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Swinging and tumbling of elastic capsules in shear flow

Published online by Cambridge University Press:  23 May 2008

S. KESSLER ,
R. FINKEN  and
U. SEIFERT
S. KESSLER
Affiliation:
II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
R. FINKEN
Affiliation:
II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
U. SEIFERT
Affiliation:
II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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Abstract

The deformation of an elastic micro-capsule in an infinite shear flow is studied numerically using a spectral method. The shape of the capsule and the hydrodynamic flow field are expanded into smooth basis functions. Analytic expressions for the derivative of the basis functions permit the evaluation of elastic and hydrodynamic stresses and bending forces at specified grid points in the membrane. Compared to methods employing a triangulation scheme, this method has the advantage that the resulting capsule shapes are automatically smooth, and few modes are needed to describe the deformation accurately. Computations are performed for capsules with both spherical and ellipsoidal unstressed reference shape. Results for small deformations of initially spherical capsules coincide with analytic predictions. For initially ellipsoidal capsules, recent approximate theories predict stable oscillations of the tank-treading inclination angle, and a transition to tumbling at low shear rate. Both phenomena have also been observed experimentally. Using our numerical approach we can reproduce both the oscillations and the transition to tumbling. The full phase diagram for varying shear rate and viscosity ratio is explored. While the numerically obtained phase diagram qualitatively agrees with the theory, intermittent behaviour could not be observed within our simulation time. Our results suggest that initial tumbling motion is only transient in this region of the phase diagram.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 605 , 25 June 2008 , pp. 207 - 226
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Abkarian, M., Faivre, M. & Viallat, A. 2007 Swinging of red blood cells under shear flow. Phys. Rev. Lett. 98, 188302.Google Scholar
Abkarian, M., Lartigue, C. & Viallat, A. 2002 Tank treading and unbinding of deformable vesicles in shear flow: Determination of the lift force. Phys. Rev. Lett. 88, 068103.Google Scholar
Barthès-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831.Google Scholar
Barthès-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.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.Google Scholar
Beaucourt, J., Biben, T. & Misbah, C. 2004 a Optimal lift force on vesicles near a compressible substrate. Europhys. Lett. 67, 676.CrossRefGoogle Scholar
Beaucourt, J., Rioual, F., Seon, T., Biben, T. & Misbah, C. 2004 b Steady to unsteady dynamics of a vesicle in a flow. Phys. Rev. E 69, 011906.Google Scholar
Biben, T. & Misbah, C. 2003 Tumbling of vesicles under shear flow within an advected-field approach. Phys. Rev. E 67, 031908.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Brink, D. M. & Satchler, G. R. 1968 Angular Momentum, 2nd edn. Clarendon.Google Scholar
Cantat, I. & Misbah, C. 1999 Lift force and dynamical unbinding of adhering vesicles under shear flow. Phys. Rev. Lett. 83, 880.Google Scholar
Chang, K. S. & Olbricht, W. L. 1993 Experimental studies of the deformation and breakup of a synthetic capsule in steady and unsteady simple shear-flow. J. Fluid Mech. 250, 609.Google Scholar
Finken, R. & Seifert, U. 2006 Wrinkling of microcapsules in shear flow. J. Phys.: Condens. Matter 18, L185.Google Scholar
Fischer, T. M. 2004 Shape memory of human red blood cells. Biophys. J. 86, 3304.Google Scholar
Frankel, T. 2004 The Geometry of Physics, 2nd edn. Cambridge University Press.Google Scholar
de Haas, K. H., Blom, C., van den Ende, D., Duits, M. H. G. & Mellema, J. 1997 Deformation of giant lipid bilayer vesicles in shear flow. Phys. Rev. E 56, 7132.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics, 1st edn. Martinus Nijhoff.Google Scholar
Healy, D. M., Rockmore, D. N., Kostelec, P. J. & Moore, S. 2003 FFTs for the 2-sphere-improvements and variations. J. Fourier Anal. Applics. 9, 341.Google Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers–theory and possible experiments. Z. Naturforsch. (C) 28, 693.Google Scholar
Kantsler, V., Segre, E. & Steinberg, V. 2007 Vesicle dynamics in time-dependent elongation flow: Wrinkling instability. Phys. Rev. Lett. 99, 178102.Google Scholar
Kantsler, V. & Steinberg, V. 2005 Orientation and dynamics of a vesicle in tank-treading motion in shear flow. Phys. Rev. Lett. 95, 258101.Google Scholar
Kantsler, V. & Steinberg, V. 2006 Transition to tumbling and two regimes of tumbling motion of vesicles in shear flow. Phys. Rev. Lett. 96, 036001.Google Scholar
Keller, S. R. & Skalak, R. 1982 Motion of a tank-treading ellipsoidal particle in a shear-flow. J. Fluid Mech. 120, 27.Google Scholar
Kraus, M., Wintz, W., Seifert, U. & Lipowsky, R. 1996 Fluid vesicles in shear flow. Phys. Rev. Lett. 77, 3685.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Lebedev, V. V., Turitsyn, K. S. & Vergeles, S. S. 2007 Dynamics of nearly spherical vesicles in an external flow. Phys. Rev. Lett. 99, 218101.Google Scholar
Leyrat-Maurin, A. & Barthes-Biesel, D. 1994 Motion of a deformable capsule through a hyperbolic constriction. J. Fluid Mech. 279, 135.Google Scholar
Leyrat-Maurin, A., Drochon, A. & Barthes-Biesel, D. 1993 Flow of a capsule through a constriction – application to cell filtration. J. Phys. Paris III 3, 1051.Google Scholar
Li, X. Z., Barthès-Biesel, D. & Helmy, A. 1988 Large deformations and burst of a capsule freely suspended in an elongational flow. J. Fluid Mech. 187, 179.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.Google Scholar
Marsden, J. E. & Hughes, T. J. R. 1983 Mathematical Foundations of Elasticity. Prentice-Hall.Google Scholar
Misbah, C. 2006 Vacillating breathing and tumbling of vesicles under shear flow. Phys. Rev. Lett. 96, 28104.Google Scholar
Mohandas, N. & Evans, E. 1994 Mechanical properties of the red cell membrane in relation to molecular structure and genetic defects. Annu. Rev. Biophys. Biomolec. Struct. 23, 787.CrossRefGoogle ScholarPubMed
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill.Google Scholar
Noguchi, H. & Gompper, G. 2004 Fluid vesicles with viscous membranes in shear flow. Phys. Rev. Lett. 93, 258102.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.Google Scholar
Noguchi, H. & Gompper, G. 2007 Swinging and tumbling of fluid vesicles in shear flow. Phys. Rev. Lett. 98, 128103.Google Scholar
Pozrikidis, C. 1995 Finite deformation of liquid capsules enclosed by elastic membranes in simple shear-flow. J. Fluid Mech. 297, 123.Google Scholar
Pozrikidis, C. 2001 Interfacial dynamics for Stokes flow. J. Comput. Phys. 169, 250.Google Scholar
Pozrikidis, C. 2003 a Modelling and Simulation of Capsules and Biological Cells. Chapman & Hall/CRC.Google Scholar
Pozrikidis, C. 2003 b Numerical simulation of the flow-induced deformation of red blood cells. Ann. Biomed. Engng 31, 1194.Google Scholar
Pozrikidis, C. 2006 A spectral collocation method with triangular boundary elements. Engng Anal. Bound. Elem. 30, 315.Google Scholar
Ramanujan, S. & Pozrikidis, C. 1998 Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: Large deformations and the effect of fluid viscosities. J. Fluid Mech. 361, 117.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
Rochal, S. B., Lorman, V. L. & Mennessier, G. 2005 Viscoelastic dynamics of spherical composite vesicles. Phys. Rev. E 71, 021905.Google Scholar
Rose, M. E. 1957 Elementary Theory of Angular Momentum. Wiley.CrossRefGoogle Scholar
Seifert, U. 1999 a Fluid dynamics in hydrodynamic force fields: Formalism and an application to fluctuating quasispherical vesicles in shear flow. Eur. Phys. J. B 8, 405.Google Scholar
Seifert, U. 1999 b Hydrodynamic lift on bound vesicles. Phys. Rev. Lett. 83, 876.Google Scholar
Skotheim, J. M. & Secomb, T. W. 2007 Red blood cells and other nonspherical capsules in shear flow: Oscillatory dynamics and the tank-treading-to-tumbling transition. Phys. Rev. Lett. 98, 078301.Google Scholar
Sukumaran, S. & Seifert, U. 2001 Influence of shear flow on vesicles near a wall: A numerical study. Phys. Rev. E 64, 11916.Google Scholar
Swarztrauber, P. N. & Spotz, W. F. 2004 Spherical harmonic projectors. Math. Comput. 73, 753.Google Scholar
Vlahovska, P. M. & Gracia, R. S. 2007 Dynamics of a viscous vesicle in linear flows. Phys. Rev. E 75, 016313.Google Scholar
Walter, A., Rehage, H. & Leonhard, H. 2001 Shear induced deformation of microcapsules: shape oscillations and membrane folding. Colloid Surf. A 183–185, 123.Google Scholar
Wang, Y. C. & Dimitrakopoulos, P. 2006 A three-dimensional spectral boundary element algorithm for interfacial dynamics in Stokes flow. Phys. Fluids 18.Google Scholar
Watanabe, N., Kataoka, H., Yasuda, T. & Takatani, S. 2006 Dynamic deformation and recovery response of red blood cells to a cyclically reversing shear flow: Effects of frequency of cyclically reversing shear flow and shear stress level. Biophys. J. 91, 1984.Google Scholar