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Rheology of a dilute two-dimensional suspension of vesicles

Published online by Cambridge University Press:  22 April 2010

GIOVANNI GHIGLIOTTI ,
THIERRY BIBEN  and
CHAOUQI MISBAH
GIOVANNI GHIGLIOTTI
Affiliation:
Laboratoire de Spectrométrie Physique, UMR 5588, 140 Avenue de la Physique, Université Joseph Fourier Grenoble I, and CNRS, 38402 Saint Martin d'Hères, France
THIERRY BIBEN
Affiliation:
Université de Lyon, Laboratoire PMCN, Université Claude Bernard-Lyon I et CNRS, 43 bvd du 11 Novembre1918, 69622 Villeurbanne, France
CHAOUQI MISBAH*
Affiliation:
Laboratoire de Spectrométrie Physique, UMR 5588, 140 Avenue de la Physique, Université Joseph Fourier Grenoble I, and CNRS, 38402 Saint Martin d'Hères, France
*
†Email address for correspondence: chaouqi.misbah@ujf-grenoble.fr
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Abstract

The rheology of a dilute two-dimensional suspension of vesicles (closed bags of a lipid bilayer membrane) is studied by numerical simulations. The numerical methods used are based on the boundary integral formulation (Green's function technique) and the phase field approach, which has become a quite popular and powerful tool for the numerical study of free-boundary problems. The imposed flow is an unbounded linear shear. The goal of the present study is to elucidate the link between the rheology of vesicle suspensions and the microscopic dynamics of the constituent particles (tank-treading and tumbling motions). A comparison with emulsion rheology reveals the central role played by the membrane. In particular, at low viscosity ratio λ (defined as the viscosity of the internal fluid over that of the ambient one), the effective viscosity decreases with λ, while the opposite trend is exhibited by emulsions, according to the classical Taylor result. This fact is explained by considering the velocity field of the ambient fluid. The area-incompressibility of the vesicle membrane modifies the surrounding velocity field in a quite different manner than what a drop does. The overall numerical results in two dimensions are in reasonable agreement with the three-dimensional analytical theory derived recently in the small deformation limit (quasi-spherical shapes). The finding that the simulations in two dimensions capture the essential features of the three-dimensional rheology opens the way for extensive and large-scale simulations for semi-dilute and concentrated vesicle suspensions. We discuss some peculiar effects exhibited by the instantaneous viscosity in the tumbling regime of vesicles. Finally, the rheology is found to be relatively insensitive to shear rate.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 653 , 25 June 2010 , pp. 489 - 518
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Abkarian, M. & Viallat, A. 2008 Vesicles and red blood cells in shear flow. Soft Matter 4, 653–657.Google Scholar
Bagchi, P., Johnson, P. C. & Popel, A. S. 2005 Computational fluid dynamic simulation of aggregation of deformable cells in a shear flow. J. Biomech. Engng 127, 1070–1080.Google Scholar
Barthès-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.Google Scholar
Batchelor, G. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (3), 545–570.CrossRefGoogle Scholar
Beaucourt, J., Biben, T. & Misbah, C. 2004 a Optimal lift force on vesicles near a compressible substrate. Europhys. Lett. 67, 676–682.Google Scholar
Beaucourt, J., Rioual, F., Séon, T., Biben, T. & Misbah, C. 2004 b Steady to unsteady dynamics of a vesicle in a flow. Phys. Rev. E 69, 011906.Google Scholar
Belzons, M., Blanc, R., Bouillot, J.-L. & Camion, C. 1981 Viscosité d'une suspension diluée et bidimensionnelle de sphères. C. R. Acad. Sci. Paris 292, 5.Google Scholar
Biben, T. 2005 Phase-field models for free-boundary problems. Eur. J. Phys. 26, 47–55.Google Scholar
Biben, T., Kassner, K. & Misbah, C. 2005 Phase-field approach to three-dimensional vesicle dynamics. Phys. Rev. E 72, 041921.CrossRefGoogle ScholarPubMed
Biben, T. & Misbah, C. 2002 An advected-field method for deformable entities under flow. Eur. Phys. J. B 29, 311–316.Google Scholar
Biben, T. & Misbah, C. 2003 Tumbling of vesicles under shear flow within an advected-field approach. Phys. Rev. E 67, 031908.CrossRefGoogle ScholarPubMed
Boskovic, S., Chon, J. W. M., Mulvaney, P. & Sader, J. E. 2002 Rheological measurements using microcantilevers. J. Rheol. 46 (4), 891–899.CrossRefGoogle Scholar
Brady, J. 1984 The Einstein viscosity correction in n dimensions. Intl J. Multiphase Flow 10, 113–114.Google Scholar
Canham, P. B. 1970 The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theoret. Biol. 26, 61–81.Google Scholar
Cantat, I. 1999 Dynamique de vésicules en adhésion. PhD thesis, Université Joseph Fourier, Grenoble.Google Scholar
Cantat, I., Kassner, K. & Misbah, C. 2003 Vesicles in haptotaxis with hydrodynamical dissipation. Eur. Phys. J. E 10, 175–189.CrossRefGoogle Scholar
Cantat, I. & Misbah, C. 1999 Dynamics and similarity laws for adhering vesicles in haptotaxis. Phys. Rev. Lett. 83 (1), 235–238.Google Scholar
Coupier, G., Kaoui, B., Podgorski, T. & Misbah, C. 2008 Noninertial lateral migration of vesicles in bounded Poiseuille flow. Phys. Fluids 20, 111702.CrossRefGoogle Scholar
Danker, G., Biben, T., Podgorski, T., Verdier, C. & Misbah, C. 2007 Dynamics and rheology of a dilute suspension of vesicles: higher-order theory. Phys. Rev. E 76, 041905.Google Scholar
Danker, G. & Misbah, C. 2007 Rheology of a dilute suspension of vesicles. Phys. Rev. Lett. 98, 088104.Google Scholar
Danker, G., Verdier, C. & Misbah, C. 2008 Rheology and dynamics of vesicle suspension in comparison with droplet emulsion. J. Non Newton. Fluid Mech. 152 (1–3), 156–167 (Fourth International workshop on Nonequilibrium Theromdynamics and Complex Fluids).Google Scholar
Danker, G., Vlahovska, P. M. & Misbah, C. 2009 Vesicles in Poiseuille flow. Phys. Rev. Lett. 102 (14), 148102.Google Scholar
Deschamps, J., Kantsler, V. & Steinberg, V. 2009 Phase diagram of single vesicle dynamical states in shear flow. Phys. Rev. Lett. 102 (11), 118105.Google Scholar
Drochon, A. 2003 Rheology of dilute suspensions of red blood cells: experimental and theoretical approaches. Eur. Phys. J. AP 22 (2), 155–162.CrossRefGoogle Scholar
Edidin, M. 2003 Lipids on the frontier: a century of cell-membranes bilayer. Nature 4, 414–418.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
Einstein, A. 1906 Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 19, 289–306.Google Scholar
Einstein, A. 1911 Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 34, 591–592.CrossRefGoogle Scholar
Evans, E. A., Waugh, R. & Melnik, L. 1976 Elastic area compressibility modulus of red cell membrane. Biophys. J. 16, 585–595.Google Scholar
Finken, R., Lamura, A., Seifert, U. & Gompper, G. 2008 Two-dimensional fluctuating vesicles in linear shear flow. Eur. Phys. J. E 25, 309–321.Google Scholar
Folch, R., Casademunt, J., Hernández-Machado, A. & Ramiírez-Piscina, L. 1999 Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. Part 1. Theoretical approach. Phys. Rev. E 60, 1724–1733.Google Scholar
Frankel, N. A. & Acrivos, A. 1970 The constitutive equation for a dilute emulsion. J. Fluid Mech. 44 (1), 65–78.CrossRefGoogle Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. 28, 693–703.Google Scholar
Hohenberg, P. & Halperin, B. 1977 Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479.Google Scholar
Jamet, D. & Misbah, C. 2007 Towards a thermodynamically consistent picture of the phase-field model of vesicles: local membrane incompressibility. Phys. Rev. E 76, 051907.Google Scholar
Jamet, D. & Misbah, C. 2008 a Thermodynamically consistent picture of the phase-field model of vesicles: elimination of the surface tension. Phys. Rev. E 78, 041903.Google Scholar
Jamet, D. & Misbah, C. 2008 b Toward a thermodynamically consistent picture of the phase-field model of vesicles: curvature energy. Phys. Rev. E 78, 031902.CrossRefGoogle Scholar
Kantsler, V., Segre, E. & Steinberg, V. 2008 Dynamics of interacting vesicles and rheology of vesicle suspension in shear flow. Europhys. Lett. 82 (5), 58005.CrossRefGoogle Scholar
Kantsler, V. & Steinberg, V. 2005 Orientation and dynamics of a vesicle in tank-treading motion in shear flow. Phys. Rev. Lett. 95, 258101.CrossRefGoogle ScholarPubMed
Kantsler, V. & Steinberg, V. 2006 Transition to tumbling and two regimes of tumbling motion of a vesicle in shear flow. Phys. Rev. Lett. 96, 036001.Google Scholar
Kaoui, B., Biros, G. & Misbah, C. 2009 Why do red blood cells have asymmetric shapes even in a symmetric flow? Phys. Rev. Lett. 103, 188101.Google Scholar
Kaoui, B., Ristow, G., Cantat, I., Misbah, C. & Zimmermann, W. 2008 Lateral migration of a 2D vesicle in unbounded Poiseuille flow. Phys. Rev. E 77, 021903.Google Scholar
Keller, S. & Skalak, R. 1982 Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120, 27–47.CrossRefGoogle Scholar
Kennedy, M. R., Pozrikidis, C. & Skalak, R. 1994 Motion and deformation of liquid drops, and rheology of dilute emulsions in simple shear flow. Comput. Fluids 23 (2), 251–278.Google Scholar
Kobayashi, R. 1993 Modeling and numerical simulations of dendritic crystal growth. Physica D 63, 410–423.CrossRefGoogle Scholar
Kraus, M., Wintz, W., Seifert, U. & Lipowsky, R. 1996 Fluid vesicles in shear flow. Phys. Rev. Lett. 77, 3685–3688.CrossRefGoogle ScholarPubMed
Lac, E., Morel, A. & Barthès-Biesel, D. 2007 Hydrodynamic interaction between two identical capsules in simple shear flow. J. Fluid Mech. 573, 149–169.CrossRefGoogle Scholar
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.Google Scholar
Lebedev, V. V., Turitsyn, K. S. & Vergeles, S. S. 2008 Nearly spherical vesicles in an external flow. New J. Phys. 10 (4), 043044.Google Scholar
MacMeccan, R. M., Clausen, J. R., Neitzel, G. P. & Aidun, C. K. 2009 Simulating deformable particle suspensions using a coupled lattice-Boltzmann and finite-element method. J. Fluid Mech. 618, 13–39.Google Scholar
Mader, M.-A., Vitkova, V., Abkarian, M., Viallat, A. & Podgorski, T. 2006 Dynamics of viscous vesicles in shear flow. Eur. Phys. J. E 19, 389–397.CrossRefGoogle ScholarPubMed
Maitre, E., Milcent, T., Cottet, G.-H., Raoult, A. & Usson, Y. 2009 Applications of level set methods in computational biophysics. Math. Comput. Model. 49, 2161–2169.Google Scholar
Mauroy, B. 2008 Following red blood cells in a pulmonary capillary. ESAIM Proc. 23, 48–65.CrossRefGoogle Scholar
McWhirter, J. L., Noguchi, H. & Gompper, G. 2009 Flow-induced clustering and alignment of vesicles and red blood cells in microcapillaries. Proc. Natl Acad. Sci. USA 106, 6039–6043.Google Scholar
Messlinger, S., Schmidt, B., Noguchi, H. & Gompper, G. 2009 Dynamical regimes and hydrodynamic lift of viscous vesicles under shear. Phys. Rev. E 80 (1), 011901.Google Scholar
Misbah, C. 2006 Vacillating breathing and tumbling of vesicles under shear flow. Phys. Rev. Lett. 96, 028104.Google Scholar
Noguchi, H. & Gompper, G. 2005 Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc. Natl Acad. Sci. USA 102, 14159–14164.Google Scholar
Olla, P. 2000 The behaviour of closed inextensible membranes in linear and quadratic shear flows. Physica A 278, 87–106.Google Scholar
Pal, R. 2000 Shear viscosity behaviour of emulsions of two immiscible liquids. J. Colloid Interface Sci. 225, 359–366.Google Scholar
Penrose, O. & Fife, P. 1990 Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43, 44–62.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
Pozrikidis, C. 1993 On the transient motion of ordered suspensions of liquid drops. J. Fluid Mech. 246, 301–320.Google Scholar
Pozrikidis, C. 2001 Interfacial dynamics for stokes flow. J. Comput. Phys. 169, 250–301.Google Scholar
Pozrikidis, C. 2003 Numerical simulation of the flow-induced deformation of red blood cells. Ann. Biomed. Engng 31, 1194–1205.CrossRefGoogle ScholarPubMed
Rallison, J. M. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89 (1), 191–200.Google Scholar
Rioual, F., Biben, T. & Misbah, C. 2004 Analytical analysis of a vesicle tumbling under a shear flow. Phys. Rev. E 69, 061914.Google Scholar
Schowalter, W. R., Chaffey, C. E. & Brenner, H. 1968 Rheological behaviour of a dilute emulsion. J. Colloid Interface Sci. 26, 152–160.Google Scholar
Taylor, G. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138, 41–48.Google Scholar
Veerapaneni, S. K., Gueyffier, D., Zorin, D. & Biros, G. 2009 A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D. J. Comput. Phys. 228, 2334–2353.Google Scholar
Vitkova, V., Mader, M.-A., Polack, B., Misbah, C. & Podgorski, T. 2008 Micro–macro link in rheology of erythrocyte and vesicle suspensions. Biophys. J. 95, 33–35.Google Scholar
Vlahovska, P. M. & Gracia, R. S. 2007 Dynamics of a viscous vesicle in linear flows. Phys. Rev. E 75 (1), 016313.Google Scholar
van der Waals, J. 1979 The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J. Stat. Phys. 20, 200–244.Google Scholar
Wang, S., Sekerka, R., Wheeler, A., Murray, B., Coriell, S. R., Braun, R. J. & McFadden, G. B. 1993 Thermodynamically-consistent phase-field models for solidification. Physica D 69, 189–200.Google Scholar
Wheeler, A., Boettinger, W. & McFadden, G. 1993 Phase-field model of solute trapping during solidification. Phys. Rev. E 47, 1893–1909.Google Scholar
Willenbacher, N. & Oelschlaeger, C. 2007 Dynamics and structure of complex fluids from high frequency mechanical and optical rheometry. Curr. Opin. Colloid Interface Sci. 12 (1), 43–49.Google Scholar