Why and how does collective red blood cells motion occur in the bloodmicrocirculation?Giovanni Ghigliotti, Hassib Selmi, Lassaad El Asmi, and Chaouqi Misbah Citation: Phys. Fluids 24, 101901 (2012); doi: 10.1063/1.4757394 View online: http://dx.doi.org/10.1063/1.4757394 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v24/i10 Published by the American Institute of Physics. Related ArticlesAlteration of chaotic advection in blood flow around partial blockage zone: Role of hematocrit concentration J. Appl. Phys. 113, 034701 (2013) Oscillation dynamics of embolic microspheres in flows with red blood cell suspensions J. Appl. Phys. 112, 124701 (2012) Depletion layer formation in suspensions of elastic capsules in Newtonian and viscoelastic fluids Phys. Fluids 24, 061902 (2012) Separation of blood cells using hydrodynamic lift Appl. Phys. Lett. 100, 183701 (2012) A new laser Doppler flowmeter prototype for depth dependent monitoring of skin microcirculation Rev. Sci. Instrum. 83, 034302 (2012) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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PHYSICS OF FLUIDS 24, 101901 (2012)

Why and how does collective red blood cells motion occurin the blood microcirculation?

Giovanni Ghigliotti,1,a) Hassib Selmi,2 Lassaad El Asmi,2

and Chaouqi Misbah3

1Department of Mathematics, University of British Columbia, 1984 Mathematics Road,Vancouver, B.C. V6T 1Z2, Canada2Laboratoire d’Ingenierie Mathematique, Ecole Polytechnique de Tunisie, Universite deCarthage, B.P. 743 - 2078 La Marsa, Tunisia3Universite Joseph Fourier and CNRS (UMR5588), Laboratoire Interdisciplinaire dePhysique, 140 Avenue de la Physique, 38402 Saint Martin d’Heres, France

(Received 4 May 2012; accepted 17 September 2012; published online 8 October 2012)

The behaviour of red blood cells (RBCs), modelled as vesicles, in Poiseuille flow,mimicking the microvasculature, is studied with numerical simulations in two di-mensions. RBCs moving in the centre of the Poiseuille flow (as in blood capillaries)are shown to attract each other and form clusters only due to hydrodynamic inter-actions, provided that their distance at a given time is below a certain critical value.This distance depends on physical parameters, such as the flow strength. Our sim-ulations reveal that clusters are unstable above a threshold value in the number offorming RBCs, beyond which one or few cells escape the pack by a self-regulatingmechanism that select the marginally stable size. This size selection depends on theflow strength as well as on the RBC swelling ratio. The results are interpreted viathe analysis of the perturbation of the flow field induced by the vesicles and theinterplay with bending and tension forces. This sheds a novel light on the process ofcollective motion of RBCs observed in vivo. C© 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4757394]

I. INTRODUCTION

Red blood cells (RBCs) are deformable cells in charge of gas exchange in the human body.Gas exchange takes place in the arterioles and capillaries, where RBCs are submitted to a Poiseuille(parabolic) flow profile.1 How do RBCs organize themselves in the microvasculature may impactflow efficiency and oxygen transport.

RBC spatial organization is thus of great relevance to biology and medicine, and several attemptshave been made in the past to model the dynamics of an ensemble of RBCs in a capillary. Earlyattempts have been focusing on a sequence of rigid spheres.2, 3 It turns out that RBC deformabilityplays a fundamental role in the collective behaviour: contrary to sets of rigid spheres,3 RBCs incapillaries have the tendency to aggregate to form clusters.4 Clusters, in which RBCs are close oneto another in a single-file configuration shall not be confused with rouleaux, a term indicating a stackof RBCs adhering one to the other. Rouleaux occur in presence of fibrinogen, the most acceptedmechanism being the depletion forces generated in presence of high molecular weight proteins (asfibrinogen).5, 6 Clusters are due to hydrodynamical interactions between RBCs, since the distancesbetween neighbouring cells are too large to be explained by depletion forces (or even by chemicalbonds between membranes).

In order to take into account their deformability, RBCs have been in recent years extensivelymodelled as vesicles7–11 (liquid drops delimited by a lipid bilayer, that is the main constituent of themembrane of living cells) or elastic capsules.12–15 In the following, we will adopt the former model.

a)Author to whom correspondence should be addressed. Electronic mail: gghiglio@math.ubc.ca. Tel.: +1 604 827 3296.Fax: +1 604 822 6074.

1070-6631/2012/24(10)/101901/11/$30.00 C©2012 American Institute of Physics24, 101901-1

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101901-2 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

Under parabolic flow, a single vesicle assumes a stationary centered shape,16 which may enjoythe mirror symmetry with respect to the centre line, in which case it is called a parachute. If themirror symmetry is absent, the shape is non-symmetric and is called slipper. Whether the vesicleassumes one shape or another depends on its swelling (i.e., the volume to surface ratio) and on theflow intensity.

The collective motion of one-dimensional files of RBCs in capillaries has been first studiednumerically under the hypotheses of axisymmetric cell shape and periodic spatial distribution.17

This approach, based on the simulation of a single cell in a periodic domain, allowed to estimate theinfluence of the RBC volume concentration (haematocrit) on the cell shape and flow rate.

Even in the simple case in which RBCs assume a symmetric position centred in the flow, a setof RBCs shows a non-homogeneous spatial organization,4 i.e., RBCs form clusters (clearly, non-uniform spatial organization is out of reach of the one-cell approach17). While this non-homogeneitywas previously considered to be due to the polydispersity in size of the RBCs (larger ones areadvected more slowly by the flow, and thus are caught up by the following ones),4 very recentlycluster formation has been numerically observed in the case of vesicles monodisperse in size18, 19

and also experimentally observed for RBCs in vitro.20 Since the only forces implemented in thesimulations were hydrodynamical forces, these must be responsible for clustering. Those authorsstudied three systems, consisting of one, three, and six vesicles. They could observe the formationof clusters, and identified the cause in the coupling between cell deformability and hydrodynamicalinteractions.

Nevertheless, a precise link between hydrodynamical interactions and cluster forma-tion/destruction has not been established yet. Furthermore, previous simulations used periodicboundary conditions along the flow direction. So the finite size effect may interfere with the in-trinsic mechanism of the cluster formation, as hydrodynamics is of long range and it is not clearhow the appropriate cut-off has to be chosen.

In this work, we use the boundary integral method (BIM), which, besides its high precision,allows one to study infinite systems, avoiding any possible artifact related to non-physical boundaries.The nonlocal character of the BIM method sets, however, a severe limitation on the computationaltime as well as on the storage memory, both behave as O(N2)—where N is the total number ofdiscretization points of the full boundaries. To circumvent this problem, we make use of a fastmultipole method (FMM) that lowers the complexity down to O(N). Thus, our method is a coupledBIM-FMM one that will allow us investigating large clusters.

Several basic and natural yet unanswered questions arise: when does a cluster form, i.e., whatis the initial interdistance that leads to a cluster? Does each cluster maintain its size, or do someclusters loose their integrity in the course of time? If so, what are the physical mechanisms thatgovern this evolution? How does a maximum stable size (if any) depend on flow conditions andother parameters? The use of extensive numerical BIM-FMM simulations allows one to gain animportant step toward answering this set of questions.

In order to gain progressive insight into this complex question, we shall first study the casewhere the set of vesicles are not confined at all. This will serve as basic problem, which is alreadycomplex by itself. In addition, we shall show that the cluster formation is an intrinsic property, nottriggered by any confinement. We shall then briefly discuss how some of the reported phenomenaare affected by lateral boundaries (i.e., confinement), mimicking blood vessel walls.

II. THE MODEL

We consider the simplified problem of clusters of RBCs in an unbounded Poiseuille flow in twodimensions. A two-dimensional model has been favoured over a three-dimensional axisymmetricone, which would have a comparable numerical cost, due to the possible axial-symmetry breakingby the cells in a parabolic flow.16, 21 In this way axial symmetry is not imposed, so that wheneveraxisymmetric solutions are observed, they are stable solutions. The Reynolds number Re is, in thephysiological conditions and in the available experiments (except in arterioles or in some site ofsudden aneurysm where it can become of order unity,22 strongly impacting dynamics), of the orderof 10−3 ≤ Re ≤ 10−2, so that inertia is negligible compared to viscous forces. This allows us to

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101901-3 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

use the Stokes equation to describe the hydrodynamics. For simplicity, we consider the situation inwhich the viscosity η of the fluid inside the RBCs is equal to that of the outer fluid (this point isfurther addressed at the end of this section). The velocity field obeys the Stokes equations:

−∇ p + η�u = 0, (1)

∇ · u = 0. (2)

In two dimensions there is, from the mechanical point of view, no distinction between RBCs andvesicles, due to the fact that membrane plane shear elasticity (present in RBCs) looses it meaning.We take the radius R of the equivalent circle as a length unit: R = √

A/π , where A is the surface ofthe vesicle. The vesicle membrane is subject to the constraint of local inextensibility and is endowedwith bending energy. The membrane free energy can be expressed as

Eγ =∮

γ

{ζ (s) + κ[c(s)]2

}ds, (3)

where γ is the membrane position, ζ is a local Lagrange multiplier associated with the constraintof inextensibility, κ is the bending modulus, c is the membrane curvature, and ds is the perimeterelement. The variation of the free energy of Eq. (3) gives the force f by which the membrane acts onthe surrounding fluids:

f = −κ

[d2c

ds2+ 1

2c3

]n + ζcn + dζ

dst, (4)

where n and t are the outward normal and clockwise tangent vector, respectively.We consider vesicles in an unbounded Poiseuille (parabolic) flow, defined as

u = b

2y2ex . (5)

The local shear rate is γ (y) = dux/dy = by, where b is the curvature of the flow profile. In thefollowing, we will use the value of γ (y = R) = bR for the scale of flow strength.

As seen below in the mathematical integral formulation, one can handle exactly a completelyunbounded geometry. It must be, however, kept in mind that in practice the unbounded flow is takento mean “weakly confined flow” (the channel width is large enough in comparison to the cell size).We shall come back to this point in the section devoted to discussion.

The problem is described in terms of three dimensionless parameters:

� The number N of vesicles composing a cluster.� The reduced area α of the vesicles, defined as the ratio between the surface A of the vesicle

and the surface of a circle having the same perimeter p (i.e., α = A/[π (p/2π )2]); α, chosen tobe the same for all the vesicles in a given simulation, assumes the values α ∈ [0.65, 0.70, 0.75];it represents the swelling of the vesicles.

� The capillary number Ca that defines the relative strength of the imposed flow and the bendingforces, Ca = ηγ R3/κ . The values explored are Ca ∈ [5.0, 10.0].

It is known that by decreasing either α or Ca a single vesicle in a Poiseuille flow shows a transitionfrom a symmetric shape (parachute-like, with the centre of mass on the Poiseuille centreline) to anon-symmetric one (slipper-like, with the centre of mass out of the centreline).16 In order to avoidmixing several effects at once, for ease of a sound understanding of phenomena, the values of α

and Ca are chosen so that a single vesicle is stable on the Poiseuille centreline with a parachute-likeshape.

We consider sets of vesicles aligned along the Poiseuille centreline, with varying numbers ofcells. These quasi one-dimensional configurations are found to be stable, i.e., all the vesicles stay onthe centreline, in the explored range of α and Ca.

Vesicles are initialized as ellipses elongated in the direction of the velocity gradient and with thecentre of mass on the centreline. One might ask why the vesicles are initialized in such an artificialshape instead of the more natural choice of the equilibrium shape of a single vesicle in a parabolic

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101901-4 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

flow (parachute shape). Indeed it turns out that the elliptical initial shapes allow for a faster drainageof the fluid between two vesicles, thus reducing the relaxation time toward the equilibrium parachuteshapes (note that these shapes differ from the single vesicle one due to hydrodynamical interactions).In this sense, this initial condition facilitates the cluster formation and minimizes the possibility thata cluster does not form due to the choice of the initial configuration.

In most of simulations the initial distance between the centers of consecutive vesicles is chosento be equal to 1.6R, corresponding to a membrane-to-membrane distance of approximately 0.4R.This distance is on one side small enough to allow a strong interaction between vesicles, and on theother is big enough to avoid transverse elastic instabilities on the cluster (i.e., the lateral expulsion ofone or several vesicles from the cluster) during the initial transient. Reasonable variations of the valueof the initial distance do not affect the final configurations. We have determined the maximal value ofthe interdistance below which a cluster form, and have found that it can significantly exceeds 1.6R,as discussed in Sec. IV. By virtue of the initial condition (vesicles centered on the flow centreline),the vesicles are not subject to any major shearing, in particular there is no tank-treading movement ofthe membrane at any moment during the whole time evolution of the system. As a consequence, theviscosity of the inner fluid plays only a very minor role in the first part of the dynamics (relaxationfrom ellipse to parachute), and no role at all in the stationary states, in which the inner fluid isquiescent.

III. THE NUMERICAL METHODS

The simulations are carried out in two dimensions by using the boundary integral method23, 24

with the implementation detailed elsewhere.25 Moreover, we make use of a parallel implementationof the fast multipole method that sensibly decreases the computing time.26 In this section, we give ashort description of the two methods.

BIM allows to solve Eqs. (1) and (2) by computing integrals on the surfaces delimiting differentfluid domains, so that the evolution of a two-dimensional system can be performed through one-dimensional integrals. In the case treated here these surfaces are the membranes of the vesicles. Ifthe viscosity is uniform throughout the system, the equation for the velocity of a point x0 lying onthe vesicle membrane is

u(x0) = u∞(x0) +∮

Ni=1γi

G(x − x0)f(x)d S(x), (6)

where u∞ represents the imposed velocity field (in our case, an unbounded parabolic profile) andthe integral represents the perturbation created by the vesicles. Each γ i is the position of one of theN vesicles, f is the force exerted by the membrane on the fluids and dS is the membrane arc-length.G is the free-space Green tensor, given by the expression

Gi j (r) = δi j ln r + rir j

r2. (7)

The direct solution of the integral Eq. (6) requires O(N2) computations, where N is the numberof discretization points. We make use of FMM to compute an approximate solution with O(N)computations. FMM is based on a double multipolar expansion with respect to the source points xand the target points x0 of Eq. (6).26, 27

In fact, the FMM method replaces the classical matrix-vector product by a multipole product,that allows to obtain, by a fast algorithm, a good approximation of the solution. The basic idea ofthe fast mutipole method is the separation of variables in the Green function. This Green functionis rewritten differently in an expression that determines the form of the moments and the transferfunctions, multipolar and local moments are then calculated. The adaptation of the FMM methodto this problem is explained elsewhere.26 Truncation errors of the resulting scheme and comparisonwith the classical BIM method are presented in Ref. 27. The implementation of the resulting methodachieves a high accuracy in a reasonable time, and thus allows us to study complex phenomena.

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101901-5 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

IV. RESULTS

In this section, we present the results obtained through the numerical simulations. We firstdescribe the behaviour for fixed α and Ca, and we vary N. This first case will allow to present theoverall features of the observed dynamics.

A. General behaviour

We first consider the case of sets of vesicles with a reduced area α = 0.70 at a capillary numberCa = 10 (Fig. 1); other values have been explored (see below). This value of Ca is bigger thanthe critical capillary number C∗

a at which the slipper-to-parachute transition occurs (C∗a � 4 for α

= 0.70), so all the vesicles assume a parachute shape. We run simulations by increasing the numberN. For 1 ≤ N ≤ 10, we observe that the vesicles always attract each other and form a cluster, providedthat the initial interdistance is below a certain value (to be anaylzed below). A cluster is definedas a set of cells whose interdistances approach finite and constant values after an initial transient.The clusters are entirely due to hydrodynamical interactions, since hydrodynamics is the only forceimplemented in the simulations (in particular no attraction/repulsion potentials are present). Duringthe dynamical evolution the membranes of neighbouring cells keep at a certain finite distance andnever touch each other. The dynamics of these clusters is nontrivial. First of all, the vesicles distributein a non homogeneous way in the clusters, with the distance between consecutive vesicles beingbigger at the front than at the back. Second, although all the vesicles show a parachute shape, thewidth in the direction of the velocity gradient of every parachute differs from the others, with thevesicles in the front being more squeezed in the vertical direction.

For N ≥ 11, we observe that the clusters are not stable, and that vesicles detach from the front,one by one, until the number of vesicles in the cluster goes down to 10. This has been verified for 11≤ N ≤ 15; for example, starting from N = 13 a cluster of 13 vesicles forms, and after some time thecluster looses its integrity by successively loosing 3 vesicles (one by one) at the front. This happensvia a self-regulating mechanism, and highlights the existence of an intrinsic maximal cluster sizeN*, with N* = 10 for Ca = 10.

B. Dependence on α and Ca cluster velocity

We have investigated the behavior of the cluster upon variation of two main parameters, α andCa. The results are qualitatively similar to those presented above, with a critical cluster size abovewhich clusters are unstable and vesicles detach from the front.

The velocity of clusters as a function of these two parameters is shown in Fig. 2. The clustervelocity Vcluster is defined as the lag velocity, i.e., the difference between the velocity of the centrelineof the unperturbed parabolic flow and the cluster velocity. The cluster velocity is a decreasing functionof the number of vesicles in the cluster. This is not surprising, since they are passively transported bythe flow: when their number increases, they shield each other and the drag on each vesicle decreases,thus decreasing the cluster velocity. Moreover, the magnitude of the velocity also increases withboth α and Ca. Similarly, an increase in α leads to more curved parachutes (more squeezed in thedirection of the velocity gradient), i.e., mass distribution of the vesicle—or more precisely the inertiamoment with respect to the center line—is smaller, and hence the vesicle moves faster.

An interesting feature is the behavior of the maximal cluster length (in number of vesicles) asa function of α and Ca (Figure 3). Higher flow strength increases very significantly the maximumnumber of vesicles in a cluster. The same holds when decreasing the reduced area α (Fig. 2).

C. Cluster (in)stability via self-regulation

A cluster of a given size is found to be stable or not, depending on the flow strength (for a givenreduced area α). Why should a cluster form at all, and what are the physical mechanisms that dictatetheir stability or instability? Let us first analyze the flow field perturbation caused by a single vesicle(Fig. 4). It is clearly seen that the vesicle creates two counter-rotating vortices converging at the

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101901-6 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

FIG. 1. Cluster steady states for Ca = 10, α = 0.70 for different number of vesicles N in an unbounded parabolic flow.

front and diverging at the back. If a second vesicle is introduced at the front, it will be “captured”by the converging vortices. This second vesicle (becoming now the leading one) will in turn createa similar flow pattern at the front which is capable of capturing a new vesicle ahead, and so on.The combination between the converging and diverging vortices leads to an equilibrium distancebetween two successive vesicles. The capture zone ahead of the vesicle is wide enough so that evenif the initial distance is as large as 3.1 radii, the vesicles attract each other (for α = 0.70, N = 8, Ca

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101901-7 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

0 2 4 6 8 10 12N

-1.1

-1

-0.9

-0.8

-0.7

Vcl

uste

r

. /

R0γ

α=0.65 Ca = 5α=0.65 Ca = 7.5α=0.65 Ca = 10α=0.70 Ca=5α=0.70 Ca=7.5α=0.70 Ca=10α=0.75 Ca=5α=0.75 Ca=7.5α=0.75Ca=10

FIG. 2. Cluster velocities as a function of N for different α and Ca.

= 10). Figure 5 shows the distance between the first and second vesicle in the cluster as a functionof time for different initial distances.

Two natural questions arise: how many vesicles can adhere to a given cluster? If the allowedmaximum number is finite, how and by which mechanism are extra vesicles expelled from thecluster? A detailed analysis of the velocity field around the vesicles can help answering thesequestions. As seen in Fig. 4, the flow pattern is convergent at the front and divergent at the back. Thismeans that for a cluster of two vesicles, the leading one will be compressed (in the vertical directionof the figure) by the perturbation created by the trailing one, and the trailing one will be stretchedby the leading one: vesicles act here as “hydrodynamic tweezers.” A first consequence is that the

4 6 8 10Ca

4

6

8

10

12

N*

α = 0.65α = 0.70α = 0.75

FIG. 3. Cluster size as a function of Ca for different α.

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101901-8 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

FIG. 4. Perturbation to the velocity field created by a single vesicle in an unbounded Poiseuille flow (α = 0.70, N = 1, Ca

= 10, stationary state). The presence of attraction and repulsion zones (in front of the vesicle and on its back, respectively)is evident.

0 10 20 30 40 50 60Time (a.u.)

1

2

3

4

5

6

7

8

9

10

d / R

0

d0 = 1.6

d0 = 2.7

d0 = 3.1

d0 = 3.2

d0 = 3.3

FIG. 5. Distance between the first and the second vesicles as a function of time (α = 0.70, N = 8, Ca = 10).

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101901-9 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

ves 1ves 2ves 3

(a)

FIG. 6. (a) Perturbations to the velocity field created by every of the three vesicles composing a cluster (N = 3). The vesiclecontours and perturbations are translated in such a way to superimpose the centers of mass of the vesicles. Vesicles arenumbered starting from the front (left) of the cluster. (b) Schematic view showing that the leading (resp. trailing) vesicle iscompressed (resp. stretched) by the trailing (resp. leading) one.

two vesicles do not form a “twin” due to their shape difference. In Fig. 6, we have superimposedthe perturbations produced by each vesicle in a cluster made up of N = 3 vesicles; in other words,we represent the perturbation by one vesicle as if the two others were absent (this can easily beaccomplished in a BIM formulation simply by considering in Eq. (6) only the integration over asingle vesicle). The perturbation streamlines are more elongated in the direction of the imposed flowwhile moving toward the front of the cluster. The equilibrium distance of two vesicles (named herea and b) is achieved when the force (linked to the velocity field perturbation) of a on b is equal tothat of b on a. This equality is fulfilled for a bigger inter-distance if the streamlines of either a or bare more elongated in the flow direction. This explains, qualitatively, why the vesicles on the frontof the cluster are more spaced than the vesicles at the back.

FIG. 7. Velocity field in the frame moving with the vesicles (α = 0.70, N = 10, Ca = 10), zoom on the first two. The twovortices merge on the centreline for N = N*.

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101901-10 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

2 4 6 8 10 12N*

2

2.2

2.4

2.6

2.8

3

3.2

3.4

d / R

0

α=0.65 Ca=5α=0.65 Ca=7.5α=0.65 Ca=10α=0.70 Ca=5α=0.70 Ca=7.5α=0.70 Ca=10α=0.75 Ca=5α=0.75 Ca=7.5α=0.75 Ca=10

FIG. 8. Distance d/R0 between the first and the second vesicles as a function of the number N* of vesicles in a cluster fordifferent α and Ca. The two horizontal lines emphasize that d/R0 is relatively constant for different N*.

This observation also allows us to unveil the detachment mechanism. Since the velocity ofthe fluid between two consecutive vesicles is smaller than that of the surrounding fluid (due to thenonzero shear rate of the imposed flow), vortices due to viscous drag will take place between avesicle and its neighbors (Figure 7). The size of the vortices is given, typically, by the interdistancebetween two vesicles. Since the leading vesicle is squeezed (in the vertical direction), its naturaltendency is to move further ahead in order to relax the stresses imposed by the following ones, butstill, it remains within the cluster as long as it stays within the capture zone. The larger vortices arethose lying between the first and the second vesicle, owing to the larger distance. As the numberof the vesicle increases, the additive effects of the trailing vesicles makes the leading one slightlyescaping ahead, to release further squeezing.

There is a critical distance at which the two vortices merge (Fig. 7), causing a net flow betweenthe first vesicle and the rest of the cluster. This net flow triggers the detachment of the leading vesiclefrom the cluster. The merging of the vortices happens for a critical size N = N*. The detachmentof the first vesicle is then completely determined by the space separating it from the second. Thisis why the distance d/R0 between the first and second vesicle for N = N* is more or less constant(2.85 ≤ d/R0 ≤ 3.2) when varying N* (upon variation of the control parameters Ca and α), as can beappreciated in Figure 8. Small fluctuations of d/R0 for N = N* are understandable due to the discretecharacter of N.

We have investigated the role played by bounding walls. If w denotes the channel width, Cn= R/w measures the degree of confinement. We have found that for Cn ≤ 0.05 (weak confinement)the same qualitative and quantitative results are recovered. Preliminary results show that as theconfinement increases the same type of flow pattern is observed, but the presence of the walls makesthe range of interaction smaller (due to the cut-off length D that reduces the range of the vorticesahead of the leading vesicle). As a first consequence the interdistances within the cluster becomelarger, and thus the extent of the cluster is wider, as compared to the unconfined one. A quantitativestudy, including convergence tests, is time consuming. We hope to investigate this matter further inthe future.

V. CONCLUSION

The main result of this paper is to show the mechanism of formation—and destruction—ofclusters of red blood cells, modelled as vesicles. The effect is purely hydrodynamical, and has been

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101901-11 Ghigliotti et al. Phys. Fluids 24, 101901 (2012)

explained by the analysis of the perturbation of the velocity field done by the deformable RBCs. Inparticular, the ample effect due to deformability has been put forward. A criterion for cluster breakupbased on the size of the recirculations between the first and the following RBCs has been suggested.

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