Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-11T05:33:14.077Z Has data issue: false hasContentIssue false
  • English
  • Français

Influence of membrane viscosity on capsule dynamics in shear flow

Published online by Cambridge University Press:  08 February 2013

Alireza Yazdani  and
Prosenjit Bagchi
Alireza Yazdani
Affiliation:
Department of Mechanical and Aerospace Engineering, Rutgers University, The State University of New Jersey, Piscataway, NJ 08854, USA
Prosenjit Bagchi*
Affiliation:
Department of Mechanical and Aerospace Engineering, Rutgers University, The State University of New Jersey, Piscataway, NJ 08854, USA
*
†Email address for correspondence: pbagchi@jove.rutgers.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Most previous numerical studies on capsule dynamics in shear flow have ignored the role of membrane viscosity. Here we present a numerical method for large deformation of capsules using a Kelvin–Voigt viscoelastic model for the membrane. After introducing the model and the related numerical implementation, we present a comprehensive analysis of the influence of the membrane viscosity on buckling, deformation and dynamics. We observe that the membrane viscosity leads to buckling in the range of shear rate in which no buckling is observed for capsules with purely elastic membrane. For moderate to large shear rates, the wrinkles on the capsule surface appear in the same range of the membrane viscosity that was reported earlier for artificial capsules and red blood cells based on experimental measurements. In order to obtain stable shapes, it is necessary to introduce the bending stiffness. It is observed that the range of the bending stiffness required is also in the same range as that reported for the red blood cells, but considerably higher than that estimated for artificial capsules. Using the stable shapes obtained in the presence of bending stiffness, we analyse the influence of membrane viscosity on deformation, inclination and tank-treading frequency of initially spherical capsules. Membrane viscosity is observed to reduce the capsule deformation, and introduce a damped oscillation in time-dependent deformation and inclination. The time-averaged inclination angle shows a non-monotonic trend with an initial decrease reaching a minimum and a subsequent increase with increasing membrane viscosity. A similar non-monotonic trend is also observed in the tank-treading frequency. We then consider the influence of the membrane viscosity on the unsteady dynamics of an initially oblate capsule. The dynamics is observed to change from a swinging motion to a tumbling motion with increasing membrane viscosity. Further, a transient dynamics is also observed in which a capsule starts with one type of dynamics, but settles with a different dynamics over a long time.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Biological Fluid Dynamics: Capsule/cell dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 718 , 10 March 2013 , pp. 569 - 595
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

ABAQUS/Standard Theory Manual (Version 6.3) 2002 Hibbitt, Karlsson & Sorensen, Inc.Google Scholar
Abkarian, M., Faivre, M. & Viallat, A. 2007 Swinging of red blood cells under shear flow. Phys. Rev. Lett. 98, 188302.CrossRefGoogle ScholarPubMed
Bagchi, P. & Kalluri, R. M. 2009 Dynamics of nonspherical capsules in shear flow. Phys. Rev. E 80, 016307.CrossRefGoogle ScholarPubMed
Barthès-Biesel, D 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831–853.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, 211–222.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
Barthès-Biesel, D. & Sgaier, H. 1985 Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in simple shear flow. J. Fluid Mech. 160, 119–135.Google Scholar
Boal, D. H. 2002 Mechanics of the Cell. Cambridge University Press.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, 609–633.Google Scholar
Diaz, A., Barthès-Biesel, D. & Pelekasis, N. 2001 Effect of membrane viscosity on the dynamic response of an axisymmetric capsule. Phys. Fluids 13, 3835–3838.Google Scholar
Dill, E. H. 2012 The Finite Element Method for Mechanics of Solids with ANSYS Applications. CRC.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.CrossRefGoogle 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. & Hochmuth, R. 1976 Membrane viscoelasticity. Biophys. J. 16, 1–11.CrossRefGoogle ScholarPubMed
Fedosov, D. A., Caswell, B. & Karniadakis, G. E. 2010 A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys. J. 98, 2215–2225.Google Scholar
Finken, R. & Seifert, U. 2006 Wrinkling of microcapsules in shear flow. J. Phys.: Condens. Matter 18, L185–L191.Google Scholar
Fischer, T. M. 1980 On the energy dissipation in a tank-treading red blood cell. Biophys. J. 32, 863–868.CrossRefGoogle Scholar
Fischer, T. M., Stöhr-Liesen, M. & Schmid-Schönbein, H. 1978 The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow. Science 202, 894–896.Google Scholar
Foessel, E., Walter, J., Salsac, A.-V. & Barthès-Biesel, D. 2011 Influence of internal viscosity on the large deformation and buckling of a spherical capsule in a simple shear flow. J. Fluid Mech. 672, 477–486.CrossRefGoogle Scholar
Goldsmith, H. L. & Marlow, J. 1972 Flow behaviour of erythrocytes: I. Rotation and deformation in dilute suspensions. Proc. R. Soc. Lond. B 182, 351–384.Google Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C 28 (11), 693–703.Google Scholar
Hénon, S., Lenormand, G., Richert, A. & Gallet, F. 1999 A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers. Biophys. J. 76, 1145–1151.Google Scholar
Holzapfel, G. A. 2000 Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley.Google Scholar
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
Kessler, S., Finken, R. & Seifert, U. 2008 Swinging and tumbling of elastic capsules in shear flow. J. Fluid Mech. 605, 207–226.Google Scholar
Lac, E., Barthès-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
Le, D. V. 2010 Effect of bending stiffness on the deformation of liquid capsules enclosed by thin shells in shear flow. Phys. Rev. E 82, 016318.CrossRefGoogle ScholarPubMed
Li, X. & Sarkar, K. 2008 Front tracking simulation of deformation and buckling instability of a liquid capsule enclosed by an elastic membrane. J. Comput. Phys. 227 (10), 4998–5018.CrossRefGoogle Scholar
Mohandas, N. & Evans, E. 1994 Mechanical properties of the red cell membrane in relation to molecular structure and genetic defects. Annu. Rev. Biophys. Biomol. Struct. 23, 787–818.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 Dynamics of fluid vesicles in shear flow: effect of membrane viscosity and thermal fluctuations. Phys. Rev. E 72, 011901.Google Scholar
Pozrikidis, C. 1995 Finite deformation of liquid capsules enclosed by elastic membranes in simple shear-flow. J. Fluid Mech. 297, 123–152.Google Scholar
Pozrikidis, C. 2001 Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J. Fluid Mech. 440, 269–291.Google Scholar
Pozrikidis, C. 2003 Modelling and Simulation of Capsules and Biological Cells. Chapman & Hall/CRC.Google Scholar
Pranay, P., Anekal, S. G., Hernandez-Ortiz, J. P. & Graham, M. D. 2010 Pair collisions of fluid-filled elastic capsules in shear flow: effects of membrane properties and polymer additives. Phys. Fluids 22, 123103.Google Scholar
Puig-de-Morales-Marinkovic, M., Turner, K. T., Butler, J. P., Fredberg, J. J. & Suresh, S. 2007 Viscoelasticity of the human red blood cell. Am. J. Physiol. Cell Physiol. 293, C597–C605.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 capsule viscosity. J. Fluid Mech. 361, 117–143.Google Scholar
Scheffer, L., Bitler, A., Ben-Jacob, E. & Korenstein, R. 2001 Atomic force pulling: probing the local elasticity of the cell membrane. Eur. Biophys. J. 30, 83–90.Google Scholar
Shrivastava, S. & Tang, J. 1993 Large deformation finite element analysis of nonlinear viscoelastic membranes with reference to thermoforming. J. Strain Anal. 28 (1), 31–51.Google Scholar
Skalak, R., Tozeren, A., Zarda, P. R. & Chien, S. 1973 Strain energy function of red blood cell membranes. Biophys. J. 13, 245–264.CrossRefGoogle ScholarPubMed
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
Sui, Y., Low, H. T., Chew, Y. T. & Roy, P. 2008 Tank-treading, swinging, and tumbling of liquid-filled elastic capsules in shear flow. Phys. Rev. E 77 (1), 016310.CrossRefGoogle ScholarPubMed
Tran-Son-Tay, R., Sutera, S. & Rao, P. 1984 Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion. Biophys. J. 46, 65–72.CrossRefGoogle ScholarPubMed
Tryggvason, G., Bunner, B., Esmaeeli, A., Al Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759.CrossRefGoogle Scholar
Vlahovska, P. M., Young, Y.-N., Danker, G. & Misbah, C. 2011 Dynamics of a non-spherical microcapsule with incompressible interface in shear flow. J. Fluid Mech. 678, 221–247.Google Scholar
Walter, A., Rehage, H. & Leonhard, H. 2000 Shear-induced deformations of polyamide microcapsules. Colloid Polym. Sci. 278, 169–175.Google Scholar
Walter, A., Rehage, H. & Leonhard, H. 2001 Shear induced deformation of microcapsules: shape oscillations and membrane folding. Colloids Surf. A: Physicochem. Engng. Aspects 183–185, 123–132.Google Scholar
Walter, J., Salsac, A.-V. & Barthès-Biesel, D. 2011 Ellipsoidal capsules in simple shear flow: prolate versus oblate initial shapes. J. Fluid Mech. 676, 318–347.Google Scholar
Wang, Y. C. & Dimitrakopoulos, P. 2006 A three-dimensional spectral boundary element algorithm for interfacial dynamics in Stokes flow. Phys. Fluids 18, 1–16.Google Scholar
Yazdani, A. Z. K. & Bagchi, P. 2011 Phase diagram and breathing dynamics of a single red blood cell and a biconcave capsule in dilute shear flow. Phys. Rev. E 84, 026314.CrossRefGoogle Scholar
Yazdani, A. & Bagchi, P. 2012 Three-dimensional numerical simulation of vesicle dynamics using a front-tracking method. Phys. Rev. E 85, 056308.Google Scholar
Yazdani, A. Z. K., Kalluri, R. M. & Bagchi, P. 2011 Tank-treading and tumbling frequencies of capsules and red blood cells. Phys. Rev. E 83, 046305.CrossRefGoogle ScholarPubMed
Zhao, H., Isfahani, A. H. G., Olson, L. N. & Freund, J. B. 2010 A spectral boundary integral method for flowing blood cells. J. Comput. Phys. 229, 3726–3744.CrossRefGoogle Scholar
Zhong-can, O.-Y. & Helfrich, W. 1989 Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39 (10), 5280–5288.CrossRefGoogle Scholar

Yazdani and Bagchi supplementary movie

Formation and propagation of wrinkles on capsule surface at capillary number Ca = 0.6.

Download Yazdani and Bagchi supplementary movie(Video)
Video 1.4 MB
Supplementary material: PDF

Yazdani and Bagchi supplementary material

Supplementary data

Download Yazdani and Bagchi supplementary material(PDF)
PDF 25.6 KB

Yazdani and Bagchi supplementary movie

Formation and propagation of wrinkles on capsule surface at Ca = 0.1.

Download Yazdani and Bagchi supplementary movie(Video)
Video 1.2 MB