid, such as gravity, ρ is the fluiddensity and u is the fluid velocity in a fixed frame of ref-erence. The stress tensor σ incorporates the forces actingover the surface of an arbitrary parcel of fluid, such aspressure and internal friction, and is dependent on thetype of fluid being modelled.

For Newtonian fluids, stress is proportional to strainrate, so that the fluid viscosity depends only on temper-ature, which is assumed to be constant throughout thisstudy. In such cases,

σ = −pI + µ2ε(u), (2)

T.D. Montenegro-Johnson et al.: Fluid mechanics of cilia and flagella in reproduction and development 3

for pressure, p and strain rate ε(u) = ∇u+(∇u)T . A non-dimensionalisation of the Navier-Stokes equations that re-sult from substitution of the stress (2) into the Cauchymomentum equation (1a) shows that the relative impor-tance of viscous forces to inertial forces is given by theReynolds number

Re = ρU2L2

µUL= inertial force

viscous force. (3)

Fluid pumping and locomotion by microscopic ciliaand flagella entails typical length-scales L of O(10−5 −10−4)m and velocities U of O(10−5 − 10−4)m ⋅ s−1, withtypical fluid densities around ρ = O(103)kg ⋅ m−3 andfluid viscosities of µ = O(10−3)Pa ⋅ s or greater. Thus, theReynolds number of the flows we will consider is no higherthan Re = 10−2 ≪ 1, which indicates that viscous forcesdominate over inertial forces. Thus, for Newtonian fluids,an accurate representation of the fluid mechanics is givenby the Stokes flow equations,

µ∇2u −∇p +F = 0, ∇ ⋅u = 0. (4)

2.1 The fundamental singularity

Insight into biological flows generated by cilia and flag-ella may be gained by considering the flows that arise dueto concentrated driving forces. Consider an infinite fluidobeying the Stokes flow equations (4) that is driven by aconcentrated force per unit volume F = fδ(x−y), of mag-nitude and direction f , where δ is the Dirac delta distri-bution centered at y. The velocity solution correspondingto this fundamental singularity is given by

ui(x) =1

8πµ(δijr+ rirjr3

) fj(y) = Sij(x,y)fj(y), (5)

where ri = xi−yi for i = 1,2,3, r2 = r21+r2

2+r23 and Sij(x,y)

is known as the Stokeslet, or Oseen-Burgers tensor. TheEinstein summation convention applies throughout.

The anisotropy of the Stokeslet is an important factoraffecting the fluid mechanics of cilia and flagella. As illus-trated in fig. 3, the velocity of the flow due to a singularforce, at a distance r1 from the force, is twice as large atpoints in line with the force than at those points perpen-dicular to the force at the same distance. As a first approx-imation, the action of a slender cylinder moving through afluid may be represented by a line distribution of singulardriving forces, as shown in fig. 3. Since the Stokes flowequations (4) are linear, the corresponding velocity fieldis given by a sum of Stokeslet solutions. Thus, the dragon a slender body moving tangentially through the flowis approximately half that on an equivalent body movingnormally. This “two to one ratio”, first described in theresistive force theory of Gray and Hancock [19] is the ba-sis for flagella and ciliary propulsive dynamics. However,drag anisotropy is not itself essential for very low Reynoldsnumber propulsion if the filaments are extensible, as re-cently shown by Pak & Lauga [20].

f1

u1(r1,0,0) = f14πµr1

u1(0, r2, r3) = f18πµr1

r1

r1

U

Force ≈ F/2

U

Force = F

Fig. 3. The flow generated by a singular force, and a demon-stration of the 2:1 drag anisotropy that enables flagellar andciliary propulsion, redrawn from Blake and Sleigh [21].

Swimmers in Stokes flow move in such a way that nonet forces [1] or torques [22] act upon them, and flowsdriven by cilia protruding from epithelia can be modelledwith image systems, as discussed in sect. 2.2. In both cases,the far-fields of the flow are given by higher-order singular-ities. By taking derivatives of the Stokeslet, it is possibleto derive the flow fields generated by higher-order singu-larities, such as point stresses and point torques. These areable to provide valuable insight into the far-field behaviourof the fluid surrounding swimming cells, and into the hy-drodynamic effects arising from the inclusion of no-slipboundaries in the flow [23,24]. Fig. 4 shows the schematicrepresentation of some of these singularities.

With each increase in the order of singularity, the de-cay of the fluid velocity in the far-field is increased byO(1/r), so that Stokeslets decay withO(1/r), stokes dipolesdecay with O(1/r2) and stokes quadrupoles with O(1/r3).

An additional important singularity, familiar as thesource dipole of potential flow theory, is given by,

Dij(x,y) = −1

4π(δijr3

− 3rirj

r5) . (6)

This expression for the i-component of a velocity field dueto a source dipole oriented in the j-direction, togetherwith zero pressure field, is also a solution of the Stokesflow equations, and can be combined with the Stokeslet toformulate solutions for translating spheres, ellipsoids andslender rods. The source dipole can alternatively be iden-tified as the Laplacian of the Stokeslet—in other words, aparticular form of the stokes quadrupole.

Singularity models capture many of the essential fea-tures of cilia and flagella driven flows. Fig. 5 shows exper-imental data from recent studies by Drescher et al. [25,


Stokeslet F

Symmetricstokes dipole stresslet

Non-symmetricstokes dipole

+=

stresslet rotlet

stokesquadrupole

+=

stressletdipole

rotletscancel

Fig. 4. Singularities of Stokes flow, with forces represented byarrow vectors. Displaced forces represent higher order singu-larities.

26]. Sperm, bacteria and individual algae are too small forgravitational sedimentation to have a dominant effect onthe flow field; the zero total force condition therefore en-tails that the far-field is given by a stresslet (fig. 5a). Simu-lation modelling (fig. 6) predicts that the flow field arounda sperm is approximated well by a stokes quadrupole,given by drag components at the front and rear of the celland a propulsive component in the middle; sufficiently farfrom the cell the dominant singularity will however stillbe the stresslet.

The flow field closer to the cell is more complex; forbiflagellate algae the time averaged flow field due to twopropulsive flagella and the cell body can be representedby a three Stokeslet model [25] (fig. 5b). Larger swimmerssuch as Volvox Carteri colonies are subject to a signif-icant gravitational force, evident as a Stokeslet far-field(fig. 5c); the near-field is given by a source dipole andstresslet combination (fig. 5d).

2.2 Wall effects

The presence of a rigid boundary near a Stokeslet signif-icantly alters the resultant flow field. This may be un-derstood through the use of singularity diagrams. To en-force the zero velocity condition on the plane boundary,that may for example represent the nodal floor, an im-age system of singularities is placed the other side of theboundary [23]. The explicit form for the velocity field aris-ing from a point force located at height h above a planeboundary represented by image systems is,

Bij(x,y) =1

8πµ[(δij

r+ rirjr3

) − (δijR

+ RiRjR3

)

+2h∆jk∂

∂RkhRiR3

− (δi3R

+ RiR3

R3)] . (7)

(a) (b)

(c) (d)

Fig. 5. (a) Experimentally observed stresslet behaviour inthe far-field of a swimming bacterium [26] (reprinted withpermission from Drescher et al., Proc. Natl. Acad. Sci. USA108:10940–10945, 2011.) (b) Experimentally observed flowfield around a swimming biflagellate Chlamydomonas Rein-hardtii, showing that the data are fitted accurately by a three-Stokeslet model [25] (c,d) Experimentally observed flow fieldsaround a larger swimmer, the Volvox Carteri algal colony[25]. The larger swimmer is subject to a non-negligible grav-itational force, evident as a far-field Stokeslet. (d) Showsthe near-field when the Stokeslet field is subtracted, ob-served to resemble a source dipole and stresslet [25] (b,c,d© 2010 APS, reprinted with permission from Drescher etal., Phys. Rev. Lett., 105:168101, 2010, online abstract athttp://prl.aps.org/abstract/PRL/v105/i16/e168101).

stokes dipole stokes dipole stokes quadrupole

drag propulsion drag

dragdragdrag

propulsion

+ =

Fig. 6. An approximate singularity representation of the flowfield surrounding a human sperm, redrawn from Smith andBlake [27]. The quadrupole representation was suggested aftercalculation of the force distribution in the tail with slenderbody theory.

The tensor ∆jk takes value +1 for j = k = 1,2, value −1for j = k = 3, and zero if j ≠ k. The image location is givenby R1 = r1, R2 = r2 and R3 = −h.

The image systems for the two cases correspondingto a force orientated parallel and perpendicular to theboundary are shown in fig. 7. The far-field arising from aStokeslet near a plane boundary is of higher order than thecorresponding Stokeslet in an infinite fluid, being O(1/r2)in the parallel case and O(1/r3) for the perpendicularcase. The boundary therefore has the effect of shieldingthe fluid in the far-field from the effect of the Stokeslet.


It is this shielding effect, illustrated in fig. 8, that allowsnodal cilia to generate directional fluid flow by tilting theiraxis of rotation in the posterior direction in mice [28]; wewill investigate the resultant fluid mechanics in more de-tail in sect. 3.

−h

h

x1

x2

x3

images: Stokeslet stokes sourcedipole dipole

−F1 2hF1 −4µh2F1

Stokeslet: F1

Far-field: stresslet, O(1/r2)(a)

−h

h

x1

x2

x3

images: Stokeslet stokes sourcedipole dipole

−F3 −2hF3 4µh2F3

Stokeslet: F3

Far-field: stokes quadrupole + source dipole, O(1/r3)(b)

Fig. 7. A diagram illustrating the image systems for aStokeslet orientated parallel (a) and perpendicular (b) to a no-slip plane boundary. Note that the wall induces a more rapiddecay in the far-field, which is of O(1/r2) in the parallel caseand O(1/r3) in the perpendicular case. Reprinted with kindpermission from Springer Science+Business Media, originallypublished by Springer and Journal of Engineering Mathemat-ics [18, fig. 4(a,b)] © Springer Science+Business Media B. V.2010.

For further detailed review of low Reynolds numberbiofluiddynamics, see Dillon et al. [29], Lauga and Powers[30] and Gaffney et al. [31]. We now consider two prob-

Far–field: stresslet, O(1/r2)

Near–field: O(1/r)

Inner–field:O(log r)

(a) (b)

Fig. 8. The wall effect : the zones of influence of a whirlingnodal cilium protruding from a cell surface during (a) the ef-fective stroke and (b) the recovery stroke. The far-field strengthdecays more rapidly than the inner- and near-fields due tothe wall influence; moreover, during the effective stroke, thecilium is further from the cell surface than during the re-covery stroke, resulting in the near-field having greater ex-tent, and propelling more fluid. This effective-recovery strokeasymmetry results in net fluid propulsion in the directionof the effective stroke. Reprinted with kind permission fromSpringer Science+Business Media, originally published bySpringer and Journal of Engineering Mathematics [18, fig.5(a,b)] © Springer Science+Business Media B. V. 2010.

lems which can be analysed using singularity approaches.For the first, Newtonian symmetry-breaking flow in thenode, we apply slender body theory and the method ofregularised Stokeslets to model the cilia and the node ge-ometry respectively, discussed in sect. 3. For the second,swimming in non-Newtonian fluids, we use a recently de-veloped hybrid of singularity methods with the finite ele-ment method [13], which we have dubbed the ‘method offemlets’, discussed in sect. 4.

3 Nodal cilia

Embryonic nodal cilia were first discovered in the 1990s,and a series of experimental studies confirmed that ciliamotility produced a fluid flow essential for the breaking ofleft-right symmetry, resulting in the normal asymmetricplacement of the internal organs in vertebrates. One of themost remarkable aspects of this work was the resolution ofthe long standing clinical question of how cilia dysfunctionand situs inversus, reversal of the internal body plan, oftenappear together [32,33]. In this section we review recentfluid mechanical models of this process, before present-ing new results inspired by recent biological observationson the developmental progression of cilia placement andconfiguration.

3.1 Geometry of the embryonic mouse node

Vertebrate development requires the establishment of threebody axes. In order of appearance they are: dorsal-ventral,anterior-posterior, and left-right. Following a series of ex-perimental studies, it is now known that cilia motion con-


verts the already-established anterior-posterior axis infor-mation with an intrinsic chirality [34], i.e. rotational di-rection, into left-right asymmetric flow. Cilia perform a‘whirling’ clockwise motion, viewed tip-to-base, with axisof rotation tilted towards the posterior. This model wasfirst advanced by Cartwright et al. [35], with the fluidmechanics of cell surface interaction being discussed byBrokaw [36] and analysed in more detail by Smith et al.[18,28,37]. Hashimoto et al. [38] established that the pos-terior tilt is generated because the basal body of the cil-ium migrates towards the posterior side of the convex cellsurface.

This process takes place in the ventral node, shown infig. 2. In the most extensively-studied species, the mouse,the node is an approximately triangular depression mea-suring 50–100µm in width and 10–20µm in depth [32],forming on the ventral side of the embryo at 7–9 days post-fertilisation (dpf). The node is covered with a membraneand filled with extraembryonic fluid, which we model asNewtonian, motivating an approach based on the Stokesflow theory described in sect. 2. The dorsal surface of thenode is covered with a few hundred nodal pit cells thattypically have one or two cilia protruding from them intothe node. Each cilium is approximately 3–5µm in lengthand 0.3µm in diameter.

Hashimoto et al. [38] characterised morphologicalchanges during the developmental process that primarilyaffect cilia numbers and positions, which we summarisebriefly: at the late bud stage, occurring at approximately7.5–8 dpf (for details of developmental stages, see Downs& Davies [39]), cilia exhibit a distribution of tilt angle, butno overall bias. At a slightly later stage of development,early headfold, also occurring in the period approximately7.5–8 dpf and subsequently late headfold, at approximately8 dpf, cells exhibit a significant posterior bias, and a sig-nificantly greater posterior bias in the peripheral regionsrelative to the centre. In the later developmental stagesknown as 1 somite and 3 somite, occurring at approxi-mately 8 dpf, posterior tilted cilia are extremely domi-nant.

During development, cilia migrate towards the poste-rior side of the nodal pit cell, which results in an increase oftilt angle in the posterior direction. These features will bemodelled using different sets of cilia positions and parame-ters. As observed by Hashimoto et al. [38] certain mutantembryos with Dvl1, Dvl2, Dvl3 genes ‘knocked out’ donot exhibit cilia migration and consequently do not pro-duce a directional fluid flow. The experimental study ofHashimoto et al. provides information on instantaneousslices of the two-dimensional flow fields using microscopicparticle imaging velocimetry and how this is altered by ge-netic knockouts; a complementary technique to study theflow generation and the influence of cilia tilt is to formulatea mathematical model, allowing the prediction of featureswhich are not yet available experimentally, for examplethree-dimensional particle tracks. We will describe brieflya computation model of Stokes flow generated by tilted ro-tating cilia published previously [18] and then apply the

model to interpret the fluid mechanics of the changingdevelopmental stages described by Hashimoto et al.

3.2 Modelling the stages of development in theembryonic mouse node

We will represent the ciliated surface, or nodal floor, by theplane x3 = 0 with cilia protruding into the region x3 > 0.The x1-axis will correspond to the left-right axis, withpositive x1 being towards the ‘left’, and the x2-axis willbe the anterior-posterior axis, with negative x2 being to-wards the posterior. In this configuration, posterior tiltcorresponds to the negative x2-direction.

Each cilium is tilted towards the already-establishedposterior direction by an angle θ and performs a conicalrotation with semi-cone angle ψ and angular frequency ω.The centreline at arclength s and time t is therefore givenby

ξ1(s, t) =s sinψ cos(ωt), (8a)

ξ2(s, t) =s(− sinψ sin(ωt) cos θ − cosψ sin θ), (8b)

ξ3(s, t) =s(− sinψ sin(ωt) sin θ + cosψ cos θ), (8c)

with the restriction θ + ψ < 90 so that each cilium doesnot come into contact with the ciliated surface, see fig. 9.The slenderness ratio is defined as η = a/L, where a isthe cilium radius and L is the cilium length. For nodalcilia η is approximately 0.1. The relative slenderness ofthe cilium indicates that slender body theory can be usedto represent both the near- and far-field flows accurately,however it is also necessary to take into account both theno-slip condition on the plane boundary representing theciliated surface, and the membrane that encloses the node,known as Reichert’s membrane.

V

A

P

L

R

θ

ψ

Fig. 9. The configuration of a tilted straight rod by an angle ofθ, where ψ is the semi-cone angle. Axis notation, V, ventral; A,anterior; P, posterior; L, left and R, right. Reprinted with kindpermission from Springer Science+Business Media, originallypublished by Springer and Journal of Engineering Mathematics[18, fig. 2(b)]© Springer Science+Business Media B. V. 2010.

The no-slip condition on the plane boundary is sat-isfied through the Stokeslet and image system given ineq. (7). The no-slip condition on the cilium is satisfied by


a Stokeslet and quadratically-weighted source dipole dis-tribution. In this formulation the surface of a cilium ismodelled as a slender prolate ellipsoid given by,

Xα(s, t) =ξ(s, t)

+ a¿ÁÁÀ1 − (s − 1/2)2

a2 + 1/4 (n(s, t) cosα + b(s, t) sinα). (9)

The vectors n(s, t) and b(s, t) are normal and binormalrespectively, the azimuthal angle α ranges from 0 to 2π,and ξ(s, t) is the straight centreline of the ellipsoid de-fined by eq. (8). To ensure that ui(Xα(s0, t)) ≈ ∂tξi(s0, t)uniformly, including towards the cilium ends and for all α,we combine a centreline distribution of Stokeslets with aquadratically-weighted distribution of source dipoles [40].

The flow due to a single cilium ui is then representedby the integral equation,

ui(x, t) =∫1

0[Bij(x;ξ(s, t))fj(s, t)

+Dij(x;ξ(s, t))gj(s, t)] ds. (10)

The function fj(s, t) is the force per unit length on a cil-ium and gj(s, t) is a source dipole distribution. An imagesystem is not required for the source dipole because theadditional terms needed to satisfy the no-slip conditiondecay rapidly; an expression for the source dipole imagesystem can also be derived [41]. The source dipole distri-bution, originally derived by Chwang & Wu [42] in thecontext of exact solutions to Stokes flow and later foundasymptotically by Johnson [40], takes the form

gj(s, t) = −a2s(1 − s)

µfj(s, t). (11)

Due to the linearity of Stokes’ equations the contribu-tion to the velocity field from each cilium is given by thesum of slender body integrals,

uciliai (x, t) =

M

∑m=1∫

L

0Gij(x,ξ(m)(s, t))f (m)j (s, t)ds

+O(η2). (12)

The total number of cilia is denoted by M , the parameterL is the length of each cilium, and the Green’s functionGij = Bij − a2s(1 − s)Dij/µ is a combination of Stokesletand plane boundary image system with the quadratically-weighted source dipole. The no-slip condition on the sur-face of each cilium is preserved provided that the cilia donot approach each other too closely, a satisfactory approx-imation for nodal cilia [28]. The centreline of the mth cil-

ium is denoted ξ(m)(s, t), defined by eq. (8) with a rangeof base positions. The a priori unknown force per unit

length on the mth cilium is denoted f(m)j (s, t).

To incorporate the covering membrane of the mousenode a surface mesh of a cube is rearranged into a smooth,approximately triangular shape S, as described in Smithet al. [18] and shown in fig. 10. The contribution to the

x1 x2

x3S

Fig. 10. A view of the mesh used to enclose the node denotedby S. The x1 axis represents the left-right axis with positivex1 being towards the left of the embryo. The x2 axis representsthe anterior-posterior axis with negative x2 being towards theposterior. The x3 axis represents the dorsal-ventral axis withpositive x3 being the ventral direction.

velocity field from the membrane can be approximated bya single-layer boundary integral of regularised Stokesletsand plane boundary images over the membrane surfaceS. To obtain a regular flow field throughout domains con-taining singularity distributions, Cortez [43] developed themethod of regularised Stokeslets. A regularised Stokesletis defined as the exact solution to eq. (4) where F is givenas a smoothed point-force, F = fψε(x − y). The symbolψε(x − y) denotes a cut-off function with regularisationparameter ε. For the choice ψε(x − y) = 15ε4/(8πµr7

ε ),Cortez et al. [44] showed that the regularised Stokeslethas the form,

Sεij(x,y) =1

8πµ(δij(r

2 + 2ε2) + rirjr3ε

) , (13)

where r2ε = r2 + ε2 and the velocity due to a regularised

Stokeslet in an infinite domain is then ui = Sεijfj . Ain-ley et al. [45] then derived the regularised image systemStokeslet that satisfies the no-slip boundary condition whichcan be written in index notation for a plane boundary atx3 = 0 as

Bεij(x,y) =1

8πµ(δij(r

2 + 2ε2) + rirjr3ε

− δij(R2 + 2ε2) +RiRj

R3ε

+ 2h∆jk [∂

∂RkhRiR3ε

− δi3(R2 + 2ε2) +RiR3

R3ε

−4πhδikφε(R)] − 6hε2

R5ε

(δi3Rj − δijR3)) (14)

where R2ε = R2 + ε2, φε(R) = 3ε2/(4πR5

ε) and ∆jk has thesame definition as in eq. (7). The velocity contributionfrom the covering membrane is then,

umemi (x, t) =∬

SBεij(x,y)φj(y, t)dSy +O(ε2), (15)

with φj(y, t) the unknown jth component of the stress onthe membrane at y and ε a regularisation parameter, in


this case chosen to be 0.06L (for further details, see refer-ences [18,43–46]). An advantage of a regularised Stokesletformulation in this context is that the velocity field nearto, and on, the surface S is regular, and can be calculatedusing Gauss-Legendre quadrature.

The velocity in the domain can then be written as thesum,

ui(x, t) =∬SBεij(x,y)φj(y, t)dSy +O(ε2)

+M

∑m=1∫

L

0Gij(x,ξ(m)(s, t))f (m)j (s, t)ds +O(η2). (16)

It remains to approximate the unknown stress distribution

φj(y, t) and force per unit length distribution f(m)j (s, t)

for each timestep comprising a beat cycle. This is achievedby imposing the no-slip condition at discrete points oneach cilium and the surface mesh, and using a bound-ary element constant force discretisation; for full details ofthe numerical implementation, see refs. [18,46]. Once thestress and force per unit length distributions are known, itis then possible to calculate the fluid velocity from eq. (16);knowledge of the velocity field then enables simulation ofparticle transport.

3.3 Configurations of cilia during development

Since computational expense grows approximately withthe cube of the number of cilia, we model the node witha smaller number of cilia than are observed experimen-tally. However, the underlying characteristics of the prob-lem such as tilt direction and the geometry of the problemdomain will be preserved.

Building on the work of Smith et al. [18], we will fo-cus on the stages of development where the fluid flow ap-pears to change from vortical to directional; the stagesconsidered are the late bud, early headfold, late headfold,1 somite and 3 somite. For the late bud stage 17 cilia weredistributed in the centre of the node with tilt angles inthe range −10 ≤ θ ≤ 10. A negative tilt angle denotesan anterior tilt and 7/17 cilia were tilted in the anteriordirection. The early headfold stage is modelled with 21cilia with tilt angles in the range −10 ≤ θ ≤ 15 with3/21 cilia tilted in the anterior direction. The late head-fold stage is modelled with 25 cilia with tilt angles in therange −5 ≤ θ ≤ 20 with only 1/25 cilia tilted in the an-terior direction. The 1 somite stage is modelled with 28cilia with tilt angles in the range 20 ≤ θ ≤ 35 and the3 somite stage is modelled with 28 cilia with tilt anglesin the range 35 ≤ θ ≤ 45. Thus we model the increasednumbers of cilia and increases in posterior tilt occurringwith each advance with developmental stage, consistentwith the experimental observations of approximately 150cilia at the late bud stage and approximately 380 cilia atthe 3 somite stage.

3.4 Particle transport

Results presented in this section will adopt the conventionof the ‘left’ of the node on the right of the figure, with thefollowing set of scalings: length is normalised with respectto cilium length, which is typically 3–5µm for the mouse.Time is scaled with respect to a beat period, 2π/ω, whereω is the angular frequency and velocities are then scaledaccording to these length and time scalings.

Figs. 11, 12 and 13 show the results of particle trackingsimulations. In these simulations, one beat cycle takes 60timesteps and the tracking simulations were run for 20,000timesteps or approximately 333 beat cycles, correspondingto 30 seconds of cilium rotation at 10 Hz. Initial particlepositions are shown with an arrow. Particles released inthe late bud and early headfold stages of development areswept around the node, in a clockwise vortex when viewedfrom the ventral side (figs. 11, 12). This is because eachcilium has a low tilt angle, thereby generating a local vor-tex; these local vortices combine to form a larger globalvortex.

Particle paths in the late headfold stage vary with po-sition in the node. A particle released in the right of thenode at initial height x3 = 1.1 is advected initially to theright and then to the left by neighbouring cilia (fig. 11).A particle with initial height x3 = 0.5 (fig. 12) is advectedin a vortical flow around the entire node.

All particles released in the 1 somite and 3 somitestages are advected leftward by a succession of cilia (figs. 11,12). Once the paths reach the edge of the cilia array atthe left of the node they return via a rightward path closeto the covering membrane. A leftward particle path is ob-served because all cilia are tilted towards the posterior atthe 1 somite stage of development. The behaviour of par-ticles released above the cilia tips in the left region of thenode is shown in fig. 13. At all stages of development theparticle path shows the expected return flow characteris-tics of each stage.

The change from a global vortex to a directional floweffectively breaks the symmetry of the left-right axis. Thisis because particles may be moved towards the left by asuccession of posteriorly tilted cilia as opposed to beingcarried in a global vortex around the node by untiltedcilia. The mechanism for how this flow is converted toasymmetric gene expression is still under active investiga-tion.

4 Swimming in non-Newtonian fluids

For a wide class of problems, for instance internal fertili-sation in mammals, the Stokes flow equations do not givean accurate representation of the fluid environment. Insuch cases, complex fluid rheological properties can havea significant impact upon swimming speed [13], and theneed for detailed study of non-Newtonian swimming haslong been recognised [47,48]. Whilst much insight has beengained into the effects of viscoelastic rheology [20,49,50],relatively less study has been given to understanding theimpact of shear-dependent viscosity on viscous swimming.


x1−6 −3 0 3 6

x2

−6

−3

0

3

6

x1−6 0 3 6

x3

0

2

4

Fig. 11. Particle paths for the late bud (—), early headfold(—), late headfold (—), 1 somite (—) and 3 somite (—) stagesof development. Note that the right hand side of the figure(positive x1) corresponds to the eventual ‘left’ axis of the em-bryo. Cilia positions are denoted by , ◁, ▷, , , , ◻, ∗,where are present at all stages, ◁, late bud only, ▷, latebud, early headfold, late headfold only, , late bud, 1 somiteonly, , early headfold, late headfold, 1 somite, 3 somite only,, late headfold, 1 somite, 3 somite only, ◻, 1 somite, 3 somiteonly, ∗, 3 somite only. The initial position for each trajectoryis marked with an arrow at x1 = −3.00, x2 = −3.75, x3 = 1.10,showing the x1x2 projection in the upper panel and the x1x3projection in the lower panel.

We will model swimming in generalised Newtonian flu-ids for which the effective fluid viscosity, µeff , is a func-

tion of γ = ( 12εij(u)εij(u))

1/2, the second invariant of the

strain rate tensor εij = (∂jui + ∂iuj). The governing equa-tions for such fluids are given by

∇ ⋅ (µeff(γ)ε(u)) −∇p +F = 0, (17a)

∇ ⋅u = 0, (17b)

which are typically nonlinear, and thus established tech-niques involving the superposition of fundamental flow so-lutions are not appropriate.

4.1 Swimming in shear-thinning rheology

Long polymer chains in suspension tend to untangle andalign with the flow. Such suspensions are said to be ‘shear-

x1−6 −3 0 3 6

x2

−6

−3

0

3

6

x1−6 0 3 6

x3

0

2

4

Fig. 12. Particle paths for the late bud, early headfold, lateheadfold, 1 somite and 3 somite stages of development. Ciliapositions are denoted as in fig. 11. The initial position for eachtrajectory is marked with an arrow at x1 = −3.00, x2 = −3.75,x3 = 0.50, showing the x1x2 projection in the upper panel andthe x1x3 projection in the lower panel.

thinning’, since their effective viscosity decreases with fluidshear. A swimmer in shear-thinning fluid creates an enve-lope of thinned fluid around itself, which has a non-trivialeffect on its locomotion [13].

The dynamics of polymer suspensions, such as cervicalmucus, may be modelled by eq. (17) with the Carreauconstitutive law [51], for which

µeff(γ) = µ∞ + (µ0 − µ∞)(1 + (λγ)2)(n−1)/2. (18)

For Carreau fluids, the effective viscosity decreases mono-tonically between a zero strain rate viscosity, µ0, and aninfinite strain rate viscosity µ∞. The material constant λis a measure of the polymer chain relaxation time.

Using the scalings u = ωLu,x = Lx, t = 2πt/ω,f =µ∞ωLf and p = µ∞ωp, where ω is the angular beat fre-quency and L is a characteristic length, for instance thelength of the flagellum, the dimensionless momentum equa-tion governing the Carreau fluid is

∇ ⋅[(1 + [ µ0

µ∞− 1] [1 + (λωˆγ)2]

(n−1)/2) ε(u)]−∇p+F = 0.

(19)


x1−6 −3 0 3 6

x2

−6

−3

0

3

6

x1−6 −3 0 6

x3

0

2

4

Fig. 13. As fig. 12, with initial particle position at x1 = 3.00,x2 = 0.00, x3 = 1.10.

For fixed beat kinematics, the trajectory is thus dependentupon three dimensionless quantities: the ratio µ0/µ∞ ofthe zero to infinite shear rate viscosities, the product λωof the characteristic relaxation time of the fluid with theangular beat frequency, known as the Deborah numberDe, and the power-law index n.

4.2 Modelling non-Newtonian swimming with femlets

We use the method of femlets, described in Montenegro-Johnson et al. [13], to model viscous swimming in a gener-alised Newtonian fluid. The method of regularised Stokesletsand the method of femlets represent the interaction ofthe swimmer with the fluid through a set of concentrated‘blob’ forces of unknown strength and direction. While themethod of regularised Stokeslets reduces the problem tofinding the coefficients in a linear superposition of velocitysolutions of known form, the method of femlets proceedsby applying the finite element method to solve simultane-ously the fluid velocity field and the strength and directionof the forces. The use of the finite element method removesthe need for the governing equations to be linear. Hence-forth, we will continue to use dimensionless variables, butdoff hats for conciseness.

Let D be a bounded domain in Rd, where in our cased = 2. We partition the domain boundary ∂D = ∂Ddir ∪

∂Dneu into those portions on which Dirichlet and Neu-mann type boundary conditions are applied respectively.The surface of the swimmer, ∂Dswim ⊂ ∂Ddir forms a partof the Dirichlet boundary. However, we will model the in-teraction of ∂Dswim with the fluid by an immersed bodyforce distribution. Thus for our case ∂Dswim is not a do-main boundary, but rather a manifold containing pointswithin the domain.

Let H1(D) be the standard Sobolev space of weaklydifferentiable functions [52] defined on D, and

VE =w ∈ (H1(D))d ∶w∣∂Ddir= udir , (20a)

V0 =w ∈ (H1(D))d ∶w∣∂Ddir= 0 , (20b)

where udir are the Dirichlet conditions imposed on ∂Ddir.Let also Q = L2(D). Multiplying (17a), (17b) by arbitrary‘test’ functions v ∈ V0, q ∈ Q, respectively, yields the fol-lowing integral form of problem (17):

∫D

∇ ⋅ [µeff(γ)ε(u)] −∇p +F ⋅ v dx = 0, (21a)

∫Dq∇ ⋅udx = 0. (21b)

Integration by parts yields an equivalent integral formula-tion with reduced differentiability requirements for u andp. This is known as the weak (or variational) formulationof the generalised Stokes flow problem (17) and reads:

Find (u, p) ∈ VE ×Q such that ∀(v, q) ∈ V0 ×Q,

∫Dµeff(γ)ε(u) ∶ ε(v)dx − ∫

Dp∇ ⋅ v dx

+∫DF ⋅ v dx = 0, (22a)

∫Dq∇ ⋅udx = 0, (22b)

so that VE , V0 are the velocity solution and test functionspaces respectively. Applied on ∂Dneu is the open bound-ary condition first proposed by Papanastasiou et al. [53],given in our case by

∫∂Dneu

v ⋅σ ⋅ndx = 0. (23)

Existence and uniqueness for problem (22) was shown byBaranger et al. [54] for both the power law and Carreaumodels.

We consider swimmers in truncated channels of theform shown in fig. 14. The no-slip condition u = 0 is ap-plied on the the channel walls ∂Ddir, and on the bound-aries where the domain has been truncated ∂Dneu we ap-ply the open boundary condition (23). The results we willpresent were obtained in a channel of height 5 and length11, with lengths normalised to a characteristic length forthe swimmer.

As the swimmer moves through the fluid, the mov-ing boundary exerts a force distribution on the fluid thatdrives the flow. We incorporate this interaction through


D

∂Dneu

∂Ddir

∂Dswim

Fig. 14. An example domain D containing a model humansperm ∂Dswim (red), showing no-slip channel walls ∂Ddir (solid,black) and open boundaries ∂Dneu (dashed, black).

the unknown body force F , which is governed by the mo-tion of the swimmer. We approximate F by a finite num-ber of smooth immersed forces of unknown strength anddirection (femlets)

F =Nf

∑k=1

gε(x −xk)fk, (24)

for Nf femlets of strength fk, located at xk. The cut-offfunction gε(x −xk) is a regularisation function similar tothat used in the method of regularised Stokeslets [43]

Associated with each femlet are K degrees of freedom,where K is the dimensionality of the problem domain. Forexample, a swimmer in two dimensions would have thelab frame force of each femlet in the x and y directions,(f1, f2), as unknowns, resulting in 2×Nf additional scalarvariables.

To calculate the 2 × Nf force unknowns, we enforce2 × Nf constraints in the form of Dirichlet velocity con-ditions us. These are given by the swimmer’s velocity inthe body frame, in which the swimmer neither rotates nortranslates, and applied at the location of each femlet. Therelationship between the body frame and the lab frame isshown in fig. 15. The body frame velocity ub is related tothe lab frame velocity u solved for in problem (22) by

ub = u −U −Ω × (x −x0), (25)

where U ,Ω are the translational and angular velocities ofthe swimmer respectively and x0 is a fixed point on theswimmer.

The translational and angular velocities U ,Ω provideadditional unknowns which are closed by the conditionsthat zero net force and torque act on the swimmer,

Nf

∑k=1fk ∫

Dgε(x −xk)dx = 0, (26a)

Nf

∑k=1fk ×xk ∫

Dgε(x −xk)dx = 0. (26b)

All femlets are given the same cut-off function gε, and theswimming velocity conditions are applied at the function’scentroid, xk. Thus, the force and torque conditions on the

x0

x

x′

y′

x

y

Fig. 15. Schematic of the relationship between the lab frame(x, y) and the body frame (x′, y′), where x is a general pointon the swimmer, given in the lab frame and x0 is a referencepoint on the swimmer. The transformation from body framevelocity to lab frame velocity is given by eq. (25).

swimmer may be written

Nf

∑k=1fk = 0,

Nf

∑k=1fk ×xk = 0, (27)

respectively. Under these conditions, problem (22) becomes

Find (u, p) ∈ VE ×Q such that ∀(v, q) ∈ V0 ×Q,

∫Dµeff(γ)ε(u) ∶ ε(v)dx − ∫

Dp∇ ⋅ v dx

+∫D

⎡⎢⎢⎢⎢⎣

Nf

∑k=1

gε(x −xk)fk⎤⎥⎥⎥⎥⎦⋅ v dx = 0, (28a)

∫Dq∇ ⋅udx = 0, (28b)

subject to,

u(xk, t) = us(xk, t) +U(t) −Ω(t) × (xk −x0), (28c)

Nf

∑k=1fk = 0,

Nf

∑k=1fk ×xk = 0. (28d)

Note that problem (28) is nonlinear, due to the depen-dence of µeff on u. We solve this nonlinear system with the


following Picard iteration: given an initial guess (u0, p0),

For m = 0,1, . . . solve until convergence:

Find (um+1, pm+1) ∈ VE ×Q such that

∀(v, q) ∈ V0 ×Q,

∫Dµeff(γm)ε(um+1) ∶ ε(v)dx − ∫

Dpm+1∇ ⋅ v dx

+∫D

⎡⎢⎢⎢⎢⎣

Nf

∑k=1

gε(x −xk)fm+1k

⎤⎥⎥⎥⎥⎦⋅ v dx = 0 (29a)

∫Dq∇ ⋅um+1 dx = 0, (29b)

subject to

u(xk, t)m+1 = us(xk, t) +U(t)m+1

−Ω(t)m+1 × (xk −x0), (29c)

Nf

∑k=1fm+1k = 0,

Nf

∑k=1fm+1k ×xk = 0, (29d)

End.

At each iteration the system (29) is discretised byTaylor-Hood P2-P1 triangular finite elements [55] over thedomain D, and the resultant linear system M( ˙γm)zm+1 =r is solved. The iteration continues until ∣∣M(γm+1)zm+1−M(γm)zm+1∣∣ < tol, a small tolerance here set to tol =10−9, returning a solution vector z of the nonlinear swim-ming problem. The solution, z, comprises the lab framevelocity of the fluid u, the fluid pressure p, the force dis-tribution along the swimmer fk and the swimming trans-lational U and rotational Ω velocties.

4.3 A Najafi-Golestanian swimmer in a shear-thinningfluid

Perhaps the simplest conceptual model of a viscous swim-mer was proposed by Najafi and Golestanian [14]; it ishighly instructive to compare the physics of this swim-mer to more detailed models of cells equipped with ciliaand flagella. The Najafi-Golestanian swimmer comprisestwo outer spheres which move relative to a central spherewith a non-reciprocal motion.

The mechanism underlying the original Najafi-Golestanian swimmer is as follows: one of the outer sphereswill move at any given time. By force balance, leftward rel-ative motion of an outer sphere results in rightward motionof the remaining spheres through the fluid, and vice versa.The distance that the remaining spheres move in the fluiddepends on the drag of the remaining two spheres. Rela-tive leftward motion of an outer sphere occurs while theother spheres are far apart; relative rightward motion ofan outer sphere occurs while the other spheres are closetogether. Hydrodynamic interaction results in the drag ofthe other spheres being reduced when they are close to-gether. Therefore the drag of the remaining spheres is lessduring relative rightward motion of the active sphere, andso the beat cycle is slightly more effective in moving theswimmer to the left than the right. Our variant of this

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

relative position

tim

e

Fig. 16. The position of the two outer swimming spheres,modelled by femlets moving relative to the central sphere forχ = 1/4 over the course of a beat cycle.

swimmer follows the suggestion in the original paper ofNajafi & Golestanian [14] of making the sphere motionsmooth in time.

We model this swimmer by three collinear femlets. Theouter femlets move as harmonic oscillators relative to thecentral femlet, and symmetry is broken by enforcing aphase difference, χ, between them

x1 = −d + sin(t), x2 = 0, x3 = d + sin(t − χ), (30)

where d is a constant displacement. The locus of this mo-tion is shown over a beat-cycle in fig. 16. Due to the sym-metry of the problem domain and beat pattern in theline y = 0, the swimmer will move in the x-direction. Fig.17 shows the progress of this swimmer in Stokes flow forχ = 1/4, so that our example swims in the negative x-direction. Here we normalise length scales such that 2d = 1.We will now examine the effects of beat phase, as well asthe Carreau rheological parameters, on swimming speed.

For the set of rheological parameters: viscosity ratioµ0/µ∞ = 2, Deborah number λω = 1, and power-law indexn = 1/2, we first examine the effect of changing the phasedifference χ. Fig. 18 shows the progress of the swimmeras a function of χ, revealing a maximum when χ = 1/4for both Carreau and Stokes flow. This phase differencecorresponds to a continuous analogue of the beat patternproposed in ref. [14]. It should be noted that for fluidswith viscoelastic properties, this may not be the optimumphase difference, due to the dependence of the fluid stresson its deformation history. For this set of fluid parameters,the mean benefit to progress is 3%, with a standard de-viation of 0.5%. The zero progress solutions at χ = 0,1/2correspond to the cases where the swimming spheres arein phase and antiphase respectively. Since the maximumprogress is achieved for χ = 1/4, we fix this variable forthe remainder of the study.

We now wish to examine the effects of decreasing thepower-law index, n. For n = 1, the Carreau equations re-duce to the Stokes flow equations. Decreasing n leads to a


0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

time

glo

bal

posi

tion

Fig. 17. The position of the central sphere of the Najafi-Golestanian swimmer in a channel of Stokes fluid over a beatcycle for χ = 1/4.

sharper decrease in the effective viscosity for lower strainrates. For λω = 1 and µ0/µ∞ = 2, progress of the swimmerover a beat-cycle as a function of n is displayed in fig. 19,showing that as n decreases, magnitude of progress perbeat increases.

By fixing λω = 1 and n = 1/2, we examine the effectsof varying the viscosity ratio µ0/µ∞. Fig. 20 displays theprogress of the swimmer over a beat-cycle as a functionof this ratio, which shows that as the infinite shear rateviscosity decreases, magnitude of progress increases.

Finally, we consider the effect of altering the Debo-rah number De = λω. This is especially important, sincefor a given fluid, the power-law index and viscosity ratioare fixed physical parameters. However, De is a functionof beat frequency, which for artificial swimmers may becontrolled and therefore optimised, much like the phasedifference χ. For µ0/µ∞ = 2, and n = 1, progress overa beat-cycle is displayed as a function of De in fig. 21,which shows that for this particular model swimmer, op-timum progress is achieved for λω ≈ 0.8, so that the angu-lar frequency of the swimmer is approximately 4/5 of thecharacteristic relaxation time of the fluid.

4.4 A two-dimensional sperm in shear-thinning andthickening fluid

For sperm in high-viscosity fluids, such as human mucus,swimming is typified by planar flagella beating that growsin peak curvature towards the distal portion of the tail[56]. To model this waveform, we prescribe a body-frametangent angle of the form

ψ(s, t) = Cs cos(ks − t), (31)

for s the arclength along the flagellum and t time. Such aparameterisation makes sense in the context of consider-

0 5 ⋅ 10−2 0.1 0.15 0.2 0.25−4

−3

−2

−1

0

⋅10−2

phase χ

pro

gre

ssFig. 18. The progress of the Najafi-Golestanian swimmer overa single beat cycle as a function of the phase difference, χ, forNewtonian (blue,+) and Carreau fluid (orange,) with µ0/µ∞ =2, λω = 1 and n = 1/2. Negative progress denotes swimming inthe negative x-direction.

0.2 0.4 0.6 0.8 1−3.64

−3.62

−3.6

−3.58

−3.56

−3.54

−3.52⋅10−2

power-law index n

pro

gre

ss

Fig. 19. The progress of the Najafi-Golestanian swimmer overa single beat cycle in Carreau fluid as a function of the power-law index, n, for λω = 1 and µ0/µ∞ = 2. Newtonian flow corre-sponds to the value n = 1. Negative progress denotes swimmingin the negative x-direction.


1 2 3 4 5−3.75

−3.7

−3.65

−3.6

−3.55

−3.5⋅10−2

viscosity ratio µ0/µ∞

pro

gre

ss

Fig. 20. The progress of the Najafi-Golestanian swimmer overa single beat cycle in Carreau fluid as a function of the viscos-ity ratio, µ0/µ∞, for λω = 1 and n = 1/2, shown for equis-paced values of µ∞. Newtonian flow corresponds to the valueµ0/µ∞ = 1. Negative progress denotes swimming in the nega-tive x-direction.

0 0.5 1 1.5−3.62

−3.6

−3.58

−3.56

−3.54

−3.52

−3.5⋅10−2

Deborah number λω

pro

gre

ss

Fig. 21. The progress of the Najafi-Golestanian swimmer overa single beat cycle in Carreau fluid as a function of the Deb-orah number, λω, for µ0/µ∞ = 2 and n = 1/2. Newtonian flowcorresponds to the value λω = 0. Negative progress denotesswimming in the negative x-direction.

ing a bending wave propagating down the flagellum, steep-ening towards the distal end as the stiffness of the flag-ellum decreases. A representative waveform produced bythe shear angle parametrisation given by eq. (31) is shownin fig. 22.

Fig. 22. A model flagella waveform, generated by the shearangle parameterisation (31) with amplitude C = 0.9π/2 andwavenumber k = 2.5π, typical of what is observed experimen-tally in high viscosity media.

Integrating the tangent vector along the flagellum givesthe flagellar centreline

xc(s, t) =x0 + ∫s

0cos(ψ(s′, t))ds′, (32a)

yc(s, t) =y0 + ∫s

0sin(ψ(s′, t))ds′, (32b)

with corresponding centreline velocity x

xc(s, t) =∫s

0− sin(ψ)ψ ds′, (33a)

yc(s, t) =∫s

0cos(ψ)ψ ds′. (33b)

This parameterisation of the flagellum is given in thebody frame [57], in which the cell body neither rotates nortranslates. The translational and rotational velocities thatarise from this instantaneous configuration are then usedto update the position of the swimmer in the lab frame,so that the cell swims through the fluid domain.

The trajectory of the swimmer is calculated by in-tegrating the swimming velocity U ,Ω with the Adams-Bashforth multistep method [58,59], which interpolatesan nth order polynomial through the current and previousn−1 values of translational and angular velocity to give theposition. For a two-dimensional sperm-like swimmer in achannel of Newtonian fluid, with less than a 0.3% changein the position of the swimmer after a single beat cyclebetween 25 steps per beat with the second order schemeand 80 steps with the third order scheme. As such, we usethe second order scheme with 25 steps per beat. We nor-malise length scales to the flagellum length, so that onelength unit corresponds to 55µm, and time scales to theperiod of the beat. Thus, modelling a tail beating at 10 Hzwould mean that one time unit corresponds to 1/10 s.

Fig. 23 shows the sperm’s progress as a function ofDeborah number for µ0/µ∞ = 2 and n = 1/2. Note thequalitative similarity to fig. 21, with overall progressioninitially increasing with De to a maximum value aroundDe = 0.8, then decreasing at a slower rate. Since the Najafi-Golestanian swimmer comprises three forces in balance,its representation by a stokes quadrupole might shed in-sight into this link, motivated by the observations in fig.


6. These results also draw an interesting parallel with theanalysis of Teran et al. [50] who used the immersed bound-ary method to show that the progression of a waving fil-ament may be enhanced in a viscoelastic Oldroyd B fluidat Deborah numbers close to 1.

If the viscosity ratio, µ0/µ∞, is less than 1, then theeffective viscosity (18) of a Carreau fluid increases withshear rate. For such a model, the relaxation time λ nolonger has a physical interpretation in terms of polymerphysics, but the Carreau law may still be used as a regu-larisation of a shear-thickening power-law fluid. Examplesof shear-thickening fluids are custard and a mixture ofcornstarch with water known colloquially as oobleck.

Fig. 24 shows the progress of the sperm over a sin-gle beat-cycle as a function of the viscosity ratio µ0/µ∞.Stokes flow, corresponding to µ0/µ∞ = 1, is marked in or-ange, and the thickening and thinning regimes lie to its leftand right respectively. We see that whilst shear-thinningaids progress, shear-thickening inhibits it. Furthermore,fig. 24 shows an almost perfect linear relationship betweenprogress and the logarithm of µ0/µ∞.

For shear-thinning fluids, there is a gradient of thick tothin fluid from the proximal to distal portions of the flag-ellum. For shear-thickening fluids, the gradient runs fromthin to thick. Thus, the disadvantage to cell progress in ashear-thickening fluid is consistent with the hypothesis ofMontenegro-Johnson et al. [13] that differential viscositybetween the distal and proximal portions of the flagellumis responsible for changes in propulsive efficiency.

0 0.5 1 1.5 2

2.1

2.2

2.3

2.4

2.5⋅10−2

De

pro

gre

ss

Fig. 23. The unsigned total distance travelled by our two-dimensional sperm over a single beat cycle in Carreau fluidas a function of the Deborah number, λω, for µ0/µ∞ = 2 andn = 1/2. Newtonian flow corresponds to the value λω = 0.

2−1 20 211.6

1.8

2

2.2

2.4

⋅10−2

viscosity ratio µ0/µ∞

pro

gre

ssFig. 24. The progress of the two-dimensional sperm-like swim-mer over a single beat cycle in Carreau fluid as a func-tion of the viscosity ratio, µ0/µ∞, for λω = 1 and n = 1/2.Stokes flow, µ0/µ∞ = 1 is marked in orange, with the shear-thickening regime for µ0/µ∞ < 1 and the shear-thinning regimefor µ0/µ∞ > 1. The horizontal axis is displayed on a logarithmicscale µ0/µ∞.

5 Conclusions

5.1 Fundamental physics of cilia and flagella drivenflow

The linearity of the Stokes flow equations makes the methodof singular solutions possible. This approach provides sig-nificant insight into the basic nature of the physics ofpropulsion and swimming. Microscopic swimmers, for whichgravitational and inertial forces are negligible, are subjectto zero total force and torque, entailing that the far-fieldof a flagellated swimmer is stresslet in nature, the flowvelocity having inverse square decay. However, within adistance of a few cell radii or flagella lengths, the flowfield can be markedly different from a stresslet, beingcloser to a stokes quadrupole for sperm, and a stresslet-source dipole combination for single celled algae. Thesefindings are confirmed both experimentally and throughmore detailed computational models. The action of grav-ity on larger swimmers results in a more slowly-decayingO(1/r) Stokeslet flow.

The lack of explicit time dependence in the Stokes flowequations entails that time-irreversible motion is essen-tial for successful swimming and pumping. Cilia and flag-ella achieve this through a number of mechanisms. Theanisotropy of the Stokeslet results in the anisotropic draglaw for slender bodies, a property which underlies thepropulsive effect of travelling bending waves. Finally, wallinteraction, which can be understood by the method ofimages, is essential to the function of cilia. Image systemsconvert both forces and torques to higher order singu-


larities in the far-field, a mechanism which explains howrotating embryonic nodal cilia create a right-to-left flow.

5.2 Cilia in embryonic development

The hybrid computational technique combining slenderbody theory with the regularised Stokeslet boundary el-ement method allows prediction of both generic featuresof cilia-driven flow in the nodal cavity, as shown previ-ously [18]. Our simulations show the effect of develop-mental changes in cilia orientation and position, using ex-perimental data on cilia numbers and orientation [38]. Inearly stages of development (late bud, early headfold), theflow is essentially vortical, whereas in later stages whenthe majority of cilia are tilted posteriorly (late headfold,1 somite and 3 somite); an asymmetric directional flowis established that breaks the symmetry of the left-rightaxis.

Qualitative agreement with particle imaging velocime-try observations [38] suggest that this approach will beuseful for analysing this and related systems in more de-tail, providing further physical insight into the coupling ofnodal flow to subsequent morphogenesis [32,60].

5.3 Swimming in non-Newtonian fluids

The Najafi-Golestanian swimmer is able to progress be-cause of hydrodynamic interaction of spheres coupled withtime-irreversible motion, whereas sperm propel throughpropagating bending waves of a slender flagellum; how-ever the effect of shear-thinning rheology on both cells isqualitatively similar. Both models swim faster in a shear-thinning fluid, with an optimal value for Deborah numberbetween 0.5 and 1, however sperm enjoy a greater propul-sive advantage relative to the Newtonian state comparedwith the three-sphere swimmer, their increase being upto around 20% compared with around 3% in the region ofparameter space investigated. For sperm, shear-thickeningfluid rheology led to a decrease in cell progress. It wasfound that cell progress had an almost exact logarith-mic dependence on the viscosity ratio µ0/µ∞. It shouldbe noted, however, that the fluid strain-rate of a Carreaufluid is independent of its recent deformation history, andso to fully capture the behaviour of human mucus an ex-tended model is desirable.

5.4 Summary

The examples in this paper illustrate aspects of the broadrange of very low Reynolds number flow. The embryonicnodal flow is typically considered Newtonian, and entailsthe wall effect converting rotation to directional flow, whichin turn converts chiral to lateral information. The swim-ming cell problems illustrate the importance of more com-plex non-Newtonian effects typical of biological fluids suchas blood and mucus, and the qualitative insights that canbe gained from conceptual models.

Reproduction and development continue to be inspi-rational topics in the biological fluid mechanics of activematter. Future work will focus on, among other topics,the nonlinear interaction of multiple ciliated and flagel-lated cells with biological fluids and with each other. Weanticipate that the techniques described will help to forma basis for these future investigations.

Acknowledgements

TDMJ and AAS acknowledge Engineering and PhysicalSciences Research Council PhD studentships. DJS is fundedby a Birmingham Science City Research Alliance Fellow-ship. Computations in sect. 3 were performed using theUniversity of Birmingham BlueBEAR HPC service, whichwas purchased through HEFCE SRIF-3 funds.

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