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Lateral migration of a capsule in plane shear near a wall

Published online by Cambridge University Press:  20 December 2013

Rajesh Kumar Singh ,
Xiaoyi Li  and
Kausik Sarkar
Rajesh Kumar Singh
Affiliation:
Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA Biomechanics and Movement Science Program, University of Delaware, Newark, DE 19716, USA
Xiaoyi Li
Affiliation:
Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA
Kausik Sarkar*
Affiliation:
Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA Biomechanics and Movement Science Program, University of Delaware, Newark, DE 19716, USA Department of Mechanical and Aerospace Engineering, George Washington University, Washington, DC 20052, USA
*
†Email address for correspondence: [email protected]
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Abstract

The migration of a capsule enclosed by an elastic membrane in a wall-bounded linear shear is investigated using a front-tracking method. A detailed comparison with the migration of a viscous drop is presented varying the capillary number (in the case of a capsule, the elastic capillary number) and the viscosity ratio. In both cases, the deformation breaks the flow reversal symmetry and makes them migrate away from the wall. They quickly go through a transient evolution to eventually reach a quasi-steady state where the dynamics becomes independent of the initial position and only depends on the wall distance. Previous analytical theories predicted that for a viscous drop, in the quasi-steady state, the migration and slip velocities scale approximately with the square of the inverse of the drop–wall separation, whereas the drop deformation scales as the inverse cube of the separation. These power law relations are shown to hold for a capsule as well. The deformation and inclination angle of the capsule and the drop at the same wall separation show a crossover in their variation with the capillary number: the capsule shows a steeper variation than that of the drop for smaller capillary numbers and slower variation than the drop for larger capillary numbers. Using the Green’s function of Stokes flow, a semi-analytic theory is presented to show that the far-field stresslet that causes the migration has two distinct contributions from the interfacial stresses and the viscosity ratio, with competing effects between the two defining the dynamics. It predicts the scaling of the migration velocity with the capsule–wall separation, however, matching with the simulated result very well only away from the wall. A phenomenological correlation for the migration velocity as a function of elastic capillary number, wall distance and viscosity ratio is developed using the simulation results. The effects of different membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak, and Skalak – are briefly investigated to show that the behaviour remains similar for different equations.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Biological Fluid Dynamics: Capsule/cell dynamics Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 739 , 25 January 2014 , pp. 421 - 443
Copyright
©2013 Cambridge University Press 

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References

Abkarian, M. & Viallat, A. 2005 Dynamics of vesicles in a wall-bounded shear flow. Biophys. J. 89, 1055–1066.Google Scholar
Aggarwal, N. & Sarkar, K. 2007 Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear. J. Fluid Mech. 584, 1–21.Google Scholar
Aggarwal, N. & Sarkar, K. 2008a Effects of matrix viscoelasticity on viscous and viscoelastic drop deformation in a shear flow. J. Fluid Mech. 601, 63–84.Google Scholar
Aggarwal, N. & Sarkar, K. 2008b Rheology of an emulsion of viscoelastic drops in steady shear. J. Non-Newtonian Fluid Mech. 150, 19–31.Google Scholar
Bagchi, P. & Kalluri, R. M. 2009 Dynamics of nonspherical capsules in shear flow. Phys. Rev. E (80)016307.Google Scholar
Barbee, J. H. & Cokelet, G. R. 1971 Fahraeus effect. Microvasc. Res. 3, 6–16.Google Scholar
Barthes-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831–853.Google Scholar
Barthes-Biesel, D., Diaz, A. & Dhenin, E. 2002 Effect of constitutive laws for two-dimensional membranes on flow-induced capsule deformation. J. Fluid Mech. 460, 211–222.Google Scholar
Barthes-Biesel, D. & Rallison, J. M. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.CrossRefGoogle Scholar
Blake, J. R. 1971 Image system for a Stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc. 70, 303–310.Google Scholar
Callens, N., Minetti, C., Coupier, G., Mader, M. A., Dubois, F., Misbah, C. & Podgorski, T. 2008 Hydrodynamic lift of vesicles under shear flow in microgravity. Eur. Phys. Lett. (83)24002.Google Scholar
Cantat, I. & Misbah, C. 1999 Lift force and dynamical unbinding of adhering vesicles under shear flow. Phys. Rev. Lett. 83, 880–883.CrossRefGoogle Scholar
Chaffey, C. E., Brenner, H. & Mason, S. G. 1965 Particle motions in sheared suspensions. XVIII. Wall migration (theoretical). Rheol. Acta 4, 64–72.Google Scholar
Chan, P. C. H. & Leal, L. G. 1979 Motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92, 131–170.Google Scholar
Coupier, G., Kaoui, B., Podgorski, T. & Misbah, C. 2008 Noninertial lateral migration of vesicles in bounded Poiseuille flow. Phys. Fluids (20)111702.Google Scholar
Damiano, E. R., Westheider, J., Tozeren, A. & Ley, K. 1996 Variation in the velocity, deformation, and adhesion energy density of leukocytes rolling within venules. Circulat. Res. 79, 1122–1130.Google Scholar
Doddi, S. K. & Bagchi, P. 2008 Lateral migration of a capsule in a plane Poiseuille flow in a channel. Intl J. Multiphase Flow 34, 966–986.Google Scholar
Doddi, S. K. & Bagchi, P. 2009 Three-dimensional computational modelling of multiple deformable cells flowing in microvessels. Phys. Rev. E (79)046318.Google Scholar
Eggleton, C. D. & Popel, A. S. 1998 Large deformation of red blood cell ghosts in a simple shear flow. Phys. Fluids 10, 1834–1845.Google Scholar
Evans, E. A. & Skalak, R. 1979 Mechanics and thermodynamics of biomembranes. Part 2. CRC Critical Rev. Bioengng 3, 331–418.Google Scholar
Fåhraeus, R. 1929 The suspension stability of blood. Phys. Rev. 9, 241–274.Google Scholar
Farutin, A. & Misbah, C. 2013 Analytical and numerical study of three main migration laws for vesicles under flow. Phys. Rev. Lett. (110)108104.CrossRefGoogle ScholarPubMed
Grandchamp, X., Coupier, G., Srivastav, A., Minetti, C. & Podgorski, T. 2013 Lift and down-gradient shear-induced diffusion in red blood cell suspensions. Phys. Rev. Lett. (110)108101.Google Scholar
Imaeda, T. 2000 Shear-induced migration of a droplet in the presence of a plane wall. Physica A 285, 306–314.Google Scholar
Kaoui, B., Ristow, G. H., Cantat, I., Misbah, C. & Zimmermann, W. 2008 Lateral migration of a two-dimensional vesicle in unbounded Poiseuille flow. Phys. Rev. E 77, 021903.Google Scholar
Karnis, A. & Mason, S. G. 1967 Particle motions in sheared suspensions. XXIII. Wall migration of fluid drops. J. Colloid Interface Sci. 24, 164–169.Google Scholar
Kennedy, M. R., Pozrikidis, C. & Skalak, R. 1994 Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Comput. Fluids 23, 251–278.Google Scholar
Lac, E., Barthes-Biesel, D., Pelekasis, N. A. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303–334.Google Scholar
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12, 435–476.Google Scholar
Lefebvre, Y. & Barthes-Biesel, D. 2007 Motion of a capsule in a cylindrical tube: effect of membrane pre-stress. J. Fluid Mech. 589, 157–181.Google Scholar
Li, H. & Ma, G. 2010 Modelling performance of a two-dimensional capsule in a microchannel flow: long-term lateral migration. Phys. Rev. E 82, 026304.Google Scholar
Li, X. Y. & Sarkar, K. 2005 Drop dynamics in an oscillating extensional flow at finite Reynolds numbers. Phys. Fluids 17, 027103.Google Scholar
Li, X. Y. & Sarkar, K. 2006 Drop deformation and breakup in a vortex at finite inertia. J. Fluid Mech. 564, 1–23.Google Scholar
Li, X. Y. & Sarkar, K. 2008 Front-tracking simulation of a liquid capsule enclosed by an elastic membrane. J. Comput. Phys. 227, 4998–5018.Google Scholar
Lorz, B., Simson, R., Nardi, J. & Sackmann, E. 2000 Weakly adhering vesicles in shear flow: tanktreading and anomalous lift force. Europhys. Lett. 51, 468–474.Google Scholar
Mukherjee, S. & Sarkar, K. 2009 Effects of viscosity ratio on deformation of a viscoelastic drop in a Newtonian matrix under steady shear. J. Non-Newtonian Fluid Mech. 160, 104–112.Google Scholar
Mukherjee, S. & Sarkar, K. 2010 Effects of viscoelasticity on the retraction of a sheared drop. J. Non-Newtonian Fluid Mech. 165, 340–349.Google Scholar
Mukherjee, S. & Sarkar, K. 2011 Viscoelastic drop falling through a viscous medium. Phys. Fluids 23, 013101.Google Scholar
Mukherjee, S. & Sarkar, K. 2013 Effects of matrix viscoelasticity on the lateral migration of a deformable drop in a wall-bounded shear. J. Fluid Mech. 727, 318–345.Google Scholar
Olapade, P. O., Singh, R. K. & Sarkar, K. 2009 Pairwise interactions between deformable drops in free shear at finite inertia. Phys. Fluids (21)063302.Google Scholar
Olla, P. 1997a The lift on a tank-treading ellipsoidal cell in a shear flow. J. De Physique Ii 7, 1533–1540.Google Scholar
Olla, P. 1997b The role of tank-treading motions in the transverse migration of a spheroidal vesicle in a shear flow. J. Phys. A 30, 317–329.CrossRefGoogle Scholar
Olla, P. 1999 Simplified model for red cell dynamics in small blood vessels. Phys. Rev. Lett. 82, 453–456.Google Scholar
Pranay, P., Henriquez-Rivera, R. G. & Graham, M. D. 2012 Depletion layer formation in suspensions of elastic capsules in Newtonian and viscoelastic fluids. Phys. Fluids (24)061902.Google Scholar
Reinke, W., Gaehtgens, P. & Johnson, P. C. 1987 Blood-viscosity in small tubes: effect of shear rate, aggregation, and sedimentation. Am. J. Physiol. 253, H540–H547.Google Scholar
Rosenkilde, C. E. 1967 Surface-energy tensors. J. Math. Phys. 8, 84–88.Google Scholar
Sarkar, K. & Schowalter, W. R. 2000 Deformation of a two-dimensional viscoelastic drop at non-zero Reynolds number in time-periodic extensional flows. J. Non-Newtonian Fluid Mech. 95, 315–342.Google Scholar
Sarkar, K. & Schowalter, W. R. 2001 Deformation of a two-dimensional drop at non-zero Reynolds number in time-periodic extensional flows: numerical simulation. J. Fluid Mech. 436, 177–206.Google Scholar
Sarkar, K. & Singh, R. K. 2013 Spatial ordering due to hydrodynamic interactions between a pair of colliding drops in a confined shear. Phys. Fluids (25)051702.CrossRefGoogle Scholar
Seifert, U. 1999 Hydrodynamic lift on bound vesicles. Phys. Rev. Lett. 83, 876–879.Google Scholar
Shapira, M. & Haber, S. 1990 Low Reynolds number motion of a droplet in shear flow including wall effects. Intl J. Multiphase Flow 16, 305–321.Google Scholar
Shin, S. J. & Sung, H. J. 2011 Inertial migration of an elastic capsule in a Poiseuille flow. Phys. Rev. E 83, 046321.Google Scholar
Shrivastava, S. & Tang, J. 1993 Large deformation finite-element analysis of nonlinear viscoelastic membranes with reference to thermoforming. J. Strain Anal. Engng Design 28, 31–51.Google Scholar
Singh, R. K. & Sarkar, K. 2009 Effects of viscosity ratio and three-dimensional positioning on hydrodynamic interactions between two viscous drops in a shear flow at finite inertia. Phys. Fluids (21)103303.Google Scholar
Singh, R. K. & Sarkar, K. 2011 Inertial effects on the dynamics, streamline topology and interfacial stresses due to a drop in shear. J. Fluid Mech. 683, 149–171.CrossRefGoogle Scholar
Skalak, R., Tozeren, A., Zarda, R. P. & Chien, S. 1973 Strain energy function of red blood-cell membranes. Biophys. J. 13, 245–280.Google Scholar
Smart, J. R. & Leighton, D. T. 1991 Measurement of the drift of a droplet due to the presence of a plane. Phys. Fluids A 3, 21–28.Google Scholar
Sukumaran, S. & Seifert, U. 2001 Influence of shear flow on vesicles near a wall: a numerical study. Phys. Rev. E (64)011916.Google Scholar
Tangelder, G. J., Teirlinck, H. C., Slaaf, D. W. & Reneman, R. S. 1985 Distribution of blood platelets flowing in arterioles. Am. J. Physiol. 248, H318–H323.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. Series A 146, 0501–0523.Google Scholar
Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y. J. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759.Google Scholar
Uijttewaal, W. S. J. & Nijhof, E. J. 1995 The motion of a droplet subjected to linear shear flow including the presence of a plane wall. J. Fluid Mech. 302, 45–63.Google Scholar
Uijttewaal, W. S. J., Nijhof, E. J. & Heethaar, R. M. 1993 Droplet migration, deformation, and orientation in the presence of a plane wall: a numerical study compared with analytical theories. Phys. Fluids A 5, 819–825.Google Scholar