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The dynamics of a vesicle in simple shear flow

Published online by Cambridge University Press:  23 March 2011

HONG ZHAO  and
ERIC S. G. SHAQFEH
HONG ZHAO*
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
ERIC S. G. SHAQFEH
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: hongzhao@stanford.edu
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Abstract

We have performed direct numerical simulation (DNS) of a lipid vesicle under Stokes flow conditions in simple shear flow. The lipid membrane is modelled as a two-dimensional incompressible fluid with Helfrich surface energy in response to bending deformation. A high-fidelity spectral boundary integral method is used to solve the flow and membrane interaction system; the spectral resolution and convergence of the numerical scheme are demonstrated. The critical viscosity ratios for the transition from tank-treading (TT) to ‘trembling’ (TR, also called VB, i.e. vacillating-breathing, or swinging) and eventually ‘tumbling’ (TU) motions are calculated by linear stability analysis based on this spectral method, and are in good agreement with perturbation theories. The effective shear rheology of a dilute suspension of these vesicles is also calculated over a wide parameter regime. Finally, our DNS reveals a family of time-periodic and off-the-shear-plane motion patterns where the vesicle's configuration follows orbits that resemble but are fundamentally different from the classical Jeffery orbits of rigid particles due to the vesicle's deformability.

JFM classification

Low-Reynolds-number flows: Boundary integral methods Biological Fluid Dynamics: Capsule/cell dynamics Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 578 - 604
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Adams, J. C. & Swarztrauber, P. N. 1997 SPHEREPACK 2.0: a model development facility. NCAR Tech. Rep., NCAR/TN-436-STR.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.CrossRefGoogle Scholar
Biben, T., Farutin, A. & Misbah, C. 2009 Numerical study of 3D vesicles under flow: discovery of new peculiar behaviors. Available at: http://arxiv.org/abs/0912.4702v1.Google Scholar
Biben, T., Kassner, K. & Misbah, C. 2005 Phase-field approach to three-dimensional vesicle dynamics. Phys. Rev. E 72, 041921.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle ScholarPubMed
Danker, G. & Misbah, C. 2007 Rheology of a dilute suspension of vesicles. Phys. Rev. Lett. 98, 088104.CrossRefGoogle ScholarPubMed
Deschamps, J., Kantsler, V. & Steinberg, V. 2009 Phase diagram of single vesicle dynamical states in shear flow. Phys. Rev. Lett. 102, 118105.CrossRefGoogle ScholarPubMed
Farutin, A., Biben, T. & Misbah, C. 2010 Analytical progress in the theory of vesicles under linear flow. Phys. Rev. E 81, 061904.CrossRefGoogle ScholarPubMed
Freund, J. B. 2007 Leukocyte margination in a model microvessel. Phys. Fluids 19, 023301.CrossRefGoogle Scholar
Ghigliotti, G., Biben, T. & Misbah, C. 2010 Rheology of a dilute two-dimensional suspension of vesicles. J. Fluid Mech. 653, 489–518.CrossRefGoogle Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch C 28 (11), 693–703.CrossRefGoogle ScholarPubMed
Hinch, E. J. & Leal, L. G. 1979 Rotation of small non-axisymmetric particles in a simple shear flow. J. Fluid Mech. 92, 591–608.CrossRefGoogle Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.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.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.CrossRefGoogle ScholarPubMed
Kaoui, B., Farutin, A. & Misbah, C. 2009 Vesicles under simple shear flow: elucidating the role of relevant control parameters. Phys. Rev. E 80, 061905.CrossRefGoogle ScholarPubMed
Keller, S. R. & Skalak, R. 1982 Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120, 27–47.CrossRefGoogle Scholar
Kim, Y. & Lai, M. C. 2010 Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method. J. Comput. Phys. 229, 4840–4853.CrossRefGoogle Scholar
Kraus, M., Wintz, W., Seifert, U. & Lipowsky, R. 1996 Fluid vesicles in shear flow. Phys. Rev. Lett. 77 (17), 3685–3688.CrossRefGoogle ScholarPubMed
Lebedev, V. V., Turitsyn, K. S. & Vergeles, S. S. 2007 Dynamics of nearly spherical vesicles in an external flow. Phys. Rev. Lett. 99, 218101.CrossRefGoogle Scholar
Lebedev, V. V., Turitsyn, K. S. & Vergeles, S. S. 2008 Nearly spherical vesicles in an external flow. New J. Phys. 10, 043044.CrossRefGoogle Scholar
Mader, M., Vitkova, V., Abkarian, M., Viallat, A. & Podgorski, T. 2006 Dynamics of viscous vesicles in shear flow. Eur. Phys. J. E 19, 398–397.CrossRefGoogle ScholarPubMed
Messlinger, S., Schmidt, B., Noguchi, H. & Gompper, G. 2009 Dynamical regimes and hydrodynamic lift of viscous vesicles under shear. Phys. Rev. E 80, 011901.CrossRefGoogle ScholarPubMed
Misbah, C. 2006 Vacillating breathing and tumbling of vesicles under shear flow. Phys. Rev. Lett. 96, 028104.CrossRefGoogle ScholarPubMed
Noguchi, H. & Gompper, G. 2007 Swinging and tumbling of fluid vesicles in shear flow. Phys. Rev. Lett. 98, 128103.CrossRefGoogle ScholarPubMed
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2003 Numerical simulation of the flow-induced deformation of red blood cells. Ann. Biomedical 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, 191–200.CrossRefGoogle Scholar
Saad, Y. & Schultz, M. 1986 A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869.CrossRefGoogle Scholar
Seifert, U. 1999 Fluid membranes in hydrodynamic flow fields: formalism and an application to fluctuating quasispherical vesicles in shear flow. Eur. Phys. J. B 8, 405–415.CrossRefGoogle Scholar
Swarztrauber, P. N. & Spotz, W. F. 2000 Generalized discrete spherical harmonic transforms. J. Comput. Phys. 159, 213–230.CrossRefGoogle Scholar
Veerapaneni, S. K., Gueyffier, D., Biros, G. & Zorin, D. 2009 a A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D. J. Comput. Phys. 228, 2334–2353.CrossRefGoogle Scholar
Veerapaneni, S. K., Gueyffier, D., Biros, G. & Zorin, D. 2009 b A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows. J. Comput. Phys. 228, 7233–7249.CrossRefGoogle Scholar
Veerapaneni, S. K., Rahimian, A., Biros, G. & Zorin, D. 2011 A fast algorithm for simulating vesicle flows in three dimensions. J. Comput. Phys. (submitted).CrossRefGoogle Scholar
Vlahovska, P. M. & Gracia, R. S. 2007 Dynamics of a viscous vesicle in linear flows. Phys. Rev. E 75, 016313.CrossRefGoogle ScholarPubMed
Zabusky, N. J., Segre, E., Deschamps, J. & Steinberg, V. 2010 Guide to phase diagrams for vesicles in shear and rotational flow: critique and future directions. Available at: http://arxiv.org/abs/1004.4733v1.Google Scholar
Zhao, H., Isfahani, A. H. G., Olson, L. & Freund, J. B. 2010 A spectral boundary integral method for micro-circulatory cellular flows. J. Comput. Phys. 229 (10), 3726–3744.CrossRefGoogle Scholar