This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.We investigate the mechanical equilibrium state of an oblate capsule when its revolution axis is
initially off the shear plane. We consider an oblate capsule with an aspect ratio of 0.5 and a strain-hardening
membrane. The three-dimensional fluid-structure interaction problem is solved numerically by coupling a
finite element method with a boundary integral method. The capsule converges towards the same mechanical
equilibrium state whatever the initial orientation. This equilibrium depends on the capillary number Ca, which
compares the viscous to the elastic forces and on the viscosity ratio   between the internal and external fluids.
For   = 1, the tumbling and swinging motions, observed when the revolution axis is initially in the shear plane,
are mechanically stable until Ca   1; when Ca is further increased, the capsule assumes the rolling motion
that is observed when its revolution axis is initially aligned with the vorticity axis. When   is increased, the tumbling-to-swinging transition appears for higher Ca and the swinging-to-rolling transition for lower Ca. For
  5, the swinging regime completely disappears: depending on Ca, it is then either the tumbling or the rolling motion that is the mechanical equilibrium state

4th Micro and Nano Flows Conference (MNF2014)

Barthes-Biesel, D

Dupont, C

Salsac, AV

English

Brunel University Research Archive

4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014Off-plane motion of an oblate capsule in a simple shear flow.Anne-Virginie SALSAC1,∗, Claire DUPONT1,2, Fabien DELAHAYE1, Dominique BARTHES-BIESEL1∗Corresponding author: Tel.: +33 (0)3 44 23 73 38; Email: a.salsac@utc.fr1Biomechanics and Bioengineering Laboratory, UMR CNRS 7338,Université de Technologie de Compiègne, France2Solids Mechanics Laboratory, UMR CNRS 7649, Ecole Polytechnique, FranceAbstract: We investigate the mechanical equilibrium state of an oblate capsule when its revolution axis isinitially off the shear plane. We consider an oblate capsule with an aspect ratio of 0.5 and a strain-hardeningmembrane. The three-dimensional fluid-structure interaction problem is solved numerically by coupling afinite element method with a boundary integral method. The capsule converges towards the same mechanicalequilibrium state whatever the initial orientation. This equilibrium depends on the capillary number Ca, whichcompares the viscous to the elastic forces and on the viscosity ratio λ between the internal and external fluids.For λ = 1, the tumbling and swinging motions, observed when the revolution axis is initially in the shear plane,are mechanically stable until Ca ∼ 1; when Ca is further increased, the capsule assumes the rolling motionthat is observed when its revolution axis is initially aligned with the vorticity axis. When λ is increased, thetumbling-to-swinging transition appears for higher Ca and the swinging-to-rolling transition for lower Ca. Forλ ≥ 5, the swinging regime completely disappears: depending on Ca, it is then either the tumbling or therolling motion that is the mechanical equilibrium state.Keywords: oblate capsule, finite element method, boundary integral method, off plane motion1. IntroductionMicrocapsules consist of a liquid internalmedium protected within a thin membrane. Thedynamics of a spheroidal capsule in simpleshear flow has recently received a lot of at-tention (Sui et al., 2008; Walter et al., 2011;Dupont et al., 2013), owing to its relevance tothe motion of a red blood cell. Most studieshave modelled the capsule with the revolutionaxis in the shear plane, which is a special case,as it is an equilibrium configuration in Stokesflow conditions. It has then been shown that thecapsule motion depends on the capillary num-ber Ca, ratio of the viscous to elastic forces.At low Ca, the capsule has a tumbling motionand rotates like a solid particle. At higher val-ues of Ca, the capsule experiences a transitionwhere two of its in-plane major axes are almostequal. Then as Ca is further increased, the cap-sule takes a swinging motion where it oscillatesabout the straining direction, while the mem-brane rotates around the deformed shape (fluid-like motion).The off-plane motion of a prolate capsulein shear flow has been recently studied (Dupontet al., 2013; Cordasco and Bagchi, 2013). It isfound that the low Ca tumbling motion in theshear plane is mechanically unstable. Indeed,for any initial orientation, the capsule placesits revolution axis along the vorticity axis per-pendicularly to the shear plane (rolling regime).As Ca is increased, the capsule longest axismoves away from the vorticity axis and pre-cesses around it. At still higher values of Ca,the capsule longest axis goes to the shear planewhere a stable swinging regime is observed.Dupont et al. (2013) have shown that those sta-ble equilibrium states do not depend on the ini-tial orientation of the capsule revolution axisrelative to the shear plane.1The case of oblate capsules has been con-sidered by Cordasco and Bagchi (2013), whofind that at moderate Ca, the capsule tends toplace its short axis in the shear plane where ittakes different motions from tumbling, kayak-ing to swinging as a function of Ca and as-pect ratio. However, as an oblate capsule takesquite a long time to reach equilibrium, mostof the situations considered by Cordasco andBagchi (2013) were not time converged. Omoriet al. (2012) have also simulated the motion ofan oblate capsule with a strain-softening neo-Hookean membrane and with short-to-long axisratio equal to 0.6. However, they have only con-sidered two capillary numbers (Ca = 0.3 and1.0) and have not studied the stability of thetumbling motion, which takes place at low Ca.It follows that a thorough parametric studyof the dynamics of oblate capsules is neededto fully understand its behavior. The objectiveof the paper is to investigate the influence ofthe capillary number, viscosity ratio and initialorientation on the equilibrium configurations ofan oblate capsule until an equilibrium state isreached.2. MethodWe consider a capsule, which is oblate in itsreference undeformed state with aspect ratioa/b = 0.5, where 2a denotes the length of therevolution axis and 2b the length of the two or-thogonal axes. We choose as length scale ` =(ab2)1/3, the radius of the sphere with the samevolume as the capsule. The capsule is filled witha Newtonian incompressible fluid with viscos-ity λµ and suspended in an unbounded New-tonian incompressible fluid with viscosity µ,where λ is the internal to external viscosity ra-tio. The internal and surrounding fluid havethe same density, thus excluding gravity effect.The Reynolds number of the flow is assumedto be very small, so that the internal and exter-nal flows are governed by the Stokes equations.The capsule is subjected to a simple shear flowwith shear rate γ˙: the base flow velocity isv∞ = γ˙yex (1)in the laboratory reference frame. The capsuleand flow centers O are identical.At time γ˙t = 0, the capsule revolution axismakes an angle ζ0 with the vorticity axis and itsprojection in the shear plane is aligned with thex-axis (Fig. 1).M03M01exeyeze′zζ0Fig. 1: Reference configuration of the oblatecapsule subjected to a simple shear flow at γ˙t =0. The initial capsule orientation is given by ζ0,the angle between the capsule revolution axise′z and the flow vorticity axis ez.The capsule has a very thin membrane,modeled as an isotropic hyperelastic surfacewith shear modulusGs and area dilatation mod-ulus Ks. The bending resistance of the capsulemembrane is neglected. We consider that thecapsule membrane behaviour is described bythe Skalak law, a strain-hardening constitutivelaw, initially proposed by Skalak et al. (1973)to model the red blood cell membrane. In thiscase, the principal Cauchy elastic tensions τ1and τ2 are given in terms of the in-plane prin-cipal stretch ratios λ1 and λ2 byτ1 =Gsλ1λ2[λ21(λ21− 1) + C(λ1λ2)2((λ1λ2)2 − 1)].(2)with the corresponding expression for τ2. Thesurface shear modulus Gs and area dilatationmodulus Ks are related by Ks = Gs(1 + 2C),where C is a constant such that C > −1/2. Thecapsule motion and deformation are thus gov-erned by the membrane constitutive law, the ra-tio Ks/Gs, the capsule initial orientation ζ0, thecapillary number Ca = µγ˙`/Gs, which mea-sures the ratio between the viscous and the elas-tic forces and the viscosity ratio λ between theinternal and surrounding fluids. In this study,we consider C = 1 and we focus on the influ-ence of ζ0, Ca and λ on the capsule dynamics.2The problem is solved numerically by cou-pling a boundary integral technique to computethe fluid flows (inside and outside the capsule)to a finite element method (for the mechanicsof the capsule wall). A Lagrangian tracking ofthe position of the membrane material points isperformed over time (Walter et al., 2010; Foes-sel et al., 2011). One of the advantages of thisnumerical method is that all the problem un-known quantities need to be determined on thecapsule surface only. The latter is meshed bysubdividing sequentially the 20 triangular facesof an icosahedron inscribed in a sphere until thedesired number of elements is reached. Nodesare then added at the middle of all the ele-ment edges and projected onto the sphere inorder to generate second order (P2) elements.This mesh is then deformed into an ellipsoidalmesh with the desired axis ratio. Here we usea mesh with 2562 nodes and 1280 triangularcurved elements. The numerical method is sta-ble when the time step satisfies the conditionγ˙ 4 t < O(hCa), where h is the typical non-dimensional mesh size (Walter et al., 2010). Weuse γ˙4t = 5×10−3 for Ca ≥ 0.5 and decreasethe time step proportionally for lower Ca.The global geometry of the capsule is eval-uated by means of the ellipsoid of inertia of thedeformed shape. We denote Li the half lengthsof the principal axes of the ellipsoid of inertia(L1 > L2 > L3). The capsule global motion ismeasured from the position of the short axis tipM t3at time t, computed from the ellipsoid of in-ertia at each time step. The membrane rotationis measured from the motion of the point M03, aspecific mesh node initially located on the cap-sule short axis (Fig. 1). In order to identifyclearly the motion of the short axis in space, weintroduce the angle ζ¯ between the capsule shortaxis OM t3and the shear plane.3. Results and discussionAs for prolate capsules (Dupont et al., 2013),we find that the capsule converges towards aunique equilibrium motion, which does not de-pend on the initial orientation ζ0 ∈]0o, 90o[. Forexample, Fig. 2 shows, for Ca = 0.01, that theangle ζ¯ tends to zero for three different initialpositions ζ0. The equilibrium motion of the cap-sule then depends only on the capillary numberCa and viscosity ratio λ. We first focus on theequilibrium states which are obtained for λ = 1.For small to moderate flow strengths cor-responding to Ca < 0.9, the short axis mi-grates towards the shear plane for any initialorientation ζ0 (Fig. 2, 3). Some oscillationsof ζ are still observed in Fig. 2 at equilibriumstate. Although they do not completely dis-appear, they remain of very small amplitude.For Ca = 0.01, it is difficult to determine pre-cisely which axis corresponds to the short axisat equilibrium, as L2 ≈ L3. Like Omori et al.(2012), we therefore also find that the point M03migrates to the shear plane. It follows that thesituation is identical to the one considered byWalter et al. (2011) where the capsule revolu-tion axis is initially positioned in the shear plane(ζ0 = 90◦). Walter et al. (2011) find that at lowCa (Ca < 0.02), the capsule takes a quasi-solid tumbling motion. For Ca ∈ [0.02, 0.05],the capsule has a transition motion: it takes anearly isotropic shape in the shear plane withnearly equal in-plane principal axes. For highervalues of Ca (Ca > 0.5), the capsule takes aswinging motion, where the axis OM t3has aslightly varying length and orientation with re-spect to the flow streamlines. Note that the timethe capsule takes to realign itself with the shearplane is quite long and increases with the initialdistance of the capsule tip from the shear planeas shown for example in Fig. 2.-0.4-0.2000.40.250 100 150 200ζ0 = 5◦ζ0 = 75◦ζ0 = 45◦ζ¯piγ˙tFig. 2: Time evolution of the angle ζ¯ betweenthe capsule small axis OM t3and the shear planefor three initial orientations ζ0 at Ca = 0.01(λ = 1).3When Ca is further increased (Ca ≥ 0.9),the principal axes, which were in the shearplane, exhibit small oscillations both about andwithin the shear plane: the equilibrium state ofthe capsule thus experiences a transition froma swinging to a swinging-oscillating motion,where the point M03precesses around the vor-ticity axis with a constant mean inclination.Then, for Ca > 1.1, the point M03convergestowards the vorticity axis and the capsule takesa rolling motion that is identical to the one ob-served when the short axis is initially alignedwith the vorticity axis (ζ0 = 0◦). The membranetank-treads around a steady shape as shownin Fig. 4. This motion was also observed byOmori et al. (2012) for Ca = 1. Note that whenthe capsule is initially positioned with ζ0 = 0◦,the rolling motion is in principle a solution ofthe problem for any value of Ca. However, thepresent results demonstrate that this rolling mo-tion is unstable for Ca ≤ 1.1.-0.4-0.2000.40.250 100 150 200Ca = 0.01− ζ0 = 5◦Ca = 0.08− ζ0 = 30◦Ca = 0.3− ζ0 = 45◦ζ¯piγ˙tFig. 3: Time evolution of the angle ζ¯ betweenthe capsule small axis OM t3and the shear planefor different values of Ca and initial orientation(λ = 1).90 92 94 98 100O exeyFig. 4: Rolling motion over one half period atsteady state for Ca = 1.2 and ζ0 = 85◦ (λ = 1).The points M01() and M03(•) were initially atthe tip of a long and the short axis respectively(Fig. 1)In conclusion, for λ = 1 and dependingon Ca, the capsule aligns its axis in the shearplane where it takes a quasi-solid tumbling mo-tion (Ca < 0.02), a transition motion (Ca ∈[0.02, 0.05]) and a swinging motion (Ca < 0.9).For higher values of Ca, the capsule starts mov-ing out of the shear plane and finally, for Ca ≥1.1, it aligns its axis with the vorticity directionand takes a rolling motion.As the viscosity ratio increases, the Ca lim-its of the different regimes evolve with λ (Fig.5):• The minimum value of Ca for which therolling motion is mechanically stable, de-creases.• The transition between the tumbling andswinging motions in the shear plane ap-pears for higher Ca.• For λ > 5, the swinging motion com-pletely disappears and the mechanicalequilibrium states are only the swingingand the rolling motions.00 2 4 6 8 10 120.11CaλRollingSwingingTumblingFig. 5: Capsule shape evolution over one halfperiod at steady state for Ca = 1.2 and ζ0 =85◦ (λ = 1).4. ConclusionWe have investigated the motion of anoblate capsule in shear flow and found that itdepends on the flow strength and the viscosityof the capsule internal liquid as compared to thesuspending one, but not on the initial inclina-tion. We have shown that oblate capsules mayexhibit only three possible equilibrium states(tumbling, swinging, rolling). There is there-fore no procession motion at equilibrium. Itis important to note that the time required toreach an equilibrium state is quite large. For4λ = 1, it is of order γ˙t = 200 and can increasesignificantly when λ increases. In practice, thetime window to observe the motion of artificialcapsules or red blood cells experimentally isbetween 10 s and 50 s (Abkarian et al, 2007,2008). At low shear rates, the capsules moveslowly, which facilitates the recording of theirdynamics in a shear flow. However, the non-dimensional time of observation γ˙t is too smallfor the capsules to reach the mechanical equi-librium state: only the transition motion can beobserved in vitro at low shear rates. When theshear rate is increased, the time window γ˙t be-comes larger. Thus the higher the shear rate, theeasier it becomes to design experiments whereequilibrium is reached. The price to pay is thatthe capture of capsule dynamics is more diffi-cult and requires even more sophisticated visu-alization means than for low shear experiments.AcknowledgmentsThis research was funded by Ecole Poly-technique (PhD scholarship), the Conseil Ré-gional de Picardie (MODCAP grant), by theFrench Ministère de la Recherche (Pilcam2grant) and by the French Agence Nationale de laRecherche (CAPSHYDR grant ANR-11-BS09-013, Labex MS2T ANR-11-IDEX-0004-02).ReferencesCordasco, D., Bagchi, P., 2013. Orbital driftof capsules and red blood cells in shear flow.Phys. Fluids 25, 091902.Dupont, C., Salsac, A.V., Barthès-Biesel, D.,2013. Off-plane motion of a prolate capsulein shear flow. J Fluid Mech. 721, 180–198.Foessel, E., Walter, J., Salsac, A.V., Barthès-Biesel, D., 2011. Influence of internal vis-cosity on the large deformation and bucklingof a spherical capsule in a simple shear flow.J Fluid Mech 672, 477–486.Omori, T., Imai, Y., Yamaguchi, T., Ishikawa,T., 2012. Reorientation of a nonsphericalcapsule in creeping shear flow. PRL 108,138102(5).Skalak, R., Tozeren, A., Zarda, R.P., Chien, S.,1973. Strain energy function of red blood cellmembranes. Biophys. J . 13, 245–264.Sui, Y., Low, H.T., Chew, Y.T., Roy, P., 2008.Tank-treading, swinging, and tumbling ofliquid-filled elastic capsules in shear flow.PRE 77, 016310.Walter, J., Salsac, A.V., Barthès-Biesel, D.,2011. Ellipsoidal capsules in simple shearflow: prolate versus oblate initial shapes. JFluid Mech 676, 318–347.Walter, J., Salsac, A.V., Barthès-Biesel, D., LeTallec, P., 2010. Coupling of finite elementand boundary integral methods for a capsulein a Stokes flow. Int J Numer Methods Eng83, 829–850.5

Off-plane motion of an oblate capsule in a simple shear flow

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Off-plane motion of an oblate capsule in a simple shear flow

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