Blood microcirculation critically depends on the migration of red cells towards the flow centerline. We identify theoretically the ratio of the inner over the outer fluid viscosities λ as a key parameter. At low λ, the vesicle deforms into a tank-treading ellipsoid shape far away from the flow centerline. The migration is always towards the flow centerline, unlike drops. Above a critical λ, the vesicle tumbles or breaths and migration is suppressed. A surprising coexistence of two types of shapes at the centerline, a bulletlike and a parachutelike shape, is predicted

Danker, Gerrit

Misbah, Chaouqi

Vlahovska, Petia M

English

Dartmouth Digital Commons (Dartmouth College)

Dartmouth CollegeDartmouth Digital CommonsOpen Dartmouth: Faculty Open Access Articles4-10-2009Vesicles in Poiseuille FlowGerrit DankerUMRPetia M. VlahovskaDartmouth CollegeChaouqi MisbahUMRFollow this and additional works at: https://digitalcommons.dartmouth.edu/facoaPart of the Engineering Commons, and the Medicine and Health Sciences CommonsThis Article is brought to you for free and open access by Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: FacultyOpen Access Articles by an authorized administrator of Dartmouth Digital Commons. For more information, please contactdartmouthdigitalcommons@groups.dartmouth.edu.Recommended CitationDanker, Gerrit; Vlahovska, Petia M.; and Misbah, Chaouqi, "Vesicles in Poiseuille Flow" (2009). Open Dartmouth: Faculty Open AccessArticles. 3536.https://digitalcommons.dartmouth.edu/facoa/3536arXiv:0809.4028v1  [physics.bio-ph]  23 Sep 2008Vesicles in a Poiseuille flowGerrit Danker1, Petia M. Vlahovska2, and Chaouqi Misbah11Laboratoire de Spectrome´trie Physique, UMR, 140 avenue de la physique,Universite´ Joseph Fourier, and CNRS, 38402 Saint Martin d’Heres, France2 Thayer School of Engineering, Dartmouth College,8000 Cummings Hall, Hanover NH 03755, USA∗(Dated: October 31, 2018)Vesicle dynamics in unbounded Poiseuille flow is analyzed using a small-deformation theory. Ouranalytical results quantitatively describe vesicle migration and provide new physical insights. Atlow ratio between the inner and outer viscosity λ (i.e. in the tank-treading regime), the vesiclealways migrates towards the flow centerline, unlike other soft particles such as drops. Above acritical λ, vesicle tumbles and cross-stream migration vanishes. A novel feature is predicted, namelythe coexistence of two types of nonequilibrium configurations at the centreline, a bullet-like and aparachute-like shapes.PACS numbers: 87.16.Dg 83.50.Ha 87.17.Jj f3.80.Lz 87.19.TtKeywords:The motions of vesicles made of closed bilayer mem-branes in viscous flows have received increasing atten-tion in recent years because of their relevance to un-derstanding cell dynamics in the microcirculation. Theunusual mechanical properties of the lipid bilayer mem-brane, such as fluidity, incompressibility and resistance tobending, give rise to a number of fascinating nonequilib-rium features of vesicle micro-hydrodynamics For exam-ple, in linear shear flow vesicles are predicted to exhibitdifferent types of motion: (i) tank-treading (TT) (thefluid membrane rotates as a tank-tread, while the orien-tation angle of the vesicle remains fixed in time) [1, 2],(ii) tumbling (TB) [3, 4], (iii) vacillating-breathing (VB)(the long axis undergoes oscillation about the flow, whilethe shape shows breathing) [5]. Some of these behav-iors have been confirmed experimentally [6, 7, 8]. Vesicledynamics in linear flows is now fairly well understood[2, 5, 9, 10, 11, 12].Quadratic flows such as the Poiseuille flow are quitecommon, especially in the blood circulatory system, yetthe impact of flow curvature on vesicle dynamics is notwell understood. Number of studies have focused on cap-illary flows, where the vesicle diameter is comparable tothe channel size and lubrication effects control vesicle mo-tion [13, 14], or vesicles near walls, where hydrodynamicinteractions with the wall give rise to cross-stream migra-tion (in a direction perpendicular to the flow) [15, 16].Blood vessels such as arterioles, however, can be 10-50times larger than the cell diameter. In this case, effectsof flow curvature may become dominant. Vesicle dynam-ics in unbounded Poiseuille flow has been considered onlyto a limited extent, using two-dimensional numerical sim-ulations [17]. Cross–streammigration has been observed,even in the absence of wall [17]. The physical mechanismbehind this migration is related to the spatial variations∗chaouqi.misbah@ujf-grenoble.frof shear rate that exists in quadratic flows: deformableparticles tend to move towards regions with lower shearin order to minimize shape distortion [18]. Drops, forexample, migrate towards or away the flow centerline de-pending on the viscosity ratio [18]. In the case of vesicles,however, details of the migration mechanism remain elu-sive and there are number of open questions: What isthe role of flow curvature in cross-stream vesicle migra-tion? Is there relation between the migration directionand the basic modes of vesicle dynamics (i.e. TT, VBand TB)? What physical parameters control vesicle mi-gration? The development of a theory that can answerthese questions represents a challenging task because notonly the shape, but also the location of the vesicle is notknown a priori, and it must be solved for in a consistentmanner.Problem formulation and solution: In a reference framecentered in the vesicle, a channel flow is written asv0 = (−vs − γ˙y − αy2) ex, (1)where α is a measure of the curvature of the flow profile,γ˙ is the local shear rate, which depends to the distancebetween the vesicle center and the flow axis, γ˙ = 2y0α,and vs is the slip velocity (to be determined), which isthe difference between the actual velocity of the vesiclecenter in the flow direction and the unperturbed flow.At the length scales of the vesicle, water is effectivelyvery viscous. Hence, the total flow field v = v0 + u (im-posed flow v0 plus perturbation u due to the presence ofthe vesicle) obeys the Stokes equations inside and outsidethe vesicle∇p = ηi∇2v, ∇ · v = 0, (2)where ηi (i = 1, 2) is the fluid viscosity; η1, η2 are theinternal and external viscosities, and λ ≡ η1/η2 is a mea-sure of the viscosity contrast.Assuming a nearly spherical shape, the solution tothe creeping–flow equations is obtained as a regular per-turbation expansion in the excess area, which is the2difference in the areas of the vesicle and an equiva-lent sphere (with radius r0) with the same volume, i.e.,∆ = A/r20− 4π. From the definition of the area it is evi-dent that ∆ is a quadratic function of the shape deviationfrom sphere. Thus it is convenient to set ǫ = ∆1/2 as aformal expansion parameter. A time-dependent vesicleconfiguration is described byR = R0 + r0[1 + ǫf(t, θ, φ)] er, (3)where R0 = (x0, y0, z0) is the time-dependent position ofthe centroid of the vesicle and f describes shape deviationfrom the spherical shape. θ and φ are spherical coordi-nates. The vesicle shape f(θ, φ) can be decomposed in aninfinite series of spherical harmonics Ylm(θ, φ) excludingthe l = 1 mode (which describes the translation R˙0):f =∑l 6=1fl =∑l 6=1l∑m=−lFlm(t)Ylm(θ, φ). (4)The amplitudes Flm(t) describe the shape evolution. Forthe present study (leading order analysis) it is sufficientto include the l = 0 mode, which ensures volume conser-vation, and the l = 2, 3 modes, which are excited by theimposed Poiseuille flow.The velocity field is given by the classical Lamb solu-tion [19]. This solution contains integration factors whichare determined from the boundary conditions. These are:(i) Continuity of the fluid velocity across the membrane(valid if impermeability is assumed). (ii) Continuity ofthe stress across the membrane: the jump of fluid stressacross the membrane is balanced by the membrane force.The latter consists of a normal force due to resistance tobending, and a tangential (fictitious) force, which origi-nates from a Lagrange multiplier enforcing local mem-brane incompressibility. (iii) Local membrane incom-pressibility, which restricts the surface velocity field to besolenoidal. We have solved the full hydrodynamic prob-lem in the same spirit as in the calculation for the linearshear flow. Details can be found in [12]; here we focussolely on the results and discuss the physical implications.It should be noted that in certain cases, e.g., drops,the migration velocity can be obtained using a simplifiedapproach based on the Lorentz reciprocal theorem[20].However, this approach is inapplicable to vesicles becauseof the tangential membrane force.Results and Discussion: The time evolution of theshape function (∂tf) together with the motion of thevesicle center with respect to the flow (R˙0) are com-puted from the kinematic condition, which requires thatthe membrane moves with the fluid velocity. We find forthe evolution of the l = 2, 3 (which are the only excitedmodes at leading order) the following equations:ǫDtF2m = −24 (σ0 + 6κ)23λ+ 32F2m +iαy023λ+ 32N2m, (5)ǫDtF3m = −120 (σ0 + 12κ)76λ+ 85F3m +αr076λ+ 85N3m (6)with Dt = ∂t + imαy0, and N20 = N2,±1 = 0, N2,±2 =±8√30π, N30 = N3,±2 = 0, N3,±1 = ∓5√21π/3,N3,±3 = ∓5√35π. κ is the membrane bending rigidityrescaled by r30/η. The evolution equations (5, 6) con-tain the tension-like quantity σ0, which is the homoge-neous part of the Lagrange multiplier [2, 12]. Its valueis computed from the condition that the shape deforma-tion complies with the available excess area. σ0 is easilydetermined from the above equations, and the expres-sion relating excess area and the shape modes Flm, asexplained in [12].Knowledge of the shape evolution allows us to computethe lateral (cross-stream) migration along the y-directionvm =i4√30παǫ r2036λ+ 71(λ + 4)(76λ+ 85)(F22 − F2,−2)+167√21παǫ r0y023λ+ 32(F31 − F3,−1)+487√35παǫ r0y023λ+ 32(F33 − F3,−3). (7)This equation is coupled back to the shape, leading tosome interesting dynamics. First, in contrast to drops, avesicle always migrates towards the centerline. Second,the migration velocity depends in a non-trivial way onthe viscosity ratio and excess area.For λ < λc, a numerical solution of the coupled equa-tions for the shape (5, 6) and the migration (7) yieldsthe usual tank-treading ellipsoid as long as the vesicleis far from the centerline, because the linear shear dom-inates over the quadratic component of the flow. Thelatter, however, induces migration towards the center ofthe flow. As the vesicle moves its shape deforms more andmore until it becomes a parachute (or assumes anothershape, namely bullet-like, as discussed below) when thecenterline is reached (Fig. 1). The migration velocity far0 2 4 6 8 10t0123y 0FIG. 1: Evolution of the vertical position of the vesicle as afunction of time. Some snapshots of the vesicle shape alongthe trajectory are shown as insets. Parameters: α = 6, κ =1,∆ = 0.25.from the centerline is well approximated by plugging thesolution for the tank-treading ellipsoidal shape, R = ǫ/2,cos 2ψ = ǫ/2h with R and ψ defined by ǫF22 = Re−i2ψ)3into Eq. (7) [5].vm = − 17284λ2 + 66671λ+ 5584060 (λ+ 4)(23λ+ 32)(76λ+ 85)√30πα∆1/2 r20.(8)For λ ∼ 1 the migration velocity is approximately vy =−0.16α∆1/2r20. In a typical experiment in a rectangu-lar microchannel we have α ∼ 0.3 (µm·s)−1, ∆ = 0.25,r0 = 20µm, and a migration velocity of 0.75µm/s[21].Using Eq. (8), we find an expected migration velocity ofabout 1µm/s. Note also that (8) is independent of themembrane bending rigidity. This is understood as fol-lows. The local shear rate is proportional to the distancefrom the centerline: γ˙ ∼ y0. The elongational componentof the flow (which increases like the shear rate) must becompensated by the membrane tension in order to ful-fil local membrane incompressibility. It follows that thetension σ0 must also scale like y0. Then, for large y0, theterms 6κ and 12κ in Eqs. (5, 6) become negligible andthe membrane dynamics becomes independent of κ.The direction of the lateral migration of the vesicle inPoiseuille flow can intuitively be understood from the fol-lowing argument. Assume that we have solved the prob-lem for the pure shear flow, γ˙(y0)y. The solution is thetank-treading vesicle shown in Fig. (2). Next, we add theBDCAyxFIG. 2: (Color online) A vesicle in Poiseuille flow far from thecenterline. The local velocity field can be decomposed intoa local shear γ˙(y0) (responsible for the tank-treading vesicleat the shown orientation) and a (small) quadratic correction(blue arrows). Due to the inclination of the vesicle long axiswith respect to the quadratic flow field, there is a net forceacting on the vesicle membrane in the negative y direction.quadratic correction −αy2 as a perturbation. The addi-tional flow, acting on the membrane of the vesicle, canbe decomposed into two components: one tangential tothe membrane, which locally modifies the tank-treadingvelocity, and one normal to the membrane, which mod-ifies the vesicle shape and possibly cause migration. Inthe segments AB and CD the perpendicular componenttends to push the vesicle into the negative y direction,whereas in the smaller segments DA and BC, the per-pendicular component pushes the vesicle in the positive ydirection. As the segments AB and CD account for mostof the vesicle’s surface, the net force has a component inthe negative y direction (towards the flow centerline).Let us turn now to the problem of migration in thetumbling regime. From the analytical results discussed0 2 4 6 8 10λ-0.8-0.6-0.4-0.20vmTT TB, (VB)outward migrationinward migration∆ = 0.25α = 6y0 = 10FIG. 3: Migration velocity as a function of the viscosity con-trast λ for the various regimes: tank-treading (TT), tumbling(TB), and vacillating-breathing (VB).0 0.2 0.4 0.6 0.8 1∆00.10.20.30.4|v m|α = 6λ = 2.168y0 = 10FIG. 4: Migration velocity as a function of the excess areafar from the centerline. Parameters are chosen such that thetank-treading regime stops at ∆ = 0.9.above for the tank-treading regime, one finds that themigration velocity close to the tumbling regime (whichoccurs at λ = λc [5]) behaves as√λ− λc, and thus itvanishes exactly at the threshold. This analysis is validon the tank-treading side. For λ > λc, the vesicle tum-bles, and in this case we have integrated numerically theset of equations (5,6,7). The migration (averaged overtime) is again always towards the flow centerline but themagnitude of the migration velocity is very close to zero(of the order of 10−2 r0 per tumbling cycle). The resultsare shown on Fig.3. For large viscosity ratios, λ ≫ 1,the migration velocity must go to zero, since the vesiclebehaves as a rigid sphere. The kinematic reversibility ofthe Stokes equations precludes a lateral migration in thiscase. This feature is also evident from the expression forthe migration velocity (7). Indeed, the F ′ijs are finite dueto the constraint of fixed excess area, but the prefactorsO(λ−1). Thus, when λ→∞, vm → 0.An interesting feature is the non-monotonic depen-dence of the migration velocity on the excess area ∆,see Fig. 4. For a spherical object (i.e., ∆ = 0) onehas no migration, vm = 0, due to the up-down symme-try. If viscosity is large enough so that tumbling be-comes possible for a sufficiently deflated vesicle, then wesimilarly expect vm ∼ 0, for high enough ∆. Hence,40 2 4 6λ00.20.40.6ψ cα = κ = 1, ∆ = .25FIG. 5: Critical angle ψc separating the basins of attractionfor two coexisting solutions at the centerline.it follows that the absolute value of the migration ve-locity attains a maximum for a certain value of ∆ asillustrated in Fig. 4. This results also agrees with theargument given on the lift force under a linear shearflow[15]. Vesicle approach to the flow centerline alsocan proceed in a non-monotonous manner. the ratioη0R40α/κ (this is a measure of the relative strength ofhydrodynamic and bending stresses) is sufficiently large(typically above 10), the vesicle approaches the centerlinemonotonously. However, for smaller curvature of the flowfield (or higher membrane rigidity) the vesicle trajectoryexhibits damped oscillations about the centerline. Fortypical values α ∼ 0.1(µms)−1, R0 ∼ 10µm, and by us-ing the viscosity of water and κ ∼ 40kBT , one finds thatη0R40α/κ ∼ 0.1. This means these damped oscillationsshould be easily observed experimentally.A new surprising feature we have discovered is the co-existence of shape solutions at the centerline for smallenough curvature of the flow field (see Fig. 5). Moreprecisely, this occurs if η0R40α/κ is lower than about4 − 5. For one solution the longest axis of the vesicleis oriented in the flow direction (bullet-like shape), andfor the other one it is perpendicular to it (parachute-like shape). Each of these solutions has its own basinof attraction, i. e., if the initial angle at the centerlineis smaller than ψc, the bullet-like shape is attained, andotherwise the parachute-like shape. If, however, the cur-vature of the flow is strong enough, the vesicle will as-sume a pronounced parachute-like shape, and the twosolutions will be indistinguishable. We hope to reportfurther on this matter in a future work. The experimen-tal range that is relatively easily accessible is about[21]10−1 < η0R40 α/κ < 10. This means that in principlethis prediction is not devoid of experimental testability.The challenge is to orient the vesicle in the appropriatebasin of attraction. Optical tweezers constitute a possibletool for this task. The slip velocity for the parachute-likeshape is higher, while for a bullet-like vesicle is lowerthan that of a rigid sphere, vs = −1/3αr20 (as deducedfrom Faxen’s law [22]). Since the slip velocity for thebullet-shape is smaller than that of the parachute thedissipation implied by this morphology should be lower.In summary, the dynamical behavior of a vesicle in un-bounded quadratic flow has revealed number of novel fea-tures that stem from the subtle interplay between mem-brane mechanics and shear gradients due to flow cur-vature. The analysis has been also performed for an ax-isymmetric Poiseuille flow and yields the same qualitativeresults.C.M. Acknowledges financial support from CNES(Centre National d’Etudes Spatiales) and ANR (MOSI-COB project).[1] M. Kraus, W. 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Vesicles in Poiseuille Flow

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Vesicles in Poiseuille Flow

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