Squaring, Parity Breaking, and S Tumbling of Vesicles under Shear Flow

Alexander Farutin* and Chaouqi Misbah†

Laboratoire Interdisciplinaire de Physique, UMR5588, Universite Grenoble I CNRS, Grenoble F-38041, France(Received 8 August 2012; published 10 December 2012)

The numerical study of 3D vesicles with a reduced volume equal to that of human red blood cells leads

to the discovery of three types of dynamics: (i) squaring motion, in which the angle between the direction

of the longest distance and the flow velocity undergoes discontinuous jumps over time, (ii) spontaneous

parity breaking of the shape leading to cross-streamline migration, and (iii) S tumbling where the vesicle

tumbles, exhibiting a pronounced S-like shape with a waisted morphology in the center. We report on the

phase diagram within a wide range of relevant parameters. Our estimates reveal that healthy and

pathological red blood cells are also prone to these types of motion, which may affect blood micro-

circulation and impact oxygen transport.

DOI: 10.1103/PhysRevLett.109.248106 PACS numbers: 87.16.D�, 87.19.U�, 47.15.G�, 47.60.Dx

The study of red blood cells (RBCs) under flow is anessential step towards elucidating a plethora of questions inhemodynamics. Indeed, understanding RBC response toflow stresses should help to improve knowledge of bothmacro- and microcirculation of blood flow. As thisresponse depends on the healthy or pathological characterof the cells, this knowledge can also serve to developefficient microfluidic architectures for cell sorting, diag-nosis, etc. Furthermore, RBCs flowing in the humancirculatory system or in the bypass system of corporealcardiovascular devices (such as artificial blood pumps orheart valves) can suffer high stresses that may severelyalter their mechanical properties. This can lead to RBCmembrane damage or ultimately hemolysis (release ofhemoglobin into plasma due to rupture caused by hydro-dynamical forces), causing serious health disorders, suchas jaundice and pulmonary hypertension. Therefore, it ishighly desirable to explore the response of RBCs, or theirbiomimetic analogs, to flow conditions that are likely tooccur in real situations.

We shall keep the model as simple as possible (althoughit remains complex at the absolute level). This will allow usto draw a clear picture with a minimum number of parame-ters. Our model is a 3D vesicle, a closed membrane madeof a phospholipid bilayer (a simple model for RBCs). Thestudy of vesicles under flow continues to attract increasinginterest. Vesicles share several features with RBCs. Forexample, tank treading, tumbling, breathing, and swinging(or vacillating) are observed in both systems under shearflow [1,2]. Under Poiseuille flow, parachute and slippershapes (adopted by RBCs in the microvasculature) [3] areexhibited by vesicles [4], capsules [5,6], and RBCs [7,8].Apart from the fact that they show similar dynamics toRBCs, vesicles have various applications in industry andmedicine, being employed as containers for biochemicalreactions and the transport and delivery of samples [9,10],as vectors for targeted drug and gene delivery [11,12], and

as artificial cells for hemoglobin encapsulation and oxygentransport [13].Direct 3D numerical simulations of vesicles under shear

flow have been carried out recently [14,15] and havefurther quantified the regions of existence of three majortypes of motions [16–21]: tank treading (TT), vacillatingbreathing (VB) (sometimes called trembling or swinging),and tumbling (TB) (or kayaking, when the orbit is out ofthe shear plane). This diagram was obtained for vesicleswhose shape is quasispherical. However, the study ofhighly deflated vesicles has remained a challenge due tonumerical complexity, in particular.We report here that vesicles, with a reduced volume

equal to that of RBCs, reveal shapes and dynamics thatmarkedly differ from traditional ones. Three major modesare discovered, namely, (i) squaring (SQ), (ii) S tumbling(STB), and (iii) parity breaking (PB). We also found thatother dynamics can take place, characterized by out-of-plane motions, with various degrees of complexity. Thewhole set of dynamics was analyzed over a wide range ofparameter space.Problem formulation.—A vesicle is subject to a linear

shear flow V1 ¼ ð _�y; 0; 0Þ. The fluids outside (viscosity�) and inside (viscosity ��, where � is the viscositycontrast) the vesicle are described by the Stokes equationswith the following boundary conditions at the membrane:(i) stress balance, (ii) continuity of velocity at the mem-brane, and (iii) membrane inextensibility. This set of equa-tions can be converted into a boundary integral formulation[22,23]. The membrane acts on the surrounding fluid with aforce [14] which arises from the (functional) derivative ofthe Helfrich energy E ¼ R½�2 ð2HÞ2 þ Z�dA, where � is the

bending rigidity modulus, H is the mean curvature, dA isthe area element, and Z is a Lagrange multiplier whichenforces local membrane area conservation. Z is deter-mined from the condition of zero surface divergence ofthe velocity field.

PRL 109, 248106 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

0031-9007=12=109(24)=248106(5) 248106-1 � 2012 American Physical Society

The problem is described by three dimensionless num-bers: (i) the viscosity contrast �, (ii) the capillary numberCa ¼ _�R3

0�ext=�, measuring the ratio between viscous

stress and bending stress, where R0 is the radius of a spherewhose volume V equals that of the vesicle, and (iii) theexcess area of a sphere, �, defined as A ¼ R2

0ð4�þ�Þ,(A is the membrane area) characterizing the degree ofdeflation of thevesicle.We can alternatively use the reduced

volume � ¼ 3ffiffiffiffiffiffiffi4�

pVA�3=2, which is related to � by � ¼

4�½��2=3 � 1�. For a human RBC, � ¼ 0:6. Handling alarge excess area (� ¼ 0:6) turned out to be quite challeng-ing from a numerical point of view. Here, we mention onlythe essential features of the adopted numerical procedure:Membrane force was calculated by quadratic interpolationof the surface; exact identities were used to regularizeboundary integrals; the equation expressing the conditionof membrane inextensibility was solved by a modificationof the steepest descent method in order to determine thetension field Z; and finally, the quality of mesh triangleswas stabilized thanks to slight displacement of the meshpoints along the surface at each time step.

Results.The three classical dynamics: TT, VB, and TB.—The

three classical modes of dynamics were investigated first.For a small enough capillary number (Ca � 2), we foundthat the vesicle exhibited TT for � < 3:5. Beyond thisvalue, and for Ca � 1, TT underwent a saddle-node bifur-cation towards TB, in accordance with classical results. Fora higher Ca (say, for Ca ¼ 2), the TT regime lost itsstability at � ’ 4 and underwent a VB dynamic. Thebreathing mode is quite ample in this study, due to the

large excess area, as shown in Fig. 1. A movie is providedin the Supplemental Material [24]. We introduced theangle, denoted c , between the direction of the longestdistance from the vesicle center and the flow velocity.Figure 2 displays c as a function of time. A remarkablefeature was discovered at a high value of Ca (� 50), wherethe VB mode underwent transverse instability leading toout-of-plane motion. This result was not observed forquasispherical vesicles and will be described in more detailin the section devoted to the full phase diagram.SQ.—Hitherto, the VB mode was known to transform

into TB mode if � was high enough. Here, the situationturned out to be rather complex. Upon increasing � (forCa> 2), TT motion first becomes unstable in favor of VB.For higher values of �, the VB mode undergoes significantmorphological changes. Indeed, during a certain time in-terval, the vesicle exhibits a squarelike shape (we call thismorphology change SQ); see Fig. 3. This change is notonly visual but can be rigorously defined: Indeed, the anglec undergoes sudden discontinuous jumps over the courseof time, in marked contrast with the VBmode, in which theangle evolves continuously (see Fig. 2). As shown in Fig. 3,the distance from the vesicle center to the membrane hasseveral local maxima. As time elapses, one of the localmaxima becomes global, which leads to an abrupt jump ofc . A movie of the SQmode is presented in Ref. [24]. If � isincreased further, the SQ mode transforms into TB. TBbehaves in a classical manner as long as Ca remains smallenough (Ca � 2). However, the dynamic changes drasti-cally as Ca increases.STB.—For a larger Ca (� 5), successive transitions from

TT to VB to SQ and then to TB are observed as � increases.However, this tumbling mode is quite different from theusual one: When the vesicle’s long axis is oriented alongthe direction of flow (or close enough to that direction), thevesicle assumes a cigarlike shape. Conversely, when thelong axis is close to the maximal extension direction(c ¼ �=4) of the flow, the vesicle exhibits a quite pro-nounced S shape. This motion is cyclical, as shown inFig. 4. Within a time interval, the vesicle becomes waistedat the center, almost separating into two halves connected

FIG. 1 (color online). Snapshot of the VB mode Ca ¼ 2 and� ¼ 4:5. Color refers to the mean curvature. FIG. 3 (color online). Snapshot of the SQ mode Ca ¼ 2 and

� ¼ 6. Color refers to the mean curvature.

FIG. 2 (color online). The angle between the longest distanceand the flow direction for VB, SQ, and TB regimes.

FIG. 4 (color online). Snapshot of the TB regime withS-shaped vesicle. Ca ¼ 10 and � ¼ 12. Color refers to the meancurvature.

PRL 109, 248106 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

248106-2

by a neck. A movie of STB motion is included in theSupplemental Material [24].

PB.—In contrast to quasispherical shapes, deflatedvesicles show other nontrivial dynamics for a small vis-cosity contrast (� < 1) and beyond a certain capillarynumber. Traditional TT motion enjoys parity symmetry(or centrosymmetry), where the shape obtained upon inver-sion, R ! �R (where R is the position of any membranepoint measured from an arbitrary origin), is identical to theoriginal shape. We found that this symmetry can be spon-taneously broken. This occurs for small enough viscositycontrasts, and for large enough capillary numbers, as isshown quantitatively in the phase diagram presentedbelow. A typical parity-broken shape is shown in Fig. 5,right. Because of obvious reasons of symmetry, the shapeobtained from an inversion operation of the shape shown inFig. 5 (right) is also a possibility, and the prevalence of oneover the other depends on initial conditions. At short time,we observed a TT regime, which then underwent sponta-neous parity breaking. With some sets of initial conditions,we first observed that the TT vesicle split into two vesiclesof equal size, connected by a tube that continued to elon-gate until it reached a certain size. The resulting morphol-ogy, represented in Fig. 5 (left), resembles a dumbbell, andhence the word ‘‘dumbbelling’’ can be adopted. However,the dumbbell does not seem to survive as time elapses: Onehalf of the dumbbell grows in favor of the other, resultingultimately in a permanent parity-broken state (Fig. 5,right). An important consequence of the parity-breakingmode is cross-streamline migration, which is absent forsymmetric shapes. It was found that the shape shown inFig. 5, (right) migrates downward, while the one obtainedby shape inversion moves upwards (with the same magni-tude, for reasons of symmetry). For Ca ¼ 10, we found amigration velocity of 0:005R0 _� (R0 is the radius of a sphereof the same volume as the vesicle). For shear rates of100 s�1 (a quite accessible value in standard experiments),the vesicle travels a distance of approximately its radius in1 s. This is clearly not devoid of experimental testability.Note that, ordinarily, parity symmetry is broken by extrin-sic factors (e.g., shear gradients in a Poiseuille flow, linearshear bound by a substrate, etc.) leading to cross-streamline migration. Here, the symmetry breaking is in-trinsic or spontaneous. We are not aware of any previousrevelation of spontaneous cross-streamline migration in apure linear unbounded shear flow.

In order to infer the mechanism leading to dumbbellinginstability, the following heuristic argument is presented.Dumbbelling is favored by the small viscosity contrast thatconfers to vesicles a high enough tank-treading angle (andthus significant elongational flow strength), predisposingthem to this kind of instability. Each half of the vesicle ispulled in an opposite direction and experiences a Stokes-type drag force � _��R2

0. The formation of a tube of length

L and radius r implies a Helfrich-type restoring force oforder �=r. Instability occurs if the first force exceeds thesecond one, namely, if � _�R3

0=� � R0=r, or if Ca � R0=r.It should be remembered that the creation of the tube is alsoaccompanied by an increase in dissipation (due to mem-brane tank-treading motion) which is approximately�� _�2r2L, corresponding to a force of �� _�rL; therefore,high viscosity ratios discourage dumbbelling.The dumbbelling mechanism can be exploited to gain

some insight into the formation of a neck in STB regime.Indeed, during TB, the long axis of the vesicle spends afinite time with an orientation with a high extensionalcomponent of flow (in the vicinity of c ¼ �=4) that cantrigger tube formation (strong extensional flow favors theformation of a tube). During the tube development process,the vesicle continues to rotate away from high extensionalorientation and spends some time aligned with the flow (inthe vicinity of c ¼ 0), when the tube tends to retract. Wehave thus to compare the retraction time of the tube, �, withtumbling time _��1. � is estimated by balancing the Stokesforce ��R0�L=� (�L=� represents a typical velocity ofretraction) on the two spheres of the dumbbell after dis-placement of the tube by an amount �L and the membraneHelfrich force that tends to suppress the tube, ��r=r2

(a change of tube length is accompanied by a change inits radius, owing to area conservation). Assuming a con-stant tube area �rL (a fact that holds in an idealized casewith a tube of sufficient thinness and length connecting twospheres), then �L=�r� r=L. Equating the two forcesyields �� �R0rL=�. If the TB period _��1 is small enoughin comparison to retraction time, i.e., if Ca>R2

0=rL, theneck will survive. In other words, a high shear rate (highCa; note that high Ca can also be achieved thanks to achange of R0, for example) favors neck formation.Complete phase diagram.—We performed a systematic

analysis in order to determine the phase diagram for thevarious motions discussed above. We explored a widerange of parameters: Ca 2 ½0:1; 100� and � 2 ½0:1; 13�(for � ¼ 0:6). The results are reported in Fig. 6. Anotherinteresting feature is the instability of in-plane dynamicsfor a high enough Ca (for example, Ca ¼ 50) resulting incomplex kayaking (see the movie in Ref. [24]). In severalcases studied here, the kayaking motion turned out to be avery long transient preceding the transverse TT mode, inwhich the vesicle tank treads with its longest axis perpen-dicular to the shear plane [15]. Crosses are used in Fig. 6 toindicate the simulation points for which noticeable insta-bility of in-plane motion was observed.

FIG. 5 (color online). Shape of a dumbbelled vesicle (Ca¼10,� ¼ 0:1) and PB regime (Ca ¼ 10 and � ¼ 0:3). Color refers tothe mean curvature.

PRL 109, 248106 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

248106-3

Discussion and conclusion.—The study of highlydeflated vesicles has revealed new panels of behavior(SQ, PB, STB, and out-of-plane motion). Upon a changeof viscosity contrast, and within a wide range of Ca, weobtained the following bifurcation cascade: PB ! TT !VB ! SQ ! STB with the possibility of out-of-plane mo-tion. Tube formation, found here as a transient only, isknown to occur in extensional flows [25] (leading todumbbelling) and during sedimentation [26,27]. In ourcase, the shear flow induced a rotation of the dumbbellcausing its quasialignment with the flow, where the exten-sional part of the flow is very weak. This explains theinstability of the dumbbelled shape.

The range of parameters explored in these simulations iswithin the reach of experiments and is also likely to beencountered in vivo. The relevant range for the rich dy-namics reported here corresponds to Ca � 1, where strongcoupling between flow and shape deformation takes place.Ca is a ratio between the response time of the vesicle shape,�b ¼ �R3

0=�, and the shearing time _��1, Ca ¼ �b _�. Asmall enough Ca means that the shape adapts itselfquasi-instantaneously to the shear flow. In this case, thereis no delay of shape evolution with regards to the shearingforce; the dynamics are quite simple, like TT and TB [28].Conversely, ifCa � 1, shape evolution can be delayed withrespect to the flow stresses, leading to complex dynamics,as reported here. In the case of RBCs, cytoskeleton elas-ticity must be taken into account. In this situation, a newdimensionless number enters the shape evolution equa-tions, namely, Cas ¼ � _�R0=G � �s _�, where G is thecytoskeleton shear modulus (typically G� 4 N=m) and�s is the response time of the cytoskeleton. We expectnontrivial dynamics to be associated with the cytoskeletonwhen Cas � 1. This is also supported by numerical

simulations [29], where the VB mode (known to take placefor vesicles for Ca� 1 [14]) is observed for a RBC model(capsule; i.e., without bending elasticity) when Cas � 1.For a RBC, both effects (bending and shearing elasticity)are present, and each mode makes its own contribution.Taking typical values for vesicles, namely, R0 � 10 m,� ¼ 10�3 P, and �� 10�19 J, we find Ca� 10 _� , with _�in s�1 units. A shear rate of a few s�1 should be largelysufficient to trigger these dynamics. Note that the range ofviscosity ratios explored here is within the reach of experi-ments [30,31]. A probable challenge is the structural stabil-ity of highly deflated vesicles under flow. Equilibriumshapes of highly deflated vesicles have already beenstudied experimentally [32], but their behavior underflow remains to be tested. For RBCs, we found (usingR0 ¼ 3 m and � ¼ 1:2� 10�3 P for plasma viscosity)Cas � 10�3 _� , with _� in s�1 units, so a shear rate value ofabout 103 s�1 should reveal the types of dynamics reportedhere. In vivo, RBCs experience shear rates which are of thisorder and larger (for example, in the arterioles the vesselwall shear rate is about 8� 103 s�1). We must keep inmind that the above value of G is only a median value andcan exhibit large variations in reality. Optical-tweezermeasurements [33] show a range of about G ¼1:2–21 N=m for healthy cells and 0:2–24 N=m forRBCs from patients who are homozygous for the sicklecell mutation and treated with hydroxyurea. This indicatesthat even a shear rate of about 100 s�1 should be sufficientto induce shape transitions of the same kind as thosereported here. For example, in the arterioles, we speculatethat SQ and/or STB (since the viscosity contrast of a humanRBC is in the range 5–7) is likely to occur due to the highshear rate values. In vitro experiments will be needed toshed light on the impact of shape transition in the micro-vasculature. Finally, extrapolation of our results to RBCsmust, for the time being, be considered with some reser-vations before systematic incorporation of the cytoskeletonin the model. This constitutes an important and promisingtask for future research, guided by the present study.We gratefully acknowledge financial support from

CNES, ESA, and ANR (’’MOSICOB’’ project).

*alexandr.farutin@ujf-grenoble.fr†chaouqi.misbah@ujf-grenoble.fr

[1] P. Vlahovska, T. Podgorski, and C. Misbah, C.R. Physique10, 775 (2009).

[2] A. Z. K. Yazdani and P. Bagchi, Phys. Rev. E 84, 026314(2011).

[3] R. Skalak and P. I. Branemark, Science 164, 717 (1969).[4] B. Kaoui, G. Biros, and C. Misbah, Phys. Rev. Lett. 103,

188101 (2009).[5] T.W. Secomb and R. Skalak, Microvasc. Res. 24, 194

(1982).[6] C. Pozrikidis, Ann. Biomed. Eng. 33, 165 (2005).

FIG. 6 (color online). Phase diagram of different motions ofvesicles. The abbreviations are TT (tank treading), VB (vacillat-ing breathing), TB (tumbling), SQ (squaring), PB (parity break-ing), and K (kayaking; instability of in-plane dynamics).

PRL 109, 248106 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

248106-4

[7] M. Abkarian, M. Faivre, R. Horton, K. Smistrup, C. A.Best-Popescu, and H.A. Stone, Biomedical materials 3,034011 (2008).

[8] G. Tomaiuolo, M. Simeone, V. Martinelli, B. Rotoli, andS. Guido, Soft Matter 5, 3736 (2009).

[9] V. Noireaux and A. Libchaber, Proc. Natl. Acad. Sci.U.S.A. 101, 17 669 (2004).

[10] M. Karlsson, M. Davidson, R. Karlsson, A. Karlsson, J.Bergenholtz, Z. Konkoli, A. Jesorka, T. Lobovkina, J.Hurtig, M. Voinova et al., Annu. Rev. Phys. Chem. 55,613 (2004).

[11] T.M. Allen and P. R. Cullis, Science 303, 1818 (2004).[12] G. Gregoriadis, Trends Biotechnol. 13, 527 (1995).[13] D. R. Arifin and A. F. Palmer, Biomacromolecules 6, 2172

(2005).[14] T. Biben, A. Farutin, and C. Misbah, Phys. Rev. E 83,

031921 (2011).[15] H. Zhao and E. S. G. Shaqfeh, J. Fluid Mech. 674, 578

(2011).[16] C. Misbah, Phys. Rev. Lett. 96, 028104 (2006).[17] P.M. Vlahovska and R. S. Gracia, Phys. Rev. E 75, 016313

(2007).[18] V. V. Lebedev, K. S. Turitsyn, and S. S. Vergeles, Phys.

Rev. Lett. 99, 218101 (2007).[19] H. Noguchi and G. Gompper, Phys. Rev. Lett. 98, 128103

(2007).[20] G. Danker, T. Biben, T. Podgorski, C. Verdier, and C.

Misbah, Phys. Rev. E 76, 041905 (2007).

[21] A. Farutin, T. Biben, and C. Misbah, Phys. Rev. E 81,061904 (2010).

[22] C. Pozrikidis, Boundary Integral and Singularity Methodsfor Linearized Viscous Flow (Cambridge University Press,Cambridge, England, 1992).

[23] J. Beaucourt, F. Rioual, T. Seon, T. Biben, and C. Misbah,Phys. Rev. E 69, 011906 (2004).

[24] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.109.248106 formovies of VB, SQ, STB, kayaking, and dumbbellingmodes.

[25] V. Kantsler, E. Segre, and V. Steinberg, Phys. Rev. Lett.101, 048101 (2008).

[26] G. Boedec, M. Jaeger, and M. Leonetti, J. Comput. Phys.230, 1020 (2011).

[27] G. Boedec, M. Jaeger, and M. Leonetti, J. Fluid Mech.690, 227 (2012).

[28] A. Farutin, O. Aouane, and C. Misbah, Phys. Rev. E 85,061922 (2012).

[29] P. Bagchi and R.M. Kalluri, Phys. Rev. E 80, 016307(2009).

[30] G. Coupier, B. Kaoui, T. Podgorski, and C. Misbah, Phys.Fluids 20, 111702 (2008).

[31] V. Kantsler, E. Segre, and V. Steinberg, Europhys. Lett. 82,58 005 (2008).

[32] J. Majhenc, B. Bozic, S. Svetina, and B. Zeks, Biochim.Biophys. Acta 1664, 257 (2004).

[33] S. Suresh, J. Mater. Res. 21, 1871 (2006).

PRL 109, 248106 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

248106-5