ne

go, 9

Received in revised form 18 May 2013Accepted 29 May 2013Available online 12 June 2013

Keywords:LocomotionCiliaFlagellaMucus transportShear-thinningCarreau model

Newtonian uids in biology, we address mathematically the role of shear-dependent viscosities on both

phology and internal structure (the so-called axoneme), which isthe same as in all eukaryotic agella, has been conserved through-out evolution [4].

l mucusy mucus ae track [4,

the motion of epididymal uid in the efferent ducts of threproductive track [8]. Failure of the transport functionalitycan lead to serious illness of the respiratory system [4]. In addition,it is believed that cilia found in specialized brain cells (ependymalcilia) are involved in the transport of cerebrospinal uid at smallscales [9]. When cilia are anchored on a free-swimming cell, suchas in many protozoa including the oft-studied Paramecium, theircollective deformation and active transport of the surroundinguid leads to locomotion of the cell, typically at very low Reynoldsnumber [5,10].

Corresponding author. Tel.: +1 52 55 5622 4602.E-mail addresses: jvelez@iim.unam.mx, rodribonos@hotmail.com (J.R. Vlez-

Cordero), e.lauga@damtp.cam.ac.uk (E. Lauga).1 Current address: Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road,

Journal of Non-Newtonian Fluid Mechanics 199 (2013) 3750

Contents lists available at

Journal of Non-Newton

journal homepage: ht tp : / /wwwCambridge CB3 0WA, UK.biological function [2]. Cilia not only have the capacity to transportsurrounding uids by means of periodic movements but they alsoplay important sensory roles [1,3]. Remarkably, their basic mor-

ids, including the removal of tracheobronchiarespiratory track [46], the transport of ovulatorovum in the oviduct of the female reproductiv0377-0257/$ - see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jnnfm.2013.05.006in thend the7], ande maleof ciliaThe editors of the recently-created journal Cilia called theirinaugural article Cilia the prodigal organelle [1]. This is perhapsan appropriate denotation for a biological appendage found on thesurface of a variety of eukaryotic cells, with a fascinating and stillnot fully understood relationship between internal structure and

of biology, namely uid transport [46]. The motion of biologicaluids induced by the collective beating of an array of cilia is thethird major form of mechanical uid transport done by the innerorgans of superior animals, ranking only behind blood pumpingand peristaltic motion. In humans, the ow produced by the defor-mation of cilia is involved in the transport of several biological u-1. Introductionthe waving agellar locomotion and ciliary transport by metachronal waves. Using a two-dimensionalwaving sheet as model for the kinematics of a agellum or an array of cilia, and allowing for both normaland tangential deformation of the sheet, we calculate the ow eld induced by a small-amplitude defor-mation of the sheet in a generalized Newtonian Carreau uid up to order four in the dimensionless wavingamplitude. The net ow induced far from the sheet can be interpreted either as a net pumping ow or, inthe frame moving with the sheet, as a swimming velocity. At leading order (square in the waving ampli-tude), the net ow induced by the waving sheet and the rate of viscous dissipation is the same as the New-tonian case, but is different at the next nontrivial order (four in thewaving amplitude). If the sheet deformsboth in the directions perpendicular and parallel to thewave progression, the shear-dependence of the vis-cosity leads to a nonzero ow induced in the far eld while if the sheet is inextensible, the non-Newtonianinuence is exactly zero. Shear-thinning and shear-thickening uids are seen to always induce oppositeeffects.When the uid is shear-thinning, the rate of working of the sheet against the uid is always smallerthan in the Newtonian uid, and the largest gain is obtained for antiplectic metachronal waves. Consider-ing a variety of deformation kinematics for the sheet, we further show that in all cases transport by thesheet is more efciency in a shear-thinning uid, and in most cases the transport speed in the uid is alsoincreased. Comparing the order of magnitude of the shear-thinning contributions with past work on elas-tic effects as well as the magnitude of the Newtonian contributions, our theoretical results, which beyondthe Carreau model are valid for a wide class of generalized Newtonian uids, suggest that the impact ofshear-dependent viscosities on transport could play a major biological role.

2013 Elsevier B.V. All rights reserved.

The active dynamics of cilia plays a crucial role in a major aspectArticle history:Received 27 December 2012

Cilia and agella are hair-like appendages that protrude from the surface of a variety of eukaryotic cellsand deform in a wavelike fashion to transport uids and propel cells. Motivated by the ubiquity of non-Waving transport and propulsion in a ge

J. Rodrigo Vlez-Cordero , Eric Lauga 1Department of Mechanical and Aerospace Engineering, University of California, San Die

a r t i c l e i n f o a b s t r a c tralized Newtonian uid

500 Gilman Drive, La Jolla, CA 92093-0411, USA

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ian Fluid Mechanics

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The study of uid transport by cilia, and more generally byeukaryotic agella, requires an understanding of their beating pat-terns as well as the physical properties of the surrounding uid.Many biological uids in the examples above display strong non-Newtonian characteristics, including nonzero relaxation timesand shear-dependent viscosities [11]. Past work has addressed the-oretically [1218] and experimentally [1921] the effect of uidviscoelasticity on transport and locomotion, but the impact ofshear-dependance material functions on the performance of ciliaand agella has yet to be fully quantied [22].

In the present paper we mathematically address the role ofshear-dependent viscosity on the propulsion performance of atwo-dimensional undulating surface whose boundary conditionsmodel the beating strokes of a cilia array or of a waving agellum.The calculation is an extension of the classical results by Taylor[23,24] to the non-Newtonian case. In the next section we describethe physics of the cilia beating patterns, distinguish individual vs.collective motion, and introduce the mathematical models thathave been proposed to quantify cilia dynamics both in Newtonianand non-Newtonian uids.

hence displaying a three-dimensional pattern [6,28]. Cilia in therespiratory track seem, however, to remain two-dimensional [29].

2.2. Collective cilia motion and metachronal waves

If we now turn our attention to the collective motion of cilia ar-ray, the cilia are observed to beat in an organized manner such thatan undulating surface will appear to form on top of the layer of ci-lia, and to deform in a wave-like fashion. These waves, known asmetachronal waves, are due to a small phase lag between neigh-boring cilia, and are akin to waves created in sports stadiums bywaving spectators. Many biological studies have been devoted tothis collective effect [26,30]. In Table 1 we reproduce some typicalvalues of the beating parameters measured experimentally for ciliamovement in the respiratory track (tracheobronchial cilia). The ci-lia length and the distance between cilia at their base are denotedby L and d, while f =x/2p is the beat frequency and k = 2p/k is themetachronal wavelength.

Different types of metachronal waves can be classied accord-ing to the relationship between their dynamics and that of theeffective stroke of the constituting cilia [5]. When the propagativedirection of the metachronal wave is the same as the direction ofthe effective stroke, the beat coordination is called symplectic. Ifinstead both directions oppose each other then the coordinationis termed antiplectic [23]. Other types of metachronal waves thatrequires certain three-dimensional coordination have also beenidentied [6,27,28,32]. Although there is no yet agreed-uponexplanation for the origin of these different types of coordination,it seems that patterns close to the antiplectic metachrony (includ-ing tracheobronchial cilia) are more widely used than the symplec-

38 J.R. Vlez-Cordero, E. Lauga / Journal of Non-Newtonian Fluid Mechanics 199 (2013) 37502. Background

2.1. Kinematics of an individual cilium

A variety of beating patterns is displayed by cells employing ci-lia or agella, depending on the surrounding geometry, the pur-pose of the beating (locomotion, ingestion, mixing, etc.) as wellas genetic factors and biochemical regulation [2,4,5,25,26]. A de-tailed description of the stroke of cilia has been offered in the past[5,6,23,27] and we summarize the main features here. In general,the beating pattern of an individual cilium displays a two-strokeeffective-recovery motion, as illustrated in Fig. 1 (top). Duringthe effective stroke, the cilium extends into the uid draggingthe maximum volume of uid forward. In contrast, the recovery(or backward) stroke is executed by bending the cilium towards it-self and the nearest boundary, minimizing thereby the drag on thesurrounding uid in the opposite direction. It is this asymmetry inthe beating pattern that induces a net uid ow in the direction ofthe effective stroke. In some cells, the cilium not only bends to-wards the wall during the recovery stroke but also tilts with re-spect to the vertical plane described by the effective stroke,Fig. 1. Schematic representation of the stroke of an individual cilium and the envelope mdashed) represents the stroke cycle as modeled by the envelope model; in this gure thetext). Bottom: representation of the envelope model covering the cilia layer and the protic wave [33] possibly because the rst one is characterized by theseparation of consecutive cilia during the effective stoke, allowing

Table 1Beating parameters for metachronal waves of tracheobronchial cilia: cilia length (L),beating frequency (f), distance between cilia (d), and metachronal wavelength (k).These numbers are found in the references indicated in the last column.

Parameter Reference

Cilium length (L) 6 lm [31]Beating frequency (f =x/2p) 20 Hz [9]Distance between cilia (d) 0.4 lm [6]Metachronal wavelength (k = 2p/k) 30 lm [31]odel. Top: effective (solid) and recovery (dashed) stroke of a cilium. The ellipse (thindirection of the dynamics along the ellipse is clockwise, but it could be reversed (seepagation of the metachronal wave.

For mucus transport by cilia, it is in fact believed that it is a com-

Other models focused on the presence of yield stresses and the

-Newthem to propel more uid volume and thus increasing the stokeefciency [34].

2.3. Modeling cilia arrays

Two different approaches have been proposed to model theperiodic motion of cilia, namely the sub-layer and the envelopemodels [23]. In sub-layer modeling, one keeps the basic ingredi-ents of the cilia beating cycle by parametrizing each cilium shapealong its stroke period. This approach allows to compute the forcesand bending moments induced by each cilium to the surroundinguid and at the same time obtain the mean ow produced abovethe cilia layer, although the hydrodynamic description can be te-dious and usually has to be done numerically [10,34,35].

In contrast, the envelope modeling approach takes advantage ofthe formation of metachronal waves above the cilia layer. The mo-tion of the cilia array is simplied as an undulating surface thatcovers the cilia layer (Fig. 1, bottom), ignoring the details of thesub-layer dynamics. In this case, the metachronal waves are pre-scribed by setting appropriate boundary conditions on the materialpoints of the envelope. In the case where the boundary conditionsare nearly inextensible, the dynamics of the ow produced by thecilia array becomes similar to the undulating motion of atwo-dimensional agellum [5]. The simplication allowed by theenvelope model has been useful in comparing, even quantitatively,the swimming velocities obtained theoretically with those ob-served in water for a number of microorganisms [27,36]. Further-more, this model, which is originally due to Taylor [24], isamenable to perturbation analysis and allows to incorporate cer-tain non-Newtonian effects in a systematic fashion [12,15].

To address the limit of validity of the envelope model, Brennen[27] derived the set of conditions to be satised in order for theenvelope model to be approximately valid, which are m/(xL2) kdand kd < 1, where m is the kinematic viscosity of the uid. With thenumbers from Table 1, we see that the envelope model is valid forthe case of the cilia array in the respiratory track when the liquidhas a viscosity similar to that of water (and m will be actually high-er if the uid is a polymeric solution, hence the model is expectedto always be valid).

2.4. Cilia in complex uids

Because of the importance of mucus in human health, numer-ous theoretical papers have attempted to elucidate the mecha-nisms of tracheobronchial mucus transport using both modelingapproaches outlined above [31,35,37,38]. Mucus is a typical exam-ple of a non-Newtonian uid, displaying both elastic behavior andstress relaxation as well as a shear-dependent viscosity [11].

In order to incorporate a variation of the viscosity in the trans-port problem, a two-uid model has been proposed consisting of alow-viscosity uid located at the cilia layer (peri-ciliary uid) witha second high viscosity uid laying above the rst one [6,35,38,39].This hypothesis of a two-uid layer, supported by experimentalobservations of tissue samples, allows to obtain values of the mu-cus ow similar to those measured in experiments [6,28].

An alternative modeling approach proposed to use a uid with aviscosity whose value changes linearly with the distance from thecell wall [37]. Such approximation, however, is not a formalmechanical model that rigorously takes into account the rheologi-cal properties of the uid. Later work considered formally the elas-tic properties of mucus by introducing the convective Maxwellequation in the theoretical formulation [31]. The values of themean velocity of the uid in this case were found to be much smal-

J.R. Vlez-Cordero, E. Lauga / Journal of Nonler than the experimental ones, indicating that the shear-thinningproperties of mucus also need to be considered. Elastic effects werethe focus on a number of studies on agellar locomotion, both the-heterogeneity in the surrounding environment [4244]. The pres-ence of a yield stress in the limit of small amplitude oscillationshinders the mean propulsion or transport velocity whereas theinclusion of obstacles mimicking an heterogeneous polymericsolution enhances propulsion.

2.5. Outline

In this paper we solve for the envelope model in a generalizedNewtonian uid using a domain perturbation expansion. The uidis the Carreau model for generalized Newtonian uids where theviscosity is an instantaneous non-linear function of the local shearrate. We solve for the ow velocity and rate of work in powers ofthe amplitude of the sheet deformation. In particular, we computeanalytically the inuence of non-Newtonian stresses on the uidvelocity induced in the far eld: that velocity can be interpretedeither as a transport velocity in the context of uid pumping oras a propulsive speed in the context of locomotion. In Section 3we present the formulation of the envelope model and the dy-namic equations that govern the uid ow. Section 4 details theasymptotic calculations followed by Section 5 which is devotedto an analysis of our mathematical results and the illustration ofthe importance of non-Newtonian effects. Finally in Section 6 wediscuss the impact of our results in the context of biological loco-motion and transport and conclude the paper. In Appendix A theapplicability of our results to a wider class of generalized Newto-nian uid models is examined.

3. Formulation

3.1. The envelope model

In the envelope model approach, the cilia tips, attached to a sta-bination of shear-thinning, which can reduce the uid ow resis-tance at the cilia layer, and elastic effects, which maintain a semi-rigid surface at the upper mucus layer, which allow for the optimaltransport of particles in the respiratory track [6,28,37,39,40]. A con-stitutive equation that takes into account a shear-dependent viscos-ity was formally used within the lubrication approximation in thecontext of gastropod [41,42] and agellar locomotion [43]. Thesemodels consider the locomotion due to a tangential or normal (orboth) deformation of a sheet above a thin uid layer with shear-dependent viscosity. In the case of tangential deformation of thesheet, the average locomotion speed is seen to go down for ashear-thinning uidwhereas in the casewhere normal deformationare also included the speed increases. This indicates that the impactof shear-dependent material function on transport depends on theparticular beating pattern, something ourmodel also shows. Recentstudies on the effect of shear-dependent viscosity onwavingpropul-sion showed that locomotion is unaffected by shear-thinning fornematodes (experiments, [19]), increases in a two-dimensionalmodel agellum with an increase in agellar amplitude for shear-thinning uids (computations, [22]), while decreases in the case ofa two-dimensional swimming-sheet-like swimmer model inshear-thinning elastic uids (experiments, [21]).oretically [1218] and experimentally [1921], and the consensusso far seems to be that for small-amplitude motion elastic effectsdo always hinder locomotion whereas it can potentially be en-hanced for large amplitude motion corresponding to order-oneWeissenberg, or Deborah, numbers.

tonian Fluid Mechanics 199 (2013) 3750 39tionary base, display a combination of normal and tangential peri-odic motions. Depending on the phase shift / between these twomotions and their relative amplitude (p 6 / 6 p), in general the

cilia tips will describe an elliptic shape in the xy-plane over oneperiod of oscillation (Fig. 1). We assume that the cilia tips are con-nected by a continuous waving sheet, and the variation of the dis-tance between adjacent cilia is therefore idealized as a stretchingor compression of the surface. The positions of material points(xm,ym) on the sheet are given by

xm x Aa coskxxt /; ym Ab sinkxxt; 1

where 2Aa and 2Ab are the maximum displacements of the materialpoints in the x and y directions respectively. The wave amplitude isthus A (dimensions of length) while the relative values of thedimensionless parameters a P 0 and b P 0 reect the ratio ofthe tangential and normal motions. The propagation of the metach-ronal wave in Eq. (1) occurs in the positive x direction (Fig. 1) andthe material points of the sheet can describe a cyclic motion eitherin a clockwise direction (symplectic), or a counterclockwise direc-tion (antiplectic).

We nondimensionalize lengths by k1 and times by x1 to getthe nondimensionalized form of Eq. (1) as

xm x a cosx t /; ym b sinx t; 2

where we have dened = Ak = 2pA/k. Assuming that the cilia arraypropagates metachronal waves with a single characteristicamplitude much smaller than its wavelength, we can use a standarddomain perturbation method to solve for the pumping velocity inthe limit 1. In the remaining text we will use for conveniencez = x t and n = x t /. Using the formula cos(z /) = coszcos/+ sinzsin/, we will also dene b = acos/, c = asin/, and a2 = b2 + c2.

rium is simply given by r r = 0 where r is the total stress tensor.

Denoting p the pressure and s the stress tensor component due touid deformation (deviatoric) we have

rp r s; 3which is associated with the incompressibility condition, r u = 0,where u denotes the velocity eld.

In our model, the uid problem is two-dimensional, and we cantherefore reduce the number of scalars used in the formulation byintroducing a streamfunction, w(x,y, t). The incompressibility con-dition is satised everywhere in the uid provided we use

u @w@y

; v @w@x

: 4

Using Eq. (4) and the time derivatives of the material position ofthe undulating sheet, u = @xm/@t and v = @ym/@t, we can identify thevelocity components of the uid on the sheet assuming the no-slipboundary condition

rwjxm ;ym b cosx tex a sinx t /ey: 5In the far eld, y?1, the ow moves at the unknown steady

speed, Uex, whose sign and value will depend on the particularmovement of the material points on the sheet and on the uid rhe-ology. The main goal of this paper is to derive analytically the non-Newtonian contributions to U.

3.3. Constitutive equation the Carreau model

As discussed above, many biological uids exhibit non-Newto-

02

ate

) avagwas

40 J.R. Vlez-Cordero, E. Lauga / Journal of Non-Newtonian Fluid Mechanics 199 (2013) 3750106 104 1

102

101

100

101

102

103

104

shear r

visc

osity

(Pa

s)

Fig. 2. Shear-thinning viscosity of mucus reproduced from published data: Steady (( ) of pig small intestine mucus [46]; steady measurements ( ) of human cervicomicro-rheological measurements ( ) of human cervical mucus [48]. When the data3.2. Governing equations of the uid ow and boundary conditions

Neglecting any inertial effects in the uid, mechanical equilib-the steady values [49]. The continuous line denotes a power-law tting, g 10:5 _c0:8go = 2187.5, kt = 2154.6, n = 0.08) and the cervical human mucus ( : go = 145.7, kt = 63indicates the typical oscillation frequencies of cilia [5].nian rheology. In Fig. 2 we reproduce the data from several pub-lished experiments on the shear-dependance of mucus viscosity[4548]. The plot shows the value of the shear viscosity, g, as afunction of the shear rate, _c. Measurements are either steady-stateor, for oscillatory data, their steady-state values are estimatedusing the CoxMerz rule [49]. Clearly, mucus is very strong

100 102

(1/s)

nd oscillatory ( ) measurements of human sputum [45]; oscillatory measurementsinal mucus [47]; micro-rheological measurements ( ) of human lung mucus [48],reported in terms of oscillatory moduli, the CoxMerz rule was applied to estimate

. The dashed lines denote a tting to the Carreau model for the lung mucus ( :1.04, n = 0.27). The shear-rate range enclosed by the vertical lines, 5K _cK50 s1,

means that the ow up to order 2 is going to be the same as the

-Newshear-thinning, the viscosity dropping by three orders of magni-tude on the range of shear rates 103K _cK102.

The dependance of the viscosity with the shear-rate can bemodeled assuming a Carreau generalized Newtonian uid whoseviscosity is given by [49]

g g1go g1

1 kt j _cj2n12; 6

where g1 is the innite-shear-rate viscosity, go the zero-shear-rateviscosity, n the dimensionless power-law index, and kt a time con-stant associated with the inverse of the shear-rate at which the vis-cosity acquires the zero-shear-rate value. In Eq. (6), the magnitudeof the shear-rate tensor, j _cj, is equal to (P/2)1/2, where the secondinvariant of the shear rate tensor, P, is given by

P X2i1

X2j1

_cij _cji

4 @u@x

2 2 @u

@y

2 4 @u

@y@v@x

2 @v@x

2 4 @v

@y

2: 7

Based on the data reproduced in Fig. 2 it is clear that go g1,and this is the limit we will consider here. If we nondimensionalizestresses and shear-rates by gox and x, respectively, we obtain thenondimensional form of the deviatoric stress,

s 1 Cuj _cj2N _c; 8wherewe have dened the non-Newtonian indexN n 1=2 andwhere Cu =xkt is the Carreau number. When n = 1 (equivalently,N 0), the Newtonian limit is recovered in Eq. (8). Clearly, otherclassical models used to describe inelastic uids may be used to tthe rheological data of mucus. We chose the Carreau model due toits wide range of applicability; in Appendix A we discuss the classof generalized Newtonian uids for which our analysis remains va-lid. We demonstrate in particular that the model we chose allows usto formally derive the leading-order effects of shear-dependence onthe ow. Any other model, provided that it is well-dened in thezero-shear-rate limit, would either give results mathematicallyequivalent to ours, or would predict an impact on the ow generatedby the sheet arising at a higher-order in the sheet amplitude.

4. Asymptotic solution for small wave amplitudes

Having posed the mathematical problem, we proceed in thissection to compute its solution order by order in the dimensionlesswave amplitude, [12,23,24]. Our goal is to calculate the leading-order inuence of the shear-dependence of the viscosity on theow eld and, in particular, the ow induced at innity, Uex. Wewill also compute its inuence on the rate of work done by thesheet in order to deform and the resulting transport efciency.

We therefore write a regular perturbation expansion of theform

fu;w; s; p; _c; j _cjg fu1;w1; s1;p1; _c1; j _cj1g 2fu2;w2; s2; p2; _c2; j _cj2g . . . 9

Using the denition of the magnitude of the shear-rate tensor,j _cj P=2p , the term j _cj2 in Eq. (8) can be expanded in terms of asj _cj2 2P

1

2 23j _c1jj _c2j O4: 10

Writing the viscosity in Eq. (8) as

2" #N

J.R. Vlez-Cordero, E. Lauga / Journal of Nong 1 2 Cu2

P1 23Cu2j _c1jj _c2j O4 ; 11Newtonian one, and thus we expect the non-Newtonian effectsto come in only at order 4 in U (or higher, see Appendix A). Aswe see below, this means that we have to compute the ow eldeverywhere up to order 3.

4.1. Solution of the velocity eld at O and O2

Is is straightforward to get from Eq. (12) that s1;2 _c1;2 andthus the solutions at order one and two are the same as theNewtonian ones. The general procedure to derive these solutionsconsists in taking the divergence and then the curl of Eq. (3) toeliminate the pressure, substituting the constitutive relationfor the stress, and using the denition of the streamfunction, Eq.(4), together with the denition of the shear-rate, _c ruruT ,to obtain the biharmonic equation satised by the streamfunction

r4wp 0; p 1;2: 13The general solution of Eq. (13) at order Op is already known

to be [23,50]

wp Upy

X1j1

Apj Bpj y

sinjz Cpj Dpj y

cosjzh i

ejy; 14

which already satises the boundary conditions in the far eld,u = @w/@yj(x,1) = U(p) and v = @w/@xj(x,1) = 0. The values of thecoefcients in Eq. (14) are computed using the boundary conditionson the sheet, Eq. (5), where rw is expanded in terms of using aTaylor approximation of the gradients around the mid plane,xm = x and ym = 0. The streamfunctions at O and O2 are then[23]

w1 b by byey sin z cyey cos z; 15

w2 12b2 2bb a2y y

2e2yc2 b2 2bb b2 cos 2z

ye2ybc bc sin 2z; 16where the net contribution to the uid velocity in the far eld ap-pears up to O2 as

U2 12b2 2ab cos/ a2: 17

4.2. Solution of the velocity eld at O3

At order three, Eq. (12) leads to the non-Newtonian contribu-tion from the constitutive equation as

s3 NCu2

2P1 _c1 _c3: 18

The values of the tensor invariant, P(1), and the components of1and Taylor-expanding Eq. (11) up to O2, we nally obtain theconstitutive relationship as

s1 2s2 . . . 1 2NCu2

2P1

" # _c1 2 _c2 . . . : 12

Note that, because of the ? symmetry, the uid transportat innity, quantied by Uex, has to scale as an even power of .Since the perturbation in the viscosity, Eq. (11), is order 2, this

tonian Fluid Mechanics 199 (2013) 3750 41the shear rate tensor, _c , can be computed using the rst ordersolution, Eq. (15), recalling that _c ruruT , and using Eq. (7)we obtain

Newu1 eyfb b by sin z cy 1 cos zg; 19

v 1 eyfb b by cos z cy sin zg; 20and therefore

_c111 2eyfb b by cos z cy 1 sin zg; 21

_c112 _c121 2eyfb by b sin z c1 y cos zg; 22

_c122 _c111 ; 23which allows us to nally compute

P1 8e2yb2y2 2aby2 y cos/ a2y 12: 24After taking the divergence and then the curl of Eq. (18) we ob-

tain inhomogeneous biharmonic equation for the streamfunctionat order three given by

r4w3 16NCu2e3yfcb2f y 2bbcgy a2chy cos z b3iy 3b2bf y b3b2 c2gy a2bhy sin zg; 25

where the y-dependent functions are

f y 2y3 8y2 7y 1; 26

gy 2y3 10y2 13y 4; 27

hy 2y3 12y2 21y 11; 28

iy 2y3 6y2 3y: 29The homogeneous solution of this equation, w3h , is given by the

general solution in Eq. (14). The particular solution, w3p , can befound using the method of variation of parameters, leading to

w3p NCu2e3yfcb2Fy 2bbcGy a2cHyg cos z fb3Iy 3b2bFy b3b2 c2Gy a2bHyg sin z; 30

where the y-dependent functions are

Fy 12y3 1

4y2 1

16y; 31

Gy 12y3 1

4y2 1

16y 1

16; 32

Hy 12y3 3

4y2 9

16y 1

8; 33

Iy 12y3 3

4y2 9

16y 3

16: 34

In order to determine the unknown constants in the homoge-neous solution, we need to evaluate the boundary condition onthe sheet at O3. Using a Taylor expansion, we have

rw3jx;0 a cos n@

@xrw2jx;0 b sin z

@

@yrw2jx;0

12a2

cos2 n @2

@x2rw1jx;0 ab cos n sin z

@@x

@

@yrw1jx;0

12b2 sin2 z

@2

@y2rw1jx;0: 35

After substituting in Eq. (35) the solutions at order one and two

42 J.R. Vlez-Cordero, E. Lauga / Journal of Non-we obtain, after some tedious but straightforward algebra@

@xw3jx;0

183b3 6b2b bb2 3bc2 cos z 3

8b3

2b2b bb2 bc2 cos 3z 14b2c bbc sin z

34b2c bbc sin 3z; 36

and

@

@yw3jx;0

1815b2c 8bbc 3a2c cos z 1

815b2c

24bbc 9b2c 3c3 cos 3z 182b3 5b2b

4bb2 4bc2 3bc2 3b3 sin z 186b3

15b2b 12bb2 12bc2 9bc2 3b3 sin 3z: 37Applying these boundary conditions to the general solution,

w3 w3h w3p , we nd the value of the only nonzero coefcientsin Eq. (14) which are given by

A31 38

12NCu2 1

b3 1

832NCu2 1

bb2

18

12NCu2 3

bc3 1

8NCu2a2 6b2b; 38

A33 18b3 1

4b2b 1

8bb2 1

8bc2; 39

B31 18

32NCu2 1

b3 1

832NCu2 1

b2b

18

5 32NCu2

bb2 1

81 1

2NCu2

bc2 3

8bc2

1316

NCu2a2b 38b3; 40

B33 38b3 9

8b2b 9

8bb2 9

8bc2 3

8b3 9

8bc2; 41

C31 14

1 12NCu2

bbc 1

412NCu2a2 b2

c; 42

C33 14

b2c bbc

; 43

D31 18

12NCu2 15

b2c 1

412NCu2 3

bbc

14

b2 32a2 13

4NCu2a2

c; 44

D33 98b2c 3

8c3 9

8b2c 9

4bbc; 45

and, as expected, the uid velocity at innity at this order is

U3 0: 46In order to illustrate quantitatively the difference between

the Newtonian ow induced by the traveling wave and thenew non-Newtonian terms, we show in Fig. 3 the instantaneousstreamlines (dotted lines) and contours (colors and numbers) ofiso-values of the vorticity at order O3 for two different valuesof the phase, /, and three different values of the non-Newtonianterm, NCu2, assuming a = b. Changing the phase / has the effect

tonian Fluid Mechanics 199 (2013) 3750of tilting the vortical streamlines formed in the ow. Increasingthe magnitude of NCu2 leads to a signicant modication of the

-NewJ.R. Vlez-Cordero, E. Lauga / Journal of Nonow: large strong vortices are created above the wavingsheet whose sign changes with the sign of NCu2. The locationand strength of the regions of high vorticity near the sheet arealso modied. Note that even though the non-Newtonianterm is scaling linearly with N , the total ow, sum of theNewtonian plus the non-Newtonian component, is notanti-symmetric shear-thinning N < 0 vs. shear-thickeningN > 0, that is, the magnitudes of the values change as wellas the signs.

4.3. Net far-eld velocity at O4

To nish the calculation we have to compute the velocity in-duced in the far eld, U(4)ex, to quantify the role of non-Newtonianstresses on uid transport, either pumping or propulsion. This canactually be determined without having to compute the entire oweld. Indeed, just like for the previous order, the streamfunction,w(4), will satisfy an inhomogeneous biharmonic equation. Theswimming speed, which contributes to a uniform U(4)y term tothe streamfunction, does not have to be computed in the far eldbut can instead be evaluated using the boundary conditions. Theexpansion of the x-velocity, u, at this order evaluated at (x,0) is ob-tained by Taylor-expanding the boundary conditions on the sheetas

Fig. 3. Instantaneous streamlines (dotted lines) and iso-vorticity contours (color levels aphase, / = 0 (top row) and p/4 (bottom row), and three values of the non-Newtoniancolumn) and +10 (shear-thickening, right column). The solid lines denote the position of tlegend, the reader is referred to the web version of this article.)tonian Fluid Mechanics 199 (2013) 3750 43@

@yw4jx;0 b sin z

@2

@y2w3jx;0 acosn

@

@x@

@yw3jx;0

12a2 cos2 n

@2

@x2@

@yw2jx;0

abcosn sin z @@x

@2

@y2w2jx;0

12b2 sin2 z

@3

@y3w2jx;0

16a3 cos3 n

@3

@x3@

@yw1jx;0

12a2bcos2 n sin z

@2

@x2@2

@y2w1jx;0

12ab2 cosn sin2 z

@

@x@3

@y3w1jx;0

16b3 sin3 z

@4

@y4w1jx;0:

47The rst term of the right-hand side of Eq. (47) is the only one

that provides a nonzero non-Newtonian term after evaluating thederivatives at y = 0. Computing the mean velocity over one periodof oscillation, we obtain a Newtonian plus a non-Newtonian contri-bution of the form

12p

Z 2p0

b sin z @2

@y2w3jx;0 dt

18NCu23b3b 6b2b2

2b2c2 17a2bb 116

b4

4b3b 9b2b2 5b2c2 6a2bb: 48

nd iso-value numbers) at order 3 obtained for the case a = b for two values of theterm, NCu2 0 (Newtonian uid, left column), 10 (shear-thinning uid, middlehe traveling wave at t = 0. (For interpretation of the references to color in this gure

NewThe contribution to the mean velocity of the eight other termsappearing in the right-hand side of Eq. (47) are the same as forthe Newtonian solution and the nal result can be found in Blake[32]. Adding all the terms and expressing themean velocity at orderfour in terms of the original parameters a, b, and /, we are nallyable to write the O(4) velocity in the far eld as the sum of a New-tonian component, U4N , and a new non-Newtonian term, U

4NN , as

U4 U4N U4NN; 49with

U4N 12b4 a2b2 1

4ab3 a3b cos/; 50

U4NN 18NCu24a2b2 3ab3 17a3b cos/ 2a2b2

cos 2/: 51The Newtonian component, Eq. (50), is Blakes solution [32]. The

new term, Eq. (51), which quanties the leading-order effect of ashear-dependent viscosity on the uid transport, is the main resultof our paper.

Two interesting features can be noted at this point by a simpleinspection of Eq. (51). First we see that if the sheet does not displaynormal deformation (b = 0) or no tangential deformation (a = 0)then U4NN 0; in order for the shear-dependent viscosity to affecttransport, it is this important that both modes of deformation bepresent, a 0, b 0. The second interesting point is that the effectscales linearly with N , and thus with the difference between thepower-law index, n, and 1. Consequently, when they have an inu-ence on transport, shear-thinning and shear-thickening uids al-ways act in opposite direction and if one type of uid hinderstransport the other uid will facilitate it.

4.4. Rate of work

We now turn to the calculation of the rate of work, W, done bythe sheet to transport the uid (per unit area of the sheet). SinceWscales quadratically with the shear rate, the leading order rate ofwork appears at O2 and is identical to the one for a Newtonianuid [23,32]. Using the nondimensionalization W gox2/k wehave in dimensionless variables

hW 2i 2a2 b2; 52where we have used h. . .i to denote mean over a period ofoscillation.

In order to compute the rate of work at higher order we recallthat, in the absence of inertia and without the presence of externalforces, the rate of work W done by a waving surface S against auid is equal to the value of the energy dissipation over the en-closed volume V

W ZSu r n dS

ZVr : ru dV : 53

Using that the stress tensor and velocity vector have been ex-panded in powers of the wave amplitude , the rate of work atthird and fourth order are

W 3 ZV

r1 : ru2 r2 : ru1 dV ; 54W 4

ZVr1 : ru3 r2 : ru2 r3 : ru1 dV : 55

44 J.R. Vlez-Cordero, E. Lauga / Journal of Non-At rst and second orders in , the constitutive equations for thestress tensor are the Newtonian ones, r1;2 p1;2I _c1;2. Theconstitutive equation at third order is in contrastr3 p3I NCu22 P1 _c1 _c3. Substituting these into Eqs. (54)and (55) we see that the pressure at any order multiplies the diver-gence of the velocity which is zero by incompressibility. The meanrates of work per unit area of the sheet at orderO3 andO4 arethus given by

hW 3i 12p

Z 2p0

Z 10

_c1 : _c2

dydz; 56

hW 4i 12p

Z 2p0

Z 10

_c1 : _c3 12

_c2 : _c2 NCu2

4P1 _c1 : _c1

!dydz:

57Substituting the values of the shear-rate tensor and its second

invariant at the respective orders in , we obtain that hW(3)i = 0,which could have been anticipated by symmetry ? . Themean rate of work per unit area at O4 is found to be the sumof a Newtonian term plus a non-Newtonian contribution as

hW 4i hW 4iN hW 4iNN; 58where

hW 4iN 14a4 b4 5a2b2 ab3 3a3b cos/

12a2b2 cos 2/; 59

hW 4iNN 18NCu215a4 3b4 4a2b2 8a3b cos/ 2a2b2

cos 2/: 60Similarly to the non-Newtonian contribution to the velocity, Eq.

(51), the effect of the shear-dependent viscosity on W scales line-arly with N , and thus shear-thinning and shear-tickening uids al-ways contribute in opposite sign to the rate of working of thesheet. In contrast however with the uid velocity, we see thatmodes with pure normal (a = 0) or tangential (b = 0) motion docontribute to the rate of work at this order.

4.5. Transport efciency

We nally consider the efciency of transport, E, measured asthe ratio between the useful work done in the uid to create theow at innity and the total dissipation, written E U2=hWi indimensionless variables [10]. Given that we have U = 2U(2) + 4U(4)

and hWi = 2hW(2)i + 4 hW(4)i, it is clear what we will obtainE 2E2 4E4 with each term found by Taylor expansion as

E2 U22

hW 2i ; E4 U

22

hW 2i 2U4

U2 hW

4ihW 2i

!: 61

The rst term, E2, is the same as theNewtonian onewhereas thesecond one, E4, includes a non-Newtonian contribution, given by

E4NN U22

hW 2i 2U4NNU2

hW4iNN

hW 2i

!: 62

Non-Newtonian stresses will lead to an increase in the ef-ciency of transport if the term on the right-hand side of Eq. (62)is positive, which we will investigate below for the various kine-matics considered.

5. Do shear-thinning uids facilitate locomotion and transport?

tonian Fluid Mechanics 199 (2013) 3750With our mathematical results we are now ready to address theimpact of non-Newtonian stresses on ow transport. We focus onfour aspects. We rst show that, within our mathematical frame-

-Newwork, shear-thinning always decrease the cost of ow transport.We then demonstrate that the maximum energetic gain is alwaysobtained for antiplectic metachronal waves. When the sheet isviewed as a model for agellar locomotion in complex uids, forexample spermatozoa, we then show that, at the order in whichwe carried out the calculations, there is no inuence of theshear-dependent viscosity on the kinematics of swimming. We -nally address the more general case of viewing the sheet as anenvelope for the deformation of cilia array, and address the impactof non-Newtonian stresses on the magnitude of the ow induced inthe far eld for a variety of sheet kinematics.

5.1. Shear-thinning uids always decrease the cost of transport

We start by considering the non-Newtonian contribution to therate of working by the sheet, Eq. (60). We can writehW 4iNN NCu2Fa; b;/ where the function F is

Fa; b;/ 1815a4 3b4 4a2b2 8a3b cos/ 2a2b2

cos 2/: 63We are going to show that this function is always positive,

meaning that for shear-thinning uids (N < 0), the sheet expendsless energy than in a Newtonian uid with the same zero-shearrate viscosity.

In order to show that FP 0 we calculate its minimum value andshow that it is positive. The derivative of F with respect to thephase is given by

dFd/

a2b sin/a b cos/; 64

while its second derivative is

d2F

d/2 a2ba cos/ b cos 2/: 65

The local extrema of F are reached when dF/d/ = 0 which occursat three points: / = 0, p and, when a < b, cos/ = a/b. When aP b,only the points / = 0, p are extrema, and plugging them into Eq.(65) we see that the minimum of F is obtained at / = 0 (positivesecond derivative). From Eq. (63) we get that this minimum valueis Fmin = [7a4 + 3b4 + 6a2b2 + 8a3(a b)]/8 which is positive sinceaP 0, bP 0, and aP b. In the case where a < b it is straightfor-ward to show that the minimum of F occurs at the phase / satisfy-ing cos/ = a/b. Plugging that value of the phase into Eq. (63), andusing that cos2/ = 2a2/b2 1 we obtain that Fmin = [11a4 + 3b4 +2a2b2]/8 which is also always positive.

We therefore always have FP 0 and the non-Newtonian contri-bution to the rate of working by the sheet, hW(4)iNN, always has thesame sign as N . For shear-thinning uids N < 0 and therefore thecost of uid transport is always reduced a result which remainstrue even if the non-Newtonian contribution to the ow at innity,U4NN , is equal to zero. As a consequence, the last term in Eq. (62),hW(4)iNN/hW(2)i, will always be negative for shear-thinning uids,indicating a contribution to an increase in the efciency. The oppo-site is true for shear-thickening uids which always add to theenergetic cost. This result is reminiscent of earlier work on the costof transport by gastropod showing that shear-thinning uids areadvantageous, although for a completely different reason (mechan-ical work vs. mucus production) [41].

As a nal note, it is worth pointing out that, had the ow kine-matics been unchanged at all orders compared to the Newtonianones, then this result of reduction in W would have in fact been

J.R. Vlez-Cordero, E. Lauga / Journal of Nonobvious. Indeed, if the shear is the same everywhere in the uid,and if the viscosity decreases, then the total dissipation should alsodecrease. However, as detailed in Section 4.4, the dissipation rateat order four in depends on the third-order kinematics in the uidwhich are different from what they would have been in the ab-sence of shear-dependence. The fact that our calculations showthat shear-thinning uids always decrease the cost of transport istherefore not obvious.

5.2. Antiplectic metachronal waves offer the largest non-Newtonianenergy saving

Another general result can be gained from a close inspection ofthe function F in Eq. (63). For given values of both amplitudes a andb, it is clear that F it is maximized when cos / = 1 and cos2/ = 1,which occurs when / = p. This is thus the value of the phase be-tween normal and tangential motion leading to the maximumnon-Newtonian decrease in the energy expanded by the sheet totransport the uid. For this particular value of the phase difference,the kinematics of the material point at x = 0, is xm = acost, ym =bsintwhich means that material points follow elliptical trajecto-ries of semi axes a and b in the anticlockwise direction. Viewingthe sheet as a model for the motion of cilia tips this is therefore awave for which the effective stroke is in the x direction while therecovery stroke occurs in the +x direction. Given that the metach-ronal waves are assumes to propagate in the positive direction, thismeans that antiplectic metachronal waves offer the largest non-Newtonian energy saving in a shear-thinning uid.

5.3. The sheet as a model for agellar locomotion

We have seen in Section 4 that the presence of non-Newtonianstresses affects the uid transported by the waving sheet only ifboth oscillatory motions, normal (b 0) and tangential (a 0),are present. In the frame moving with the uid at innity, this in-duced speed can be also interpreted as the swimming speed of afree-moving sheet, which prompted Taylors original idea to pro-pose it as the simplest model of agellar locomotion [24]. The dif-ference however between the waving kinematics of a agellumand the one we have studied in this paper is that a agellum isinextensible. Consequently, the agellum has O() motion only inthe normal direction, while the tangential deformation is O(2) asa result from the requirement of inextensibility (material pointson a agellum describe a typical gure-8 motion). Higher-orderkinematics can also be described similarly [23,24,51].

To address the impact of a shear-dependent viscosity on loco-motion we have followed the same procedure as above and com-puted the swimming velocity systematically up to O4 usingthe Carreau model and enforcing inextensibility condition. We ob-tain formally that U4NN 0: the shear-dependence of the viscositydoes not affect the locomotion speed of the sheet at this order.Inextensible swimmers, such as those employing agella, anddeforming in a constant, small-amplitude waving motion of thetype modeled here are thus expected to be hardly affected by theshear-dependence of the uid. Since the rate of working decreasesfor shear-thinning uids, swimmers are thus more efcient than inNewtonian uids (see Eq. (62)). Our theoretical result is consistentwith experiments by Shen and Arratia [19] which have shown thatthe swimming velocity of a nematode (C. elegans) waving freely ina shear-thinning uid (xanthan gum solution) is the same as thatobserved in a Newtonian uid with similar viscosity.

5.4. The sheet as a model for uid transport and pumping

Returning to the sheet as amodel for uid transport by the tips ofcilia arrays, we investigate here the effect of the shear-dependent

tonian Fluid Mechanics 199 (2013) 3750 45viscosity on the pumping performance as a function of the valuesof a, b, and /. Recall that the magnitude of the non-Newtonian con-tribution, U4NN , scales with the absolute value ofNCu2 while its sign

is that of N , which is positive if the uid is shear-thickening, andnegative is the uid is shear-thinning. Among the many combina-tions of a, b, and / that we may choose, we limit the study to threerelevant cases: a = b, b a, and a b, and in each case consider fourillustrative values of/, namely,/ = 0,/ = p/2, and/ = p. In Fig. 4wedisplay the three different kinematics of the sheets in the casewherethe phase is / = p/4. Below we reproduce the results for the fourth-order swimming speed, U(4), and discuss its relation to the leading-order Newtonian contribution at order 2. We also compute themean rate of working, hW(4)i. The main results are summarized inTable 2 in the discussion section of the paper.

5.4.1. Fluid transport with both normal and tangential motionIn the case where the normal and tangential amplitudes are

similar, a = b c, we obtain

i U4 10c4

4NCu2; for / 0; the O2

velocity is U2 c2; 66

ii U4 c4

21 1

2NCu2

; for / p=2; U2 0; 67

iii U4 c41NCu2; for / p; U2 c2: 68We see that for all the values of the phase /, the uid transport

is increased, i.e. the second order velocity and the non-Newtoniancontribution have the same sign, if the uid is shear-thinning,N < 0. This results remains true for / = p/2 when the mean veloc-

i hW 4i c4 8 2NCu2

; for / 0; 69

ii hW 4i c4 5 52NCu2

; for / p=2; 70

iii hW 4i 4c4NCu2; for / p; 71and the maximum reduction of the rate of work is given when / = p,as anticipated above. In all cases, the transport efciency in Eq. (62)is increased for two reasons: more uid is being transported, and

considered in this work: (1) normal plus tangential motion of the same magnitude, a = b;

Table 2Effect of a shear-thinning viscosity on the mean transport velocity and efciency forthree different wave kinematics: normal plus tangential deformation, motiondominated by the normal mode, and motion dominated by the tangential mode.Four different values of the phase between normal and tangential motion areconsidered. The sign + (resp. ) indicates a situation where a shear-thinning uidimproves on (resp. hinders) the velocity or efciency; all signs would be reversed in ashear-thickening uid.

Normal + tangentiala = b

Normal modeb a

Tangential modea b

/ = 0 Velocity: + Velocity: Velocity: Efciency: + Efciency: + Efciency: +

/ = p/2, p/2 Velocity: + Velocity: + Velocity: Efciency: + Efciency: + Efciency: +

/ = p Velocity: + Velocity: + Velocity: +Efciency: + Efciency: + Efciency: +

46 J.R. Vlez-Cordero, E. Lauga / Journal of Non-Newtonian Fluid Mechanics 199 (2013) 3750ity at O2 is zero as in this case the non-Newtonian contributionhas the same sign as the Newtonian term at order four whenN < 0.

On the other hand, the mean rate of work per unit lengthbecomes

Fig. 4. Representation of the sheet kinematics for the three different types of motion

(2) motion dominated by the normal mode, b a; (3) motion dominated by the tangentiindicate the position of a material point during a stroke cycle. In (1) and (2) the positionpositions of the material points are shown at t = 0, p/2, p, and 3p/2.transport occurs at a lower cost.

5.4.2. Motion dominated by the normal modeIn the case where the sheet kinematics is dominated by the nor-

mal deformation, b a, we have U(2) b2/2. Keeping all terms in a/al mode, a b. In all cases displayed a value of / = p/4 was chosen and the red dotsof the wave is shown at time t = 0 (solid line) and p (dashed solid line). In (3) the

in wativeon t

-Newb up to the leading-order non-Newtonian contribution, we obtainthat the velocity at O4 is given by

i U4 b4

2 14ab3

32NCu2 1

; for / 0; 72

ii U4 b4

2 a2b2 1 1

4NCu2

; for / p=2; 73

iii U4 b4

2 14ab3 1 3

2NCu2

; for / p; 74

while at leading-order in a/b we have

hW 4i b4 3NCu2 1

; 75

Fig. 5. In the wave conguration space a/b vs. /we use colors to indicate the regionscomponent at order O4, i.e. jU4NN j > jU4N j (from Eqs. (50) and (51)) for three illustrjU4NN j=jU4N j 20 is chosen to account for the zeros of U4N . The color scheme, showncolor in this gure legend, the reader is referred to the web version of this article.)

J.R. Vlez-Cordero, E. Lauga / Journal of Non4 2

for all values of the phase. In the cases where / = p/2, and p, thevelocity induced by the motion of the sheet is increased if the uidis shear-thinning. In contrast, for in-phase motion, / = 0, the non-Newtonian term acts in the direction opposite to the leading-orderNewtonian term, U(2). However, the non-Newtonian contribution tothe velocity, Eq. (72), scales as N ab3 while the contribution to theenergetics, Eq. (75), scales as N b4, and thus in the limit b a con-sidered here, Eq. (62) leads to a positive value for E4NN: the motion isstill more efcient in a shear-thinning uid.

5.4.3. Motion dominated by the tangential modeThe last case we consider is the one where a b and the kine-

matics is dominated by the tangential deformation. In this case, thesecond order velocity is U(2) a2/2. Keeping all terms in b/a up tothe leading-order non-Newtonian contribution in U(4) we obtain

i U4 14ba3 1 17

2NCu2

; for / 0; 76

ii U4 a2b2 1 14NCu2

; for / p=2; 77

iii U4 14ba3 1 17

2NCu2

; for / p; 78

while the rate of work is given byhW 4i a4

4152NCu2 1

: 79

This time, for a shear-thinning uid, the non-Newtonian contribu-tion acts in the direction of the leading-order Newtonian ow onlyfor / = p while a ow in the opposite direction is created for / = 0and p/2. However, similarly to the discussion in the previous sec-tion, since the non-Newtonian effect on the rate of working is N a4 (Eq. (79)) while the impact on the ow speed is N ba3(Eq. (76)) and N a2b2 (Eq. (77)), the transport efciency, Eq.(62), is systematically decreased in the shear-thinning case.

5.4.4. Numerical approachWith the solution for both the net Newtonian ow speed, U4N

(Eq. (50)), and the non-Newtonian one, U4NN (Eq. (51)), it is straight-

hich the magnitude of the net non-Newtonian ow dominates that of the Newtonianvalues of the non-Newtonian coefcient, NCu2 (0.1, 1 and 10). The cut-off value ofhe right, gives the iso-values of jU4NN j=jU4N j. (For interpretation of the references to

tonian Fluid Mechanics 199 (2013) 3750 47forward to numerically compare their values. In Fig. 5 we plot theregions in which the ratio in magnitude between both terms,jU4NN=U4N j, is above 1 in the wave conguration space a/b vs. /,for three values of the non-Newtonian term NCu2 (0.1, 1, and10). The gure indicates therefore the regions where the non-New-tonian contribution dominates the Newtonian one. The colorscheme for the iso-values of the velocity ratio is indicated on theright of the gure, and the cut-off of the ratio to account for thezeros of U4N is chosen to be 20. When the ow is almost NewtonianNCu2 0:1 the non-Newtonian contribution can almost every-where be neglected, except near the region where the Newtonianterm exactly cancels out. As NCu2 increases, the non-Newtoniancontribution to the net ow becomes predominant over almostall conguration space; the only region remaining white (indicat-ing a ratio of less than 1) is at the bottom of each panel indicatingthe prevalence of the Newtonian term for waves dominates by nor-mal modes (a/b 1).

6. Discussion

In this paper we propose a mathematical modeling approach toaddress the role of shear-dependent viscosity on agellar locomo-tion and transport of mucus by cilia array. We employ the envelopemodel originated by Taylor [23,24] and allow for both normal andtangential deformation of a two-dimensional waving sheet. Wecompute the ow eld induced by a small-amplitude deformationof the envelope in a generalized Newtonian Carreau uid with

estimate the contribution of the shear-thinning effects. Indeed,

1 De2

Newtonian Fluid Mechanics 199 (2013) 3750power index n up to order 4 in the dimensionless waving ampli-tude, . The net ow induced at innity can be interpreted eitheras a net pumping ow or, in the frame moving with the sheet, asa swimming velocity.

At leading order, i.e. at order 2, the ow induced by the sheet,and the rate of working by the sheet to induce this ow, is thesame as the Newtonian solution. The non-Newtonian contributionsin both the induced net ow and the rate of working appear at thenext order, i.e. O(4). In both cases, the effect is linear in n 1meaning that shear-thinning uids (n < 1) and shear-thickeninguids (n > 1) always induce opposite effects. The leading-ordernon-Newtonian contribution to the ow created at innity is non-zero only if the sheet deforms both in the direction normal andtangential to the wave direction. The non-Newtonian contributionto the rate of working is found to always be negative in the case ofshear-thinning uids, which are thus seen to systematically de-crease the cost of transport independently of the details of thewave kinematics. The maximum gain in dissipated energy is ob-tained for antiplectic waves where the direction of the propagatingwave and that of the effective cilia stroke are opposite.

The sheet kinematics can be interpreted as a model for the loco-motion of a agellated microorganism. In this case, and making thebiologically-realistic assumption that the agellum is inextensible,we nd that the swimmer is more efcient in a shear-thinninguid but that the non-Newtonian contribution to the swimmingspeed of the sheet is exactly equal to zero. This result is consistentwith recent experiments [19] showing that the swimming speed ofa nematode waving freely in a shear-thinning uid is same as thatobserved in a Newtonian uid. In contrast, computations withswimmers deforming their agella with increasing amplitudeshowed that the locomotion speed was enhanced by shear thin-ning. The effect was attributed to a viscosity gradient induced bythe increased beating agellar amplitude, a result which couldpotentially be tackled theoretically with an approach similar toours adapted to their kinematics [22]. In contrast, recent experi-ments with a two-dimensional swimmer model showed a decreaseof the swimming speed in a shear-thinning elastic uid [21], a re-sult not explained by our theoretical approach but perhaps due tothe importance of higher-order terms for large-amplitude swim-ming or due to non-negligible elastic effects.

Viewing the sheet as a model for the transport of non-Newto-nian uids by cilia arrays, we address three different types of kine-matics to understand the impact of shear-thinning uids on theow transport: kinematics dominated by normal deformation,those dominated by tangential motion, and kinematics where nor-mal and tangential beating were of the same order of magnitude. Inall three cases we consider four different values of the phase be-tween the normal and tangential waving motion, and the resultsare summarized in Table 2 where we used + to denote instanceswhen the shear-thinning uid improves on the performance (owvelocity or transport efciency) and otherwise. In all cases, ashear-thinning uid renders the ow transport more efcient,and in eight out of the twelve cases it also increases the transportspeed in particular for the biologically relevant case where themagnitudes of the normal and tangential deformation are of thesame order. A further numerical investigation in Section 5.4.4 alsoidenties the regions in which the non-Newtonian term is strongerthan the Newtonian one.

In this paper we have assumed that the uid has no memorybut instead its viscosity is an instantaneous function of the shearrate a class of models known as generalized Newtonian uids.Under that limit, we saw that the shear-thinning property of mu-cus does facilitate its transport by cilia arrays. The constitutive

48 J.R. Vlez-Cordero, E. Lauga / Journal of Non-model we used in this work then serves as an alternative modelto the previously-studied two-uid model and allows us to treatthe mucus layer as a continuous uid. We still need, however, towhere the Deborah number, De, is dened as De = krx, with kr beingthe relaxation time of the uid, and gs/g is the ratio of the solventviscosity with the viscosity of the polymeric solution (typically gs/g 1). For mucus we have kr 30 s [52], and thus, using the typicalfrequency in Table 1 for tracheobronchial cilia, we haveDe 4 103 1. For a solvent with the same viscosity as water,we have gs/g 105 at zero shear rate (see Fig. 2) and thus De2gs/g 1. Elastic effects lead therefore to a decrease of the transportspeed by a factor of gs/g.

To compare simply the elastic and shear-thinning contributions,consider a viscoelastic uid being deformed by a low-amplitudeundulating surface with normal and tangential motion of the sameorder of magnitude. The magnitude of the shear-dependent contri-bution to the transport velocity scales as 4jN jCu2 while theelastic contribution scales as 2gs/g. With jN j 0:4, kt 2000 s(Fig. 2), and x = 2p 20 Hz (Table 1), the ratio of shear-dependentto elastic effects varies as 2jN jCu2=gs=g 1015 2. Despite thedifferent scaling with the motion amplitude , and although our re-sults are only strictly valid in the mathematical limit ? 0, we ex-pect that the shear-dependent viscosity will play the mostimportant role on the ability of cilia to transport mucus-like uids.In addition, a similar scaling ratio can be computed to compare thenon-Newtonian contribution derived in our paper, U4NN , to theNewtonian speed at leading order, U(2), and we ndU4NN=U

2 4jN jCu2=2 1010 2, another potentially very largeratio.2 It is therefore clear that the rheological properties of the sur-rounding uid will play important roles in the uid ow produced bycilia arrays [6,28,40].

Beyond the assumption on the uid rheology, the model pro-posed in this paper has been carried out under a number of math-ematical assumptions. The calculation was two-dimensional andas such does not address the ow in the direction perpendicularto the plane of the wave propagation. For the sheet as a modelfor agellar locomotion, the nite-length of the agellum is ex-pected to play an important role in non-linear uids. And for cilia,the discrete nature of the cilia in a dense array will also lead tonontrivial ow in the sublayer with contributions to both transportand energetics still unquantied. It is hoped that these limitationsdo not prevent the current results to still be relevant to bothlocomotion and transport studies, for example mucus transportin the respiratory track, but instead suggest potentially interestingdirections with future work.

Acknowledgements

R. Vlez acknowledges CONACyT-Mxico and UC-MEXUS fortheir nancial support during his postdoctoral stay at UCSD. Wealso thank the support of the NSF (Grant CBET-0746285 to E.L.).

Appendix A. Application to other generalized Newtonian uids

In this paper we have used a specic empirical model (Carreauuid) to mathematically describe the relationship between viscos-

2 We emphasize again that the calculations presented in this paper are only valid inthey do not play any role at leading order but only contribute atO4. Elastic stresses, in contrast, do contribute at order O2and hinder the uid transport by the prefactor [12,19]:

1 De2gs=g ; 80the mathematical limit e? 0. The order-of-magnitude estimates presented above areonly formally true when e is small and are used to point out how easy it is for theshear-dependent terms to become predominant.

-Newity and shear rate. How applicable are our results to uids de-scribed by other rheological laws, g _c? In the asymptotic limit as-sumed this work, we need a uid model able to describe thebehavior of ows in the limit of low shear rates. We rst requirethe viscosity, g, to be a smooth function of the shear rate, _c, leadinga nonzero value of g in the limit _c! 0, denoted g(0). This require-ment rules out the use of the power-law model, valid only in thelimit of high shear rates [49]. Due to the ? symmetry, theshear-dependent viscosity needs to be an even function of theshear rate. The only even function at order one in is the absolutevalue function, which is not smooth (differentiable) at _c 0.Hence, the viscosity necessarily has to be a function of the shearrate square. As described in the text, we have

j _cj2 2P1

2O3: A:1

In the limit of small wave amplitudes, the viscosity then takes thegeneral form

gj _cj2 g 2P1

2O3

" # g0 2P

1

2dgd _c2

j0; A:2

using a Taylor expansion near zero. The shear-dependent viscositywill thus give an effect at order O2 at best, and thus at O3 inthe ow eld. If the derivative in Eq. (A.2) is zero then the effect willbe at higher order. If instead the derivative is not dened, then themodel is not appropriate to describe ow behavior at small shearrates. In the case of the Carreau model considered above, the deriv-ative dg=d _c2j0 takes a nite value, so we indeed obtain a nonzero ef-fect at second order. There are other models that could potentiallybe used as well to t rheological data. One of such model is thatof a Cross uid, given in a dimensionless form by

g1 1 Cuj _cj2N : A:3In that case, it is rst straightforward to show that only N 6 0

(shear-thinning uid) leads to a nite value of the viscosity at zeroshear rate. Computing the derivative in Eq. (A.3) to perform theTaylor expansion it is then easy to show that only the specic caseN 1 leads to a nite value for dg=d _c2j0. When N 1 the re-sults obtained for a Cross uid would thus be similar to the resultsdescribed in the paper; whenN > 1 the Cross uid is not well de-ned near the zero-shear-rate limit while when N < 1 the deriv-ative of the viscosity is zero, and therefore non-Newtonian effectson the uid ow would appear at a higher order. A similar analysiscan be carried out for the Ellis uid, given by

g1 1 jsj2=s21=2h iN

; A:4

with similar results (jsj is the magnitude of the stress tensor and s(1/2) is the shear at which g = 1/2). Consequently, for any generalizedNewtonian uid with a well-dened behavior in the zero-shear-ratelimit, the asymptotic results are either exactly the same as the onesin our paper, or they predict a non-Newtonian impact onlyoccurring at higher-order in the wave amplitude (and are thus iden-tical to the Newtonian results up to order 4 at least).

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Waving transport and propulsion in a generalized Newtonian fluid1 Introduction2 Background2.1 Kinematics of an individual cilium2.2 Collective cilia motion and metachronal waves2.3 Modeling cilia arrays2.4 Cilia in complex fluids2.5 Outline

3 Formulation3.1 The envelope model3.2 Governing equations of the fluid flow and boundary conditions3.3 Constitutive equation the Carreau model

4 Asymptotic solution for small wave amplitudes4.1 Solution of the velocity field at ? and ? 4.2 Solution of the velocity field at ? 4.3 Net far-field velocity at ? 4.4 Rate of work4.5 Transport efficiency

5 Do shear-thinning fluids facilitate locomotion and transport?5.1 Shear-thinning fluids always decrease the cost of transport5.2 Antiplectic metachronal waves offer the largest non-Newtonian energy saving5.3 The sheet as a model for flagellar locomotion5.4 The sheet as a model for fluid transport and pumping5.4.1 Fluid transport with both normal and tangential motion5.4.2 Motion dominated by the normal mode5.4.3 Motion dominated by the tangential mode5.4.4 Numerical approach

6 DiscussionAcknowledgementsAppendix A Application to other generalized Newtonian fluidsReferences