Hostname: page-component-848d4c4894-89wxm Total loading time: 0 Render date: 2024-07-06T03:43:23.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Ellipsoidal capsules in simple shear flow: prolate versus oblate initial shapes

Published online by Cambridge University Press:  04 April 2011

J. WALTER ,
A.-V. SALSAC  and
D. BARTHÈS-BIESEL
J. WALTER
Affiliation:
Laboratoire de Biomécanique et Bioingénierie (UMR CNRS 6600), Université de Technologie de Compiègne, BP 20529, 60205 Compiègne, France
A.-V. SALSAC
Affiliation:
Laboratoire de Biomécanique et Bioingénierie (UMR CNRS 6600), Université de Technologie de Compiègne, BP 20529, 60205 Compiègne, France
D. BARTHÈS-BIESEL*
Affiliation:
Laboratoire de Biomécanique et Bioingénierie (UMR CNRS 6600), Université de Technologie de Compiègne, BP 20529, 60205 Compiègne, France
*
†Email address for correspondence: dbb@utc.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The large deformations of an initially-ellipsoidal capsule in a simple shear flow are studied by coupling a boundary integral method for the internal and external flows and a finite-element method for the capsule wall motion. Oblate and prolate spheroids are considered (initial aspect ratios: 0.5 and 2) in the case where the internal and external fluids have the same viscosity and the revolution axis of the initial spheroid lies in the shear plane. The influence of the membrane mechanical properties (mechanical law and ratio of shear to area dilatation moduli) on the capsule behaviour is investigated. Two regimes are found depending on the value of a capillary number comparing viscous and elastic forces. At low capillary numbers, the capsule tumbles, behaving mostly like a solid particle. At higher capillary numbers, the capsule has a fluid-like behaviour and oscillates in the shear flow while its membrane continuously rotates around its deformed shape. During the tumbling-to-swinging transition, the capsule transits through an almost circular profile in the shear plane for which a long axis can no longer be defined. The critical transition capillary number is found to depend mainly on the initial shape of the capsule and on its shear modulus, and weakly on the area dilatation modulus. Qualitatively, oblate and prolate capsules are found to behave similarly, particularly at large capillary numbers when the influence of the initial state fades out. However, the capillary number at which the transition occurs is significantly lower for oblate spheroids.

JFM classification

Biological Fluid Dynamics: Capsule/cell dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 676 , 10 June 2011 , pp. 318 - 347
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Abkarian, M., Faivre, M. & Viallat, A. 2007 Swinging of red blood cells under shear flow. Phys. Rev. Lett. 98 (18), 188302.CrossRefGoogle ScholarPubMed
Bagchi, P. & Kalluri, R. M. 2009 Dynamics of nonspherical capsules in shear flow. Phys. Rev. E 80 (1), 016307.CrossRefGoogle ScholarPubMed
Barthès-Biesel, D. 2003 Flow-induced capsule deformation. In Modeling and Simulation of Capsules and Biological Cells, pp. 1–31. Chapman & Hall/CRC.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.CrossRefGoogle 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.CrossRefGoogle Scholar
Barthès-Biesel, D., Walter, J. & Salsac, A.-V. 2010 Flow-induced deformation of artificial capsules. In Computational Hydrodynamics of Capsules and Biological Cells, pp. 35–70. Taylor & Francis.CrossRefGoogle Scholar
Carin, M., Barthès-Biesel, D., Edwards-Lévy, F., Postel, C. & Andrei, D. 2003 Compression of biocompatible liquid-filled HSA-alginate capsules: determination of the membrane mechanical properties. Biotechnol. Bioengng. 82, 207–212.CrossRefGoogle ScholarPubMed
Cerda, E. & Mahadevan, L. 2003 Geometry and physics of wrinkling. Phys. Rev. Lett. 90 (7), 074302.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Cook, R. D., Malkus, D. S., Plesha, M. E. & Witt, R. J. 2001 Concepts and Applications of Finite Element Analysis, 4th edn. Wiley.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.CrossRefGoogle ScholarPubMed
Doddi, S. K. & Bagchi, P. 2008 Lateral migration of a capsule in a plane Poiseuille flow in a channel. Intl J. Multiphase Flow 34 (10), 966–986.CrossRefGoogle Scholar
Dyn, N., Hormann, K., Kim, S.-J. & Levin, D. 2001 Optimizing 3D triangulations using discrete curvature analysis, In Mathematical Methods for Curves and Surfaces: Oslo 2000, pp. 135–146. Vanderbilt University.Google Scholar
Finken, R., Kessler, S. & Seifert, U. 2010 Micro-capsules in shear flow. J. Phys.: Condens. Matter arXiv:1004.4879.CrossRefGoogle Scholar
Finken, R. & Seifert, U. 2006 Wrinkling of microcapsules in shear flow. J. Phys.: Condens. Matter 18 (15), L185–L191.Google Scholar
Goldsmith, H. L. & Marlow, J. 1972 Flow behaviour of erythrocytes. Part I. Rotation and deformation in dilute suspensions. Proc. R. Soc. Lond. B 182 (1068), 351–384.Google Scholar
Hammer, P. C., Marlowe, O. J. & Stroud, A. H. 1956 Numerical integration over simplexes and cones. Math. Tables Aids Comput. 10 (55), 130–137.Google Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. (C) 28 (11), 693–703.CrossRefGoogle ScholarPubMed
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. 2006 Transition to tumbling and two regimes of tumbling motion of a vesicle in shear flow. Phys. Rev. Lett. 96 (3), 036001.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
Kessler, S., Finken, R. & Seifert, U. 2008 Swinging and tumbling of elastic capsules in shear flow. J. Fluid Mech. 605, 207–226.CrossRefGoogle Scholar
Lac, É., Barthès-Biesel, D., Pelekasis, N. A. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flow: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303–334.CrossRefGoogle Scholar
Le, D.-V. & Tan, Z. 2010 Large deformation of liquid capsules enclosed by thin shells immersed in the fluid. J. Comput. Phys. 229 (11), 4097–4116.CrossRefGoogle Scholar
Lefebvre, Y., Leclerc, E., Barthès-Biesel, D., Walter, J. & Edwards-Levy, F. 2008 Flow of artificial microcapsules in microfluidic channels: A method for determining the elastic properties of the membrane. Phys. Fluids 20 (12), 123102.CrossRefGoogle Scholar
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
Liu, L., Yang, J.-P., Ju, X.-J., Xie, R., Yang, L., Liang, B. & Chu, L.-Y. 2009 Microfluidic preparation of monodisperse ethyl cellulose hollow microcapsules with non-toxic solvent. J. Colloid Interface Sci. 336 (1), 100–106.CrossRefGoogle ScholarPubMed
Luo, H. & Pozrikidis, C. 2007 Buckling of a pre-compressed or pre-stretched membrane. Intl J. Solids Struct. 44, 8074–8085.CrossRefGoogle Scholar
Oden, J. T. 1972 Finite Elements of Non-Linear Continua. McGraw-Hill.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle 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.CrossRefGoogle Scholar
Risso, F., Collé-Paillot, F. & Zagzoule, M. 2006 Experimental investigation of a bioartificial capsule flowing in a narrow tube. J. Fluid Mech. 547, 149–173.CrossRefGoogle Scholar
Schenk, O. & Gärtner, K. 2004 Solving unsymmetric sparse systems of linear equations with PARDISO. Future Generation Comput. Syst. 20 (3), 475–487.CrossRefGoogle Scholar
Schenk, O. & Gärtner, K. 2006 On fast factorization pivoting methods for sparse symmetric indefinite systems. Electron. Trans. Numer. Anal. 23, 158–179.Google Scholar
Schmid-Schönbein, H. & Wells, R. E. 1969 Fluid drop-like transition of erythrocytes under shear. Science 165, 288–291.CrossRefGoogle Scholar
Skalak, R., Tozeren, A., Zarda, R. P. & 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 (7), 078301.CrossRefGoogle ScholarPubMed
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
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.CrossRefGoogle Scholar
Walter, J., Salsac, A.-V., Barthès-Biesel, D. & Le Tallec, P. 2010 Coupling of finite element and boundary integral methods for a capsule in a Stokes flow. Intl. J. Numer. Meth. Engng. 83, 829–850.CrossRefGoogle Scholar
Xiang, Z. Y., Lu, Y. C., Zou, Y., Gong, X. C. & Luo, G. S. 2008 Preparation of microcapsules containing ionic liquids with a new solvent extraction system. React. Funct. Polym. 68 (8), 1260–1265.CrossRefGoogle Scholar

Walter et al. supplementary material

Evolution of the capsule shape in the shear plane in the tumbling regime. The initial shape is a prolate spheroid, a/b = 2, and the membrane follows the Sk law with C = 1. The colour scale corresponds to the normal component of the load, q.n. The red dot shows the position of a material point, originally on the short axis

Download Walter et al. supplementary material(Video)
Video 634.2 KB

Walter et al. supplementary material

Evolution of the capsule shape in the shear plane in the tumbling-to-swinging transition. The initial shape is a prolate spheroid, a/b = 2, and the membrane follows the Sk law with C = 1. Same legend as movie 1

Download Walter et al. supplementary material(Video)
Video 458.6 KB

Walter et al. supplementary material

Evolution of the capsule shape in the shear plane in the swinging regime. The initial shape is a prolate spheroid, a/b = 2, and the membrane follows the Sk law with C = 1. Same legend as movie 1

Download Walter et al. supplementary material(Video)
Video 310.3 KB

Walter et al. supplementary material

Evolution of the capsule shape in the shear plane in the tumbling regime. The initial shape is an oblate spheroid, a/b = 0.5, and the membrane follows the Sk law with C = 1. Same legend as movie 1

Download Walter et al. supplementary material(Video)
Video 213.3 KB

Walter et al. supplementary material

Evolution of the capsule shape in the shear plane in the tumbling-to-swinging transition. The initial shape is an oblate spheroid, a/b = 0.5, and the membrane follows the Sk law with C = 1. Same legend as movie 1

Download Walter et al. supplementary material(Video)
Video 360.1 KB

Walter et al. supplementary material

Evolution of the capsule shape in the shear plane in the swinging regime. The initial shape is an oblate spheroid, a/b = 0.5, and the membrane follows the Sk law with C = 1. Same legend as movie 1

Download Walter et al. supplementary material(Video)
Video 440.1 KB