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Reynolds number effects on lipid vesicles

Published online by Cambridge University Press:  31 August 2012

David Salac  and
Michael J. Miksis
David Salac*
Affiliation:
Department of Mechanical and Aerospace Engineering, University at Buffalo SUNY, Buffalo, NY 14260, USA
Michael J. Miksis
Affiliation:
Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
*
†Email address for correspondence: [email protected]
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Abstract

Vesicles exposed to the human circulatory system experience a wide range of flows and Reynolds numbers. Previous investigations of vesicles in fluid flow have focused on the Stokes flow regime. In this work the influence of inertia on the dynamics of a vesicle in a shearing flow is investigated using a novel level-set computational method in two dimensions. A detailed analysis of the behaviour of a single vesicle at finite Reynolds number is presented. At low Reynolds numbers the results recover vesicle behaviour previously observed for Stokes flow. At moderate Reynolds numbers the classical tumbling behaviour of highly viscous vesicles is no longer observed. Instead, the vesicle is observed to tank-tread, with an equilibrium angle dependent on the Reynolds number and the reduced area of the vesicle. It is shown that a vesicle with an inner/outer fluid viscosity ratio as high as 200 will not tumble if the Reynolds number is as low as 10. A new damped tank-treading behaviour, where the vesicle will briefly oscillate about the equilibrium inclination angle, is also observed. This behaviour is explained by an investigation on the torque acting on a vesicle in shear flow. Scaling laws for vesicles in inertial flows have also been determined. It is observed that quantities such as vesicle tumbling period follow square-root scaling with respect to the Reynolds number. Finally, the maximum tension as a function of the Reynolds number is also determined. It is observed that, as the Reynolds number increases, the maximum tension on the vesicle membrane also increases. This could play a role in the creation of stable pores in vesicle membranes or for the premature destruction of vesicles exposed to the human circulatory system.

JFM classification

Biological Fluid Dynamics: Capsule/cell dynamics Biological Fluid Dynamics: Membranes Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 711 , 25 November 2012 , pp. 122 - 146
Copyright
Copyright © Cambridge University Press 2012

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References

1. Abkarian, M., Faivre, M. & Viallat, A. 2007 Swinging of red blood cells under shear flow. Phys. Rev. Lett. 98 (18), 188302.CrossRefGoogle ScholarPubMed
2. 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 (6), 068103.CrossRefGoogle ScholarPubMed
3. Abkarian, M. & Viallat, A. 2005 Dynamics of vesicles in a wall-bounded shear flow. Biophys. J. 89 (2), 1055–1066.CrossRefGoogle Scholar
4. Abkarian, M. & Viallat, A. 2008 Vesicles and red blood cells in shear flow. Soft Matt. 4 (4), 653–657.CrossRefGoogle ScholarPubMed
5. Allen, T. M. & Cullis, P. R. 2004 Drug delivery systems: entering the mainstream. Science 303 (5665), 1818–1822.CrossRefGoogle ScholarPubMed
6. Bark, D. L. & Ku, D. N. 2010 Wall shear over high degree stenoses pertinent to atherothrombosis. J. Biomech. 43 (15), 2970–2977.CrossRefGoogle ScholarPubMed
7. Brown, D. L., Cortez, R. & Minion, M. L. 2001 Accurate projection methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 168 (2), 464–499.CrossRefGoogle Scholar
8. Choon, A. & Cullis, P. R. 1995 Recent advances in liposomal drug-delivery systems. Curr. Opin. Biotechnol. 6 (6), 698–708.CrossRefGoogle Scholar
9. Coupier, G., Kaoui, B., Podgorski, T. & Misbah, C. 2008 Noninertial lateral migration of vesicles in bounded Poiseuille flow. Phys. Fluids 20 (11), 111702.CrossRefGoogle Scholar
10. Danker, G., Vlahovska, P. M. & Misbah, C. 2009 Vesicles in Poiseuille flow. Phys. Rev. Lett. 102 (14), 148102.CrossRefGoogle ScholarPubMed
11. Deschamps, J., Kantsler, V., Segre, E. & Steinberg, V. 2009a Dynamics of a vesicle in general flow. Proc. Natl Acad. Sci. USA 106 (28), 11444–11447.CrossRefGoogle ScholarPubMed
12. Deschamps, J., Kantsler, V. & Steinberg, V. 2009b Phase diagram of single vesicle dynamical states in shear flow. Phys. Rev. Lett. 102 (11), 118105.CrossRefGoogle ScholarPubMed
13. Ding, E. J. & Aidun, C. K. 2000 The dynamics and scaling law for particles suspended in shear flow with inertia. J. Fluid Mech. 423, 317–344.CrossRefGoogle Scholar
14. Du, Q., Liu, C., Ryham, R. & Wang, X. 2009 Energetic variational approaches in modelling vesicle and fluid interactions. Physica D: Nonlinear Phenom. 238, 923–930.CrossRefGoogle Scholar
15. Du, Q., Liu, C. & Wang, X. 2006 Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212 (2), 757–777.CrossRefGoogle Scholar
16. Du, Q. & Zhang, J. 2008 Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations. SIAM J. Sci. Comput. 30 (3), 1634–1657.CrossRefGoogle Scholar
17. Farago, O. & Santangelo, C. D. 2005 Pore formation in fluctuating membranes. J. Chem. Phys. 122, 044901.CrossRefGoogle ScholarPubMed
18. Fraser, K. H., Taskin, M. E., Griffith, B. P. & Wu, Z. J. 2011 The use of computational fluid dynamics in the development of ventricular assist devices. Med. Engng Phys. 33 (3), 263–280.CrossRefGoogle ScholarPubMed
19. Fung, Y. C. & Zweifach, B. W. 1971 Microcirculation – mechanics of blood flow in capillaries. Annu. Rev. Fluid Mech. 3, 189–210.CrossRefGoogle Scholar
20. Ghigliotti, G., Biben, T. & Misbah, C. 2010 Rheology of a dilute two-dimensional suspension of vesicles. J. Fluid Mech. 653, 489–518.CrossRefGoogle Scholar
21. Heilmann, C., Geisen, U., Benk, C., Berchtold-Herz, M., Trummer, G., Schlensak, C., Zieger, B. & Beyersdorf, F. 2009 Haemolysis in patients with ventricular assist devices: major differences between systems. Eur. J. Cardio-thoracic Surg. 36 (3), 580–584.CrossRefGoogle ScholarPubMed
22. 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
23. Kantsler, V. & Steinberg, V. 2006 Transition to tumbling and two regimes of tumbling motion of a vesicle in shear flow. Phys. Rev. Lett. 96 (3), 036001.CrossRefGoogle ScholarPubMed
24. 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
25. Kraus, M., Wintz, W., Seifert, U. & Lipowsky, R. 1996 Fluid vesicles in shear flow. Phys. Rev. Lett. 77 (17), 3685–3688.CrossRefGoogle ScholarPubMed
26. Ku, D. N., Giddens, D. P., Zarins, C. K. & Glagov, S. 1985 Pulsatile flow and atherosclerosis in the human carotid bifurcation – positive correlation between plaque location and low and oscillating shear-stress. Arteriosclerosis 5 (3), 293–302.CrossRefGoogle ScholarPubMed
27. Laadhari, A., Saramito, P. & Misbah, C. 2012 Vesicle tumbling inhibited by inertia. Phys. Fluids 24 (3), 031901.CrossRefGoogle Scholar
28. Lebedev, V. V., Turitsyn, K. S. & Vergeles, S. S. 2008 Nearly spherical vesicles in an external flow. New J. Phys. 10, 043044.CrossRefGoogle Scholar
29. Lichtenberg, A. J. & Lieberman, M. A. 1992 Regular and Chaotic Dynamics. Springer.CrossRefGoogle Scholar
30. Mikulencack, D. R. & Morris, J. F. 2004 Stationary shear flow around fixed and free bodies at finite Reynolds number. J. Fluid Mech. 520, 215–242.CrossRefGoogle Scholar
31. Min, C. & Gibou, F. 2006 A second-order accurate projection method for the incompressible Navier–Stokes equations on non-graded adaptive grids. J. Comput. Phys. 219 (2), 912–929.CrossRefGoogle Scholar
32. Min, C. & Gibou, F. 2007 A second-order accurate level set method on non-graded adaptive Cartesian grids. J. Comput. Phys. 225 (1), 300–321.CrossRefGoogle Scholar
33. Misbah, C. 2006 Vacillating breathing and tumbling of vesicles under shear flow. Phys. Rev. Lett. 96 (2), 028104.CrossRefGoogle ScholarPubMed
34. Noguchi, H., Gompper, G. & Lubensky, T. C. 2005 Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc. Natl Acad. Sci. USA 102 (40), 14159–14164.CrossRefGoogle ScholarPubMed
35. 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, 117–143.CrossRefGoogle Scholar
36. Rand, R. P. 1964 Mechanical properties of the red cell membrane. II. Viscoelastic breakdown of the membrane. Biophys. J. 4, 303–316.CrossRefGoogle ScholarPubMed
37. Salac, D. & Miksis, M. 2010 Reynolds number effects on the behaviour of lipid vesicles. In Proceedings of the 63rd Annual Meeting of the APS Division of Fluid Dynamics, vol. 55(16), RK.6. The American Physical Society.Google Scholar
38. Salac, D. & Miksis, M. 2011 A level set projection method of lipid vesicles in general flows. J. Comput. Phys. 230 (22), 8192–8215.CrossRefGoogle Scholar
39. Sandre, O., Moreaux, L. & Brochard-Wyart, F. 1999 Dynamics of transient pores in stretched vesicles. Proc. Natl Acad. Sci. 96 (19), 10591–10596.CrossRefGoogle ScholarPubMed
40. Schwalbe, J. T., Vlahovska, P. M. & Miksis, M. J. 2010 Monolayer slip effects on the dynamics of a lipid bilayer vesicle in a viscous flow. J. Fluid Mech. 647, 403–419.CrossRefGoogle Scholar
41. Seifert, U. 1997 Dynamics of giant vesicles. Mol. Cryst. Liq. Cryst. Sci. Technol. A – Mol. Cryst. Liq. Cryst. 292, 213–225.CrossRefGoogle Scholar
42. 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 (3), 405–415.CrossRefGoogle Scholar
43. Siegenthaler, M. P., Martin, J., van de Loo, A., Doenst, T., Bothe, W. & Beyersdorf, F. 2002 Implantation of the permanent Jarvik-2000 left ventricular assist device – a single-centre experience. J. Am. College Cardiol. 39 (11), 1764–1772.CrossRefGoogle Scholar
44. Strony, J., Beaudoin, A., Brands, D. & Adelman, B. 1993 Analysis of shear-stress and hemodynamic factors in a model of coronary-artery stenosis and thrombosis. Am. J. Physiol. 265 (5, Part 2), H1787–H1796.Google Scholar
45. Tangelder, G. J., Slaaf, D. W., Arts, T. & Reneman, R. S. 1988 Wall shear rate in arterioles in vivo – least estimates from platelet velocity profiles. Am. J. Physiol. 254 (6, Part 2), H1059–H1064.Google ScholarPubMed
46. Tieleman, D. P., Leontiadou, H., Mark, A. E. & Marrink, S. J. 2003 Simulation of pore formation in lipid bilayers by mechanical stress and electric fields. J. Am. Chem. Soc. 125 (21), 6382–6383.CrossRefGoogle ScholarPubMed
47. Torchilin, V. P. 2006 Multifunctional nanocarriers. Adv. Drug Deliv. Rev. 58 (14), 1532–1555.CrossRefGoogle ScholarPubMed
48. Veerapaneni, S. K., Gueyffier, D., Biros, G. & Zorin, D. 2009a A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows. J. Comput. Phys. 228 (19), 7233–7249.CrossRefGoogle Scholar
49. Veerapaneni, S. K., Gueyffier, D., Zorin, D. & Biros, G. 2009b A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D. J. Comput. Phys. 228 (7), 2334–2353.CrossRefGoogle Scholar
50. Vlahovska, P. M. & Gracia, R. S. 2007 Dynamics of a viscous vesicle in linear flows. Phys. Rev. E 75 (1), 016313.CrossRefGoogle ScholarPubMed
51. Vlahovska, P. M., Podgorski, T. & Misbah, C. 2009 Vesicles and red blood cells in flow: from individual dynamics to rheology. C. R. Phys. 10 (8), 775–789.CrossRefGoogle Scholar
52. Wang, Z. J. & Frenkel, D. 2005 Pore nucleation in mechanically stretched bilayer membranes. J. Chem. Phys. 123, 154701.CrossRefGoogle ScholarPubMed
53. Xiu, D. B. & Karniadakis, G. E. 2001 A semi-Lagrangian high-order method for Navier–Stokes equations. J. Comput. Phys. 172 (2), 658–684.CrossRefGoogle Scholar
54. Zabusky, N. J., Segre, E., Deschamps, J., Kantsler, V. & Steinberg, V. 2011 Dynamics of vesicles in shear and rotational flows: modal dynamics and phase diagram. Phys. Fluids 23 (4), 041905.CrossRefGoogle Scholar
55. Zettner, C. M. & Yoda, M. 2001 Moderate-aspect-ratio elliptical cylinders in simple shear with inertia. J. Fluid Mech. 442, 241–266.CrossRefGoogle Scholar
56. Zhao, R., Antaki, J. F., Naik, T., Bachman, T. N., Kameneva, M. V. & Wu, Z. J. 2006 Microscopic investigation of erythrocyte deformation dynamics. Biorheology 43 (6), 747–765.Google ScholarPubMed
57. Zhao, H. & Shaqfeh, E. S. G. 2009 The dynamics of a vesicle in shear flow. Tech. Rep. Stanford University.Google Scholar
58. Zhao, H. & Shaqfeh, E. S. G. 2011 The dynamics of a vesicle in simple shear flow. J. Fluid Mech. 674 (1), 578–604.CrossRefGoogle Scholar