Analytical progress in the theory of vesicles under linear flow

Alexander Farutin,1 Thierry Biben,2 and Chaouqi Misbah1

1Laboratoire de Spectrométrie Physique, UMR5588, 140 Avenue de la Physique, Université Joseph Fourier Grenoble–CNRS,38402 Saint Martin d’Hères, France

2Université de Lyon, Lyon F-69000, Laboratoire PMCN, Université Lyon 1–CNRS, UMR 5586, F-69622 Villeurbanne Cedex, France�Received 31 December 2009; revised manuscript received 5 May 2010; published 3 June 2010�

Vesicles are becoming a quite popular model for the study of red blood cells. This is a free boundaryproblem which is rather difficult to handle theoretically. Quantitative computational approaches constitute alsoa challenge. In addition, with numerical studies, it is not easy to scan within a reasonable time the wholeparameter space. Therefore, having quantitative analytical results is an essential advance that provides deeperunderstanding of observed features and can be used to accompany and possibly guide further numericaldevelopment. In this paper, shape evolution equations for a vesicle in a shear flow are derived analytically withprecision being cubic �which is quadratic in previous theories� with regard to the deformation of the vesiclerelative to a spherical shape. The phase diagram distinguishing regions of parameters where different types ofmotion �tank treading, tumbling, and vacillating breathing� are manifested is presented. This theory revealsunsuspected features: including higher order terms and harmonics �even if they are not directly excited by theshear flow� is necessary, whatever the shape is close to a sphere. Not only does this theory cure a quite largequantitative discrepancy between previous theories and recent experiments and numerical studies, but also itreveals a phenomenon: the VB mode band in parameter space, which is believed to saturate after a moderateshear rate, exhibits a striking widening beyond a critical shear rate. The widening results from excitation offourth-order harmonic. The obtained phase diagram is in a remarkably good agreement with recent three-dimensional numerical simulations based on the boundary integral formulation. Comparison of our results withexperiments is systematically made.

DOI: 10.1103/PhysRevE.81.061904 PACS number�s�: 87.16.D�, 83.80.Lz, 83.50.Ha, 87.19.rh

I. INTRODUCTION

A vesicle is a closed simply connected membrane sepa-rating two liquids. Its dynamics in an ambient flow is animportant research target due to various medical applica-tions. For instance, they are employed as biochemical reac-tors �1,2�, as vectors for targeted drug and gene delivery�3,4�, and as artificial cells for hemoglobin encapsulation andoxygen transport �5�.

One can also regard a vesicle, from mechanical point ofview, as a simplified model of a living cell. The most promi-nent biological counterpart for a vesicle is the red blood cell�RBC�. The vesicle system allows to identify the elementaryprocesses of viscoelastic properties of cells moving passivelyin a fluid. The study of a single vesicle under a shear flow isone of the simplest nonequilibrium example. Nevertheless,this problem gives rise to a very complicated dynamics withdifferent types of vesicle motion because of the nontrivialcompetition between the interactions of straining and rota-tional parts of shear flow with the vesicle. Three main typesof motion for a vesicle in a shear flow have been identifiedso far: �i� tank treading �TT�, when shape and orientation ofthe vesicle is a steady state, �ii� tumbling �TB�, when thevesicle makes full rotations, and �iii� vacillating-breathing�VB, aka trembling or swinging�, when the longest axis ofthe vesicle oscillates around certain direction with the shapestrongly changing during these oscillation cycles. Sincemany properties of a vesicle �e.g., the effective viscosity�strongly depend on its type of motion, it is important todefine the phase diagram of a vesicle in a shear flow, pre-dicting which type of motion will be realized for each set ofparameters defining the vesicle dynamics.

A widely accepted model for a vesicle proposes the fol-lowing assumptions: zero Reynolds number limit, membraneimpermeability and local inextensibility, and the continuityof velocity field and mechanical stress across the membrane.The force exerted on the membrane by the liquids is bal-anced by the membrane rigidity force that is calculated fromthe energy of membrane bending taken as �6�

E =�

2� �2H�2dA +� ZdA , �1�

where � is the bending rigidity coefficient of the membrane,H is the mean curvature and Z is a Lagrange multiplier,ensuring the local membrane inextensibility. Under these as-sumptions, and after a certain choice of units for all vari-ables, the vesicle dynamics can be completely described bythree dimensionless numbers �see for example �7��: the vis-cosity contrast

� =�int

�ext, �2�

the excess area relative to a sphere

� = Ar0−2 − 4� , �3�

and the capillary number

Ca =�ext�̇r0

3

�. �4�

Here �int,ext are the viscosities of fluids inside and outside themembrane respectively, A is the surface area of the vesicle,

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r0= � 3V4� �1/3 is the radius of a sphere containing the same vol-

ume as the vesicle, and �̇ is the shear rate. It is convenient tochoose r0 as the unit of distance so that the volume of thevesicle is

V =4�

3. �5�

We also use �ext as the unit of viscosity and �̇−1 as the unit oftime.

Several studies have been recently dedicated to the deter-mination of the phase diagram regarding different types ofmotion of an almost spherical vesicle in a shear flow. Thesestudies were theoretical �8–11�, experimental �12–15�, andnumerical �16,17�. Those theoretical works treat the mem-brane deviation from a spherical shape as a perturbation��=�� is the small expansion parameter�, represented as aseries of spherical harmonics. Then it is suggested that oneor two first orders in the expansion of the shape evolutionequations must be retained. Generally, the resulting equa-tions differ only because the authors propose distinct rules ofneglection. The leading order theory �8� �LOT� keeps onlythe terms of order of the imposed flow and the effects of thevorticity in the final equations. This theory is precise up toO��� in the expansion scheme. Later studies have includedhigher order terms in the expansion �up to O��2��. Lebedev etal. �11� included the next order terms in the membrane Hel-frich force �as compared to LOT�, while another study �here-after called higher order theory �10� �HOT�� accounted forthe next order terms not only in the Helfrich force but also inthe hydrodynamic field as well.

In the theories of Refs. �10,11�, as well as in numericalstudies based on dissipative particles dynamics �17�, a satu-ration of the VB/TB phase border for large enough Ca wassuggested—i.e., the value of � at which the transition fromVB to tumbling occurs becomes almost independent on thecapillary number �or equivalently on shear rate�.

Recent advances in three-dimensional numerical compu-tations based on the boundary integral formulation �16� havemade it possible to study vesicle dynamics quantitatively.The results of this study show, that the loss of stability of theTT solution occurs at values of � significantly higher thanthose predicted by analytical calculations �10,11�. Further-more, no saturation of the VB/TB phase border upon increas-ing shear rate �or Ca� was found. These results provide a newinput that is worth understanding from the analytical pointview.

The main objective of this work is to investigate the rea-sons for the discrepancies between the results provided bythe recent numerical simulations �16� and analytical theories.It was pointed out in Ref. �11� that the critical values of � atwhich the steady-state solution loses its stability, scales asO��−1/2� for a fixed shear rate. However, this behavior,which also agrees with the result of Keller and Skalak �18�,does not provide the correct information about the next orderterms, as will be seen here. The next correction turns out tocause a rather big shift of the phase borders.

This peculiar behavior �i.e., that the critical � diverges atvanishing �� confers to the vesicle problem a special status:

in the small � limit, one further order and only one in theexpansion scheme in powers of � is needed as compared toprevious studies. As a consequence, it will be shown, in par-ticular, that the next order term, previously unaccounted for,survives whatever small the deviation from a sphere is. Thisis the source of deviation between the recent numerical workand the existing analytical theories. As will be seen inclusionof one more order in the expansion will allow us to extract aphase diagram that is in quantitative agreement with the re-sults of numerical simulations �16�. It will also be shown thatfourth-order spherical harmonics �in previous theories onlysecond-order harmonics were included� cannot be neglected,and especially beyond a certain shear rate. Indeed, it will berevealed that these harmonics give rise to a new importantfeature recently revealed in numerical simulations �16�. Moreprecisely, the VB/TB phase border does not saturate for fixed� when shear rate exceeds a certain value, rather a signifi-cant widening is revealed, in contrast to previous theoretical�10,11�, numerical �17�, and experimental investigations�14�.

II. SHAPE EVOLUTION EQUATIONS

A. Basic definitions and the expansion scheme

Here we perform the expansion of the shape evolutionequations one order higher as compared to the most ad-vanced of previous papers �10�. Since a slightly deflatedvesicle in the absence of flow takes a centrosymmetric shape,and the shear flow has a symmetry center in every point, weconsider only symmetric shapes.

Naturally, we take the origin of coordinates at the centerof the vesicle moving with the velocity of the undisturbedshear flow. That the vesicle moves with the same velocity asthe applied shear flow is a consequence of the symmetry ofthe problem. We use the conventional parametrization of thevesicle membrane with reference points belonging to asphere of radius unity

R = x�1 + f�x�� . �6�

The shape function f�x� is then expanded in spherical har-monics of x. Note that the shape depends also on time, butthe temporal variable will be specified only when need be.Following Refs. �9,10�, we introduce a formal expansion pa-rameter =�� which helps classifying different orders. Onlysecond-order harmonics are present to the order of O�� inthe equilibrium shape function of an almost spherical vesiclein the absence of flow. They are also the only harmonics thatcan interact with linear flow directly. For consistency consid-erations, pushing the expansion to next order implies that thefourth and zeroth order harmonics cannot be neglected. Theyenter dynamics as a result of interactions between the vesicleshape and the Helfrich force and the flow; they are of orderO�2�. The shape function does not contain spherical har-monics of odd orders owing to the centrosymmetry. Thespherical harmonics of even orders higher than 4 have anamplitude O�3� and thus are neglected. As follows from theinequality

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1 =3V

4�=� R�x�3d2x

4� �� R�x�

d2x

4��3

= R03

the average radius of the vesicle R0 is not exactly 1. There-fore a negative correction must be added to it in order tosatisfy Eq. �5�. This correction is of order O�2� and leads toa nonzero coefficient for zero-order spherical harmonic in theexpansion of f�x�, which will be denoted f0 below.

The expansion takes formally the form

f�x� = 2f0 + f2 + 2f4 + O�3� ,

fk = l=−k

k

fk,l�t�Yk,l�x� . �7�

There are various definitions of spherical harmonics, herewe use those that satisfy the following normalization condi-tions,

� Yk,l�x�Yk,l� �x�d2x =

4��k!�2

�2k + 1��k + l�!�k − l�!.

However, all equations are written in universal form and arevalid for any rescaling of spherical harmonics.

The condition �Eq. �5�� together with the definition �=2 provide two additional constraints on the coefficients ofthe expansion �Eq. �7��. We use the former to express f0through f2,l

� f0�x�d2x = −� f2�x�2d2x −

3� f2�x�3d2x + O�2� �8�

and the latter to norm the coefficients f2,l

2� f2�x�2d2x −2

3� f2�x�3d2x = 1 + O�2� . �9�

We retain one more order in these expressions compared toprevious works for the sake of consistency. Note that thevolume and the surface have been expanded to the orderO��3� in order to obtain the equalities �8� and �9�.

B. Derivation of the evolution equations of the fk from theboundary integral formulation

We shall adopt here a different spirit from that of previoustheories �8–11�. There the Lamb solution is used �a solutionof the Stokes equations inside and outside the vesicle�. Wefind it convenient to take the boundary integral formulationas a starting point. The present spirit is equivalent to usingthe Lamb solution, but it has the advantage that the boundaryconditions at the membrane are already implemented in theboundary integral formulation. Indeed, the Stokes equationstogether with the boundary conditions on the membrane, andfar away from it, can be converted into boundary integralformulation �19�. This leads to the following integral equa-tion:

vl�x��1 + ��

= 2ul�x� + 2� Gjl�R�x� − R�x���Fj�x��d2R�x�� + �1 − ��

�� Kjlm�R�x� − R�x���v j�x��Nm�x��d2R�x�� . �10�

Here v�x� is the actual velocity of point R�x�, u�x� is thevelocity of point R�x� in the imposed flow, F�x� is the mem-brane rigidity force at point R�x�, N�x� is the outward point-ing normal to the vesicle surface at point R�x�. The integra-tion is taken over the surface of the vesicle and d2R�x�� is thesurface area element. The integration kernels have the fol-lowing form:

Gij�R� =1

8��ij

R+

RiRj

R3 �, Kijk�R� =3

4�

RiRjRk

R5 .

Using the expression for the Helfrich force, we can find fromEq. �10� the velocity field on the surface of the vesicle. TheHelfrich force is given in terms of the shape as

Fi = − 2���2H�H2 − K� + �SH� − ZH Ni +�Z�x��Rj�x�

��ij − NiNj� ,

�11�

where K is the Gaussian curvature and �S is the Laplace-Beltrami operator. It can easily be checked that expression�11� vanishes for a spherical shape, and thus is of order O��.In order to balance Eq. �10� at order O�1� we need to assumethe imposed flow to be of the same order as Eq. �11�. Thisrequirement can equivalently be fulfilled, following Refs.�8,10�, by the demand that � scale as −1. We then set �=−1�̄, with �̄ of order O�1�.

The precise technical details will be given elsewhere,while here we shall present only the spirit and some interme-diate steps. We expand v�x�, u�x�, and F�x� into vectorspherical harmonics of x. For convenience we define them as

Y1,k,li �x� = eimnxm�nYk,l�x� ,

Y2,k,li �x� = �2k + 1�xiYk,l�x� − �iYk,l�x� ,

Y3,k,li �x� = �iYk,l�x� .

Here the differentiation with respect to xi is taken formally asif x were a regular three-dimensional �3D� vector not boundto a sphere of radius unity, and eimn is the Levi-Civita �unitantisymmetric� tensor. The advantage of such a definition isthat not only do these spherical harmonics constitute an or-thogonal set with respect to the integration of dot productover x, i.e., the quantity

� Y j,k,li �x�Y j�,k�,l�

i �x��d2x

is zero unless j= j�, k=k�, l= l�, but also operators G and Kare diagonal in the chosen basis for a spherical vesicle, sothat the integrals

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� � Y j,k,li �x�Gim�x − x��Y j�,k�,l�

m �x���d2xd2x�,

� � Y j,k,li �x�Kimn�x − x��Y j�,k�,l�

m �x���xnd2xd2x�,

are zero unless j= j�, k=k�, l= l�. Given the centrosymmetrywe impose, in our expansion of v�x�, u�x�, and F�x�, theabsence of Y1,k,l

i �x�, for even k, and Y2,k,li �x�, and Y3,k,l

i �x�, forodd k. In order to find the evolution equations we need toexpand the velocity field to the order of O�2�. Coefficientsfor Y j,k,l

i �x� for k 6 are O�3� and thus can be neglected. Itwill be shown later that coefficients of fifth and sixth ordersof vector spherical harmonics make corrections to the evolu-tion equations for fk,l with k�4 which are O�3�. So onlyfive first orders �starting from the zeroth one� of vectorspherical harmonics should be taken into account in the ex-pansion of any space-dependent quantity under consider-ation.

In order to fulfill the condition of local inextensibility ofthe membrane we impose zero surface divergence of the ve-locity field �8�.

dA�x�dt

=�vi�x��Rj�x�

��ij − Ni�x�Nj�x��dA�x� = 0. �12�

Here

�vi�x��Rj�x�

=�vi�x�

�xk

�xk

�Rj�x�

is the Jacobian matrix. We can take the derivatives as if xwere a regular 3D vector, because the surface divergence isfully determined by the distribution of the velocity field onthe membrane and does not depend on the continuation ofthe flow into the liquids.

We expand Eqs. �10� and �12� to order O�2� and theresulting integrals could be performed analytically. Project-ing the results of the integration on the space of vectorspherical harmonics up to the fourth order, we obtain a set ofequations satisfied by the coefficients entering the expan-sions of v�x� and Z�x�. Since the surface area is evaluated upto order O�3�, while the velocity field in Eq. �12� is ex-panded only up to order O�2�, it is not appropriate to useEq. �12� in order to ensure the conservation of the wholesurface area. Therefore, we only use the projections of Eq.�12� on Yk,l�x� for k equal to 2 or 4. Accordingly, we leavethe isotropic part of Z�x� �denoted here as Z0� undetermined.Once the final shape evolution equations are obtained weshall use the constraint that the time derivative of Eq. �9� isequal to zero in order to determine Z0. This way of reasoningwas used in previous studies �8–11�. Note that unlike othertheories �8–11�, since we expand the equations to higher or-der, it is not legitimate to replace dA�x� in Eq. �12� with d2xprior to projection on the spherical harmonics subspace.Such a substitution would imply neglecting terms of orderO�2� in final equations, and would be inconsistent with thespirit of the present theory.

Having determined the velocity field thanks to the aboveexpansions, we are in a position to obtain the final evolution

equation by making use of the kinematic equation expressingthe fact that the membrane velocity is equal to the fluid ve-locity at the membrane

� f�x��t

= vi�x�xi −�i f�x�v j�x���ij − xixj�

R�x�. �13�

Then the task is to substitute the expanded velocity field intothis equation �in terms of coefficients of the velocity field�and project the resulting expression onto the space of spheri-cal harmonics of interest. This then leads to the determina-tion of the evolution equations that must be satisfied by theshape coefficients fk,l. It can be shown that the followingidentities hold Y1,5,l

i �x�xi=0, Y2,6,li �x�xi=7Y6,l�x�, and

Y2,6,li �x�xi=6Y6,l�x�. Therefore the projection of the first part

of the right-hand side of �13� on the subspace of sphericalharmonics of orders up to four does not depend on the coef-ficients of the vector spherical harmonics of orders five andsix in the velocity field. Regarding the second part, �i f�x� isof order O�� and the coefficients for fifth and sixth orders ofvector spherical harmonics in the velocity field expansion areof order O�2� so that their contribution in the shape evolu-tion equation is of order O�3�, that is beyond our accuracy.

It is convenient to decompose the applied shear flow intoits elongational and rotational parts to simplify the finalequations. The same decomposition can be applied to generallinear flow: the quantity E2�x�=Ui�x�xi /2 defines the strain-ing part, while the vector �i=eijk� jUk�x� /2 represents thevorticity. Note that u�x� in Eq. �10� is the velocity of theimposed flow evaluated at the point R�x�, which takes thefollowing form:

ul�x� = ��lE2 + eljkxj�k��1 + f�x�� .

The shape evolution equations for the second- and fourth-order harmonics can be written in a compact form

Df2,l

Dt=� F2�x�Y2,l�x��d2x

� Y2,l�x�Y2,l�x��d2x

, �14�

2Df4,l

Dt=� F4�x�Y4,l�x��d2x

� Y4,l�x�Y4,l�x��d2x

, �15�

The left-hand side term of the equation is a special deriva-tive of fk,l that naturally arises in a nonrotating coordinatesystem, and its definition is

Dfk,l

Dt=

� fk,l

�t+� eijm�i fkxj�mYk,l�x��d2x

� Yk,l�x�Yk,l�x��d2x

. �16�

To the order at which our expansion is performed, the rota-tional velocity of the vesicle is not equal to the vorticity ofthe imposed flow �unlike lower order calculations �8–11��,

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but rather it has a new contribution originating from theshape function

� j = � j + ejkl�ikE2�il f2

2+ O�2� . �17�

Note that if only the first term �i is retained �8–11�, then Eq.�16� coincides with the Jaumann derivative.

The functions Fk can be written as

F2 = a1f2 + a2f22 + 2�a3f2

3 + a4f2f4� + b1E2 + b2E2f2

+ 2�b3f2�i f2�iE2 + b4�i f2�ij f2� jE2 + b5�i f2�ijE2� j f2

+ b6E2f4� + O�3� ,

F4 = �c1f4 + c2f22� + 2�c3f2

3 + c4f2f4� + d1E2f2

+ 2�d2f4E2 + d3f22E2 + d5f2�iE2�i f2� + O�3� .

The coefficients ai, bi, ci, and di are rational functions of �,�̄= /Ca, and Z0. The exact expressions are listed in the Ap-pendix.

III. PHASE DIAGRAM

A. Analytical phase diagram and comparison with previousworks

The phase diagram for �=0.43 is presented on the Fig. 1.The value �=0.43 is chosen for the sake of comparison withavailable numerical data �16�. Also shown on that figure arethe results obtained in two previous theories �10,11�. As canbe seen the basic structure of the phase diagram bears simi-larity with previous ones. There are however some importantdistinctions. First, the phase borders are significantly shiftedtoward the region of higher viscosity contrasts. Second, theVB/TB transition curve does not saturate upon increasingshear rate �or Ca�, rather it exhibits a striking widening. Wehave also compared the results with those obtained recentlyby full three-dimensional simulations. Remarkable agree-ment between the two approaches is found �see comparisonin Ref. �16��.

B. Basic reasons for necessity of higher order expansion

Our first concern is to understand why the inclusion of theterms of order O�2� provides such a dramatic shift of thephase borders even though the shape is almost spherical �therelative excess area is only 0.43 / �4��=0.034�. We havestudied the evolution of the phase diagram as a function of �in order to gain further insight. We have tracked the viscositycontrast � at which the loss of stability of TT motion occursin the two asymptotic limits Ca→0 �the corresponding valueis denoted as �c0� and Ca=� �denoted as �c��. It was re-ported in �11� that �c�−1. Since we expect the expansionsof �c to be analytic in , we have thus attempted the follow-ing ansatz:

�c = �c�−1�−1 + �c

�0� + O�� . �18�

To check the validity of this expansion we plot these criticalvalues as a function of �−1/2=1 /. We expect then a linearbehavior. This is presented on Fig. 2 where we also compareour results with those of previous theories. A key point to bediscussed further below is the fact that previous theories pro-vide correct values for the dominant term �c

�−1� in the expan-sions �Eq. �18��, but not for �c

�0� which does not vanish evenin the limit of almost spherical vesicles �note that the nextorder terms in the expansion Eq. �18� tend to 0 with theexcess area�. Because �c

�0� remains finite whatever small isthe deviation from a sphere, the discrepancy between the TTphase borders obtained by different theories �with one lowerorder below the present one, namely, of the order of O���persists for any excess area. In contrast the present theoryvalid to order O��2�, has the property that even if one wishesto make an expansion to the next order �i.e., order O��3��,then the shift in the borders in the phase diagram would benegligibly small, i.e., the shift would vanish in the smallexcess area limit. In other words, the present theory showsthat there is a convergence of the small deformation schemeat order O��2�. This implies that there is no need to continuethe expansion of the shape evolution equation beyond theorder O�2� since this would neither notably modify the re-sults for ��0.43 nor significantly extend the applicability tohigher values of �. Comparison of the present theory withnumerical results �see comparison provided in Ref. �16��shows good agreement for �=0.43 �the only value exploredto date by the numerical scheme�.

0 2 4 6 8 10Ca

0

5

10

15

λ

Present TheoryHigher Order TheoryLebedev et al. Theory

TB

VB

TTλc0

λc∞

FIG. 1. Continuous lines represent the phase diagram for �=0.43, the results of the higher order theory �10� and Lebedev et al.theory �11� are added in dashed and dotted lines for comparison.The TT phase borders are almost indistinguishable for last twotheories.

0 2 4 6 8 10∆-1/2

0

10

20

30

40

50

λ0

Higher Order TheoryKeller-Skalak TheoryPresent Theory

0 2 4 6 8 10∆-1/2

0

10

20

30

40

50

λ∞

Present TheoryHigher Order Theory

FIG. 2. �0 and �� as a function of �−1/2.

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C. Discussion of the tank-treading phase borders

As can be seen the dependencies shown on Fig. 2 arefairly linear for ��1 and are almost indistinguishable fromstraight lines for ��0.5. The region of 1 /�� where thecurves noticeably deviate from straight lines is wider forpresent theory than for HOT. This is most likely caused bythe introduction of fourth-order spherical harmonics, theiramplitude is not small as compared to that of the second-order harmonics for � 1. The expression �9� becomes in-applicable, and as a result f4,l�t� grow uncontrollably withtime in solutions of the differential equations. This problemcan be fixed by expanding the excess area to the order O�4�,but the results of such patching are not trustworthy becausethey go beyond the precision of the differential equations.

Note also that the second term in Eq. �17� tends to alignthe vesicle to the elongating direction of the straining part ofthe shear flow which makes the angle � /4 with the directionof the flow. In the TT phase close to the loss of the stabilityof the steady-state solution the vesicle is almost parallel tothe flow. The effective vorticity �Eq. �17�� turns out to be lessthan � in the TT phase, and it leads to the increased stabilityof the steady-state solution. The correction to the vorticity isproportional to and thus increases with � and dominatesover � in the large excess area limit. As a consequence, thetruncation �Eq. �17�� becomes illegitimate. Generally, wemay assert that ��1 is a good estimate for the applicabilitylimits of the small deformation approximation.

Some theories �8,11,18� allow for analytical extraction ofthe coefficients �c

�−1� and �c�0�, and for HOT and the present

theory the extraction is numerical by using the slope andoffset of the lines on the Fig. 2. The combined data arepresented in the Tables I and II.

HOT and Lebedev et al. theories show good agreementfor the coefficients �c

�−1� with the present calculation. Thismeans that expansion of the shape evolution equations to theorder O�� was indeed sufficient in order to capture the cor-rect value of �c

�−1� even for large values of Ca, where Leb-edev et al. theory loses its applicability. The present theoryshows that the correct value of �c

�0� is captured, thanks to thefact that the expansion scheme is pushed one step further inthe excess area. Keller-Skalak theory gives a slightly differ-ent result for �c

�−1�, the difference is caused by the fact theyused a velocity field having a surface divergence of orderO��, whereas the membrane local inextensibility requireszero surface divergence instead. On the other hand Keller-

Skalak theory provides a better estimate for the value of �c�0�

than the one proposed by HOT and this explains why itsresults are relatively close to the results of numerical simu-lations for ��1.

D. Discussion of TB/VB transition

We plan to dedicate a separate research to the propertiesof different types of vesicle motion, so here we only brieflydiscuss further implication of the new theory on the TB/VBtransition. Neglecting harmonics of orders higher than two,as done in previous theories, provides two major simplifica-tions. First, the vesicle shape has three symmetry planes,which make the definition of the orientation angle obvious,and second the in-plane motion is defined by only two inde-pendent variables. As a consequence, the dynamics relaxeswith time to a cyclic motion �which degenerates into a pointin the TT phase�. Thus only a simple limit cycle can exist,and this shows that each of the VB and TB modes has itsown region of existence in parameter space. The above as-sertions cannot be made in general, and especially when thefourth-order harmonics are included. It also turns out thatnear the transition region the vesicle tends to finish its TBquasicycles by assuming an oblate almost axisymmetricshape in the shear plane rather than an elongated shape per-pendicular to the flow. For such shape the error in the defi-nition of the orientation angle is very large.

Let us now discuss how each type of motion is deter-mined from our evolution equations. We start with some ini-tial values for fkl�t� and wait a certain time interval until theinitial data are irrelevant. Then it was checked whether dur-ing one oscillation quasicycle f�x� has a maximum for xlying in the shear plane and perpendicular to the shear ve-locity. Despite some noise due to the aforementioned com-plications it is clearly seen that unlike with previous theoriesthe VB/TB phase border does not saturate even for Ca=10,and the VB region broadens with the increase of the capillarynumber �Fig. 1�.

Note that it was suggested recently �20� that higher ordersof spherical harmonics can be excited close to the VB/TBphase border and this may cause some widening of the VBphase region �but still saturation at large Ca is found, unlikeour theory�. It can be checked that when Z0+20� /Ca�0, thefourth-order harmonics are excited �i.e., the correspondingdecay time becomes infinite to leading order�. Here, wefound that close to the VB/TB transition at some times dur-ing the oscillation the quantity

TABLE I. �c0�−1� and �c0

�0� extracted from different theories. Itshould be noted that Lebedev et al. �11� theory provided only val-ues for �c0

�−1� and we have formally extracted from their theory thevalue of �c0

�0� for comparison purpose.

Theory �c0�−1� �c0

�0�

LOT �8� 823

�30��3.38� − 3223�−1.39�

HOT �10� 3.91 −1.22

Lebedev et al. �11� 1623

�10��3.90� − 3223�−1.39�

Keller and Skalak �18� 23�10��3.74� 73

63�1.17�Present 3.89 1.54

TABLE II. �c��−1� and �c�

�0� extracted from different theories. Itshould be noted that Lebedev et al. �11� theory provided only val-ues for �c�

�−1� and we have formally extracted from their theory thevalue of �c�

�0� for comparison purpose.

Theory �c��−1� �c�

�0�

LOT �8� 823

�30��3.38� − 3223�−1.39�

HOT �10� 4.79 −1.38

Lebedev et al. �11� 1623

�15��4.78� − 3223�−1.39�

Present 4.76 1.13

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−Z0Ca

�19�

exceeds 20. This is probably the reason why the VB/TBphase border remains unsaturated for much larger valuesthan in previous analytical studies. It was proposed in �20�,however, that excitation of higher harmonics was due to ther-mal fluctuations. This contrasts in spirit with our theorywhere the fourth-order harmonic is excited as nonlinear in-teraction of second harmonics and no reference to tempera-ture is required.

IV. COMPARISON WITH EXPERIMENTS

Experiments have been performed recently regarding de-termination of the phase diagram of vesicle motions undershear flow �14�. They referred to a previous theoretical work�11� which had suggested that the phase diagram should de-pend on only two independent dimensionless control param-eters �and not three�, namely,

� =4

�30�1 +

23

32����, S =

7�

3�3

Ca

�. �20�

It has been reported �14� �with a certain degree of uncer-tainty� that the experimental data were consistent with thefact that only the above two parameters determine the phasediagram. Our results do not comply with this report, asshown on Fig. 3. Indeed, besides S and �, the excess area �plays an important role. Experiments mixed data for different�’s in the plane �S ,��. The band of the VB mode in experi-ments looks quite wide, and we believe that this reflects thesensitivity of the location of the VB band to excess area �inother words the experimental data may be viewed as juxta-position of bands each representing a value of �; see theexample of Fig. 3�. It is hoped that a study representing eachvalue of � will be performed in the future with the aim ofmaking comparison with theory clearer.

Furthermore, it must be noted that our estimates for theTT-VB phase borders are higher than those observed in ex-periments. This is not very surprising since transients arefound to be very long close to TT-VB transition �as alsodiscussed in our recent full numerical simulations �16��, and

therefore a firm conclusion on the type of motion cannot bemade on the basis of the current experimental data, which arelimited to only few periods of oscillation �while transient canexhibit up to hundreds of cycles as discussed recently �16��.In other words, this kind of long relaxation would convey theimpression that dynamics is of VB type, whereas in reality itis a TT one.

Other experiments �15� were performed under linear flowwith applied velocity field ux= �s0+�0�y, uy = �s0−�0�x, uz=0, which is different from a simple shear flow. The experi-ments were performed with no viscosity contrast and theratio of the rotational ��0� and the straining �s0� parts of theflow was varied instead. The results of the experiments wereplotted in coordinates

�̃ =4�0

�30�s01 +

23

32����, S̃ =

14�

3�3

s0

���21�

representing the generalization of Eq. �20�. It was thenclaimed �15� that the phase diagram depends on these twoparameters only �at least with no viscosity contrast�. The Fig.4 simulates this experiment using the present theory. Thediscrepancy between resulting phase diagrams for variousvalues of � is indeed not as striking as for the simple shearflow �Fig. 3�: the TT phase borders fall close to each otherand VB bands overlap for a wide range of �. At the sametime, the VB/TB phase borders still vary significantly withthe excess area. However, by fixing � and varying � we can

produce series of phases diagrams in the S̃ and �̃ plane cor-responding to different values of �0 /s0, �see Fig. 5�. Here wefind the same kind of discrepancy as on Fig. 3.

Finally, let us compare our results with other set of ex-periments in the TT regime �13�. We represent the vesicleorientation angle �0 under strong shear flow �Fig. 6� �for thesake of comparison with experiments which were performedat high shear rates�. It can be checked that the exact form ofthe bending energy becomes insignificant under strong flowsbecause the dominant contribution to the membrane forcecomes form the tension part which enforces incompressibil-ity of the membrane. Unlike for the VB/TB phase border, the

0 2 4 6 8 10 12 14 16 18 20S

1

2

3

Λ

∆=0.16∆=0.43∆=0.8Lebedev et al. Theory

FIG. 3. Phase diagrams for various values of � in S−�coordinates.

0 5 10 15 20S~

1

2

2.5

1.5

Λ

Present Theory, ∆=0.43Present Theory, ∆=1Lebedev et al. Theory

~

FIG. 4. Phase diagrams of vesicle motions under general planarlinear flows. The viscosity contrast � is fixed to 1 and the ratio ofrotating and straining parts of the flow �0 /s0 is varied. The theoryof Lebedev et al. �11� is plotted in dotted lines for comparison.

ANALYTICAL PROGRESS IN THE THEORY OF VESICLES… PHYSICAL REVIEW E 81, 061904 �2010�

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dependence of �0 on � quickly saturates with Ca. We takeCa=100 in our calculation. We use the artificial parameter ��Eq. �20�� instead of � in order to show the effect of O��2�terms taken into account by the present calculation. Note thatthe results of Lebedev it et al. theory �11� do not depend onthe excess area � in this representation, unlike our resultsand those reported by experiments �see Fig. 6�. The results ofHOT depend also on �, but that dependence is weak so thatthey hardly differ from theoretical results of Ref. �11� �we donot plot them here in order to avoid encumbering of thefigure�. Experimental results �13� are provided for four dif-ferent values of �: 0.15, 0.24, 0.42, and 1.43. We exclude�=1.43, since we do not expect our theory to be applicableto a such large value of excess area. As can be seen, theoret-ical results and experimental ones show a rather good agree-ment provided that one is not close to the TT/VB border �i.e.,if the angle is not too close to zero�. The discrepancy forsmall inclination angles may be attributed to thermal fluctua-tions or interactions with walls. However, a systematic theo-retical study of these factors should be performed beforedrawing conclusive answers.

V. DISCUSSION AND CONCLUSION

Here we have presented a small deformation theory of avesicle in shear flow keeping one more order in the expan-

sion of the shape evolution equations than prior studies. Wehave confirmed the leading term in the asymptotic expansionof critical viscosity ratios determined by previous analyticalstudies, but we also were able to determine accurately thenext term in the expansion. This term is constant and sur-vives no matter how small the excess area is. Moreover, thistheory provides a correction to the phase borders that is sig-nificant in a wide range of excess areas. Unlike previousanalytical works, but in agreement with the results of thenumerical simulations �16�, we observe an unsaturatedgrowth of the VB/TB phase border in a quite large range ofthe capillary number.

We hope that this work will incite future experimentalresearch. It will also be interesting to investigate experimen-tally the presence of widening of the VB band as a functionof shear rate. Finally, we did not include third-order sphericalharmonics in the expansion of the shape function, restrictingour calculations only to centrosymmetric vesicles. In the fullnumerical simulation �16�, no symmetry restriction is im-posed and so far no manifestation of the third-order har-monic has been observed. This confers to our assumption ofcentrosymmetry a certain legitimacy.

ACKNOWLEDGMENTS

We would like to thank G. Danker, S. S. Vergeles, and P.M. Vlahovska for helpful discussions. A.F. and C.M. ac-knowledge financial support from CNES �Centre NationalD’études Spatiales� and ANR �Agence Nationale pour la Re-cherche�; “MOSICOB project.”

APPENDIX: THE SHAPE EVOLUTION EQUATIONS ANDTHE EXPRESSION OF VARIOUS COEFFICIENTS IN

TERMS OF PHYSICAL PARAMETERS

We remind that

F2 = a1f2 + a2f22 + 2�a3f2

3 + a4f2f4� + b1E2 + b2E2f2

+ 2�b3f2�i f2�iE2 + b4�i f2�ij f2� jE2 + b5�i f2�ijE2� j f2

+ b6E2f4� + O�3� ,

F4 = �c1f4 + c2f22� + 2�c3f2

3 + c4f2f4� + d1E2f2

+ 2�d2f4E2 + d3f22E2 + d5f2�iE2�i f2� + O�3� .

The coefficients are then written as

a1 = − 24Z0 + 6�̄

23� + 32,

a2 = 24�49� + 136�Z0 + �432� + 1008��̄

�23� + 32�2 ,

b1 =120

23� + 32,

b2 = 2400� − 2

�23� + 32�2 ,

0 5 10 15 20S

1

2

2.5

1.5

3

Λ

Present Theory, ω0=1/2 (simple shear flow)

Present Theory, ω0=3/4

Present Theory, ω0=5/4

Lebedev et al. Theory, arbitrary ω0

∆=0.43, s0=1/2

~

~

FIG. 5. Phase diagrams of vesicles under general planar linearflows. The straining and rotational parts of the flow are denoted ass0 and �0 respectively.

0 0.5 1 1.5 2Λ

0

0.6

0.5

0.4

0.3

0.2

0.1

φ 0

Experiment, ∆=0.15Experiment, ∆=0.24Experiment, ∆=0.42Present Theory, ∆=0.15Present Theory, ∆=0.24Present Theory, ∆=0.42Lebedev et al. Theory

TT inclination angle for Ca=100

FIG. 6. inclination angle of vesicles in TT phase for large Ca.

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b6 = − 40241� + 344

�23� + 32�2 ,

c1 = − 40Z0 + 20�̄

19� + 20,

c2 =16

3

�92 − 3��Z0 + �1822� + 3112��̄�19� + 20��23� + 32�

,

d1 =20

3

1047� + 1072

�23� + 32��19� + 20�,

d2 = 10257� − 32

�23� + 32��19� + 20�,

d3 =1

6

− 125 055�3 + 594 716�2 + 1 107 168� + 392 320

�2� + 5��23� + 32��19� + 20�2 ,

a3 = −8

175

�2 479 595�3 + 6 703 156�2 + 18 601 472� + 16 622 592��23� + 32�3�19� + 20�

Z0

−8

175

78 181 390�3 + 390 845 256�2 + 713 624 832� + 429 094 912

�23� + 32�3�19� + 20��̄

a4 = 4�18 335�2 + 57 376� + 44 224�Z0 + �318 896�2 + 1 026 064� + 809 600��̄

�19� + 20��23� + 32�2

b3 =1

70

78 420 885�4 + 815 632 786�3 + 2 193 572 112�2 + 1 954 954 752� + 481 607 680

�2� + 5��19� + 20��23� + 32�3

b4 = −1

630

490 295 475�4 + 3 878 418 742�3 + 9 801 602 064�2 + 9 403 966 464� + 2 954 997 760

�2� + 5��19� + 20��23� + 32�3

b5 =1

315

136 559 745�4 + 385 825 454�3 + 141 580 368�2 − 26 564 352� + 98 232 320

�2� + 5��19� + 20��23� + 32�3

c3 = −4

63

2 552 583�3 + 1 777 744�2 − 913 584� + 315 392

�23� + 32�2�19� + 20�2 Z0

−16

63

32 203 871�3 + 118 114 286�2 + 154 157 080� + 69 803 008

�23� + 32�2�19� + 20�2 �̄

c4 = 2�21 383�2 + 63 844� + 44 928�Z0 + 48�13 411�2 + 35 444� + 23 040��̄

�23� + 32��19� + 20�2

d4 =1

42

41 587 815�4 + 95 846 332�3 − 88 522 016�2 − 295 841 536� − 152 842 240

�2� + 5��23� + 32�2�19� + 20�2

ANALYTICAL PROGRESS IN THE THEORY OF VESICLES… PHYSICAL REVIEW E 81, 061904 �2010�

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