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Motion of a deformable capsule through a hyperbolic constriction

Published online by Cambridge University Press:  26 April 2006

Anne Leyrat-Maurin  and
Dominique Barthes-Biesel
Anne Leyrat-Maurin
Affiliation:
Université de Technologie de Compiègne, URA CNRS 858, BP 649, 60 206 Compiègne Cedex, France
Dominique Barthes-Biesel
Affiliation:
Université de Technologie de Compiègne, URA CNRS 858, BP 649, 60 206 Compiègne Cedex, France
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Abstract

A model for the low-Reynolds-number flow of a capsule through a constriction is developed for either constant-flow-rate or constant-pressure-drop conditions. Such a model is necessary to infer quantitative information on the intrinsic properties of capsules from filtration experiments conducted on a dilute suspension of such particles. A spherical capsule, surrounded by an infinitely thin Mooney-Rivlin membrane, is suspended on the axis of a hyperbolic constriction. This configuration is fully axisymmetric and allows the entry and exit phenomena through the pore to be modelled. An integral formulation of the Stokes equations describing the flow in the internal and external domains is developed. It provides a representation of the velocity at any location in the flow as a function of the unknown forces exerted by the boundaries on the fluids. The problem is solved by a collocation technique in the case where the internal and external viscosities are equal. Microscopic quantities (instantaneous geometry, centre of mass velocity, elastic tensions in the membrane) as well as macroscopic quantities (entry time, additional pressure drop or flow rate reduction) are predicted as a function of the capsule intrinsic properties and flow characteristics. The results obtained for a capsule whose initial diameter is larger than that of the constriction throat show that the maximum energy expenditure occurs when the particle centre of mass is still upstream of the throat (typically 1 diameter away), and is thus due to the entry process. For large enough or rigid enough capsules, the model predicts entrance or exit plugging, in agreement with experimental observations. It is then possible to correlate the variation of the pore hydraulic resistance to the flow capillary number (ratio of viscous to elastic forces) and to the size ratio between the pore and the capsule.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 279 , 25 November 1994 , pp. 135 - 163
Copyright
© 1994 Cambridge University Press

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References

Barthes-Biesel, D. & Rallison, J. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.Google Scholar
Barthes-Biesel, D. & Sgaier, H. 1985 Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in shear flow. J. Fluid Mech. 160, 119–135.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
Chien, S., Schmid-Schöunbein, G. W., Sung, K. L. P., Schmalzer, E. A. & Skalak, R. 1984 Viscoelastic Properties of Leukocytes in White Cell Mechanics: Basic Science and Clinical Aspects, pp. 19–51. Alan R. Liss, Inc.
Delves, L. M. & Mohamed, J. L. 1985 Computational Methods for Integral Equations. Cambridge University Press.
Drochon, A. 1991 Détermination des propriétés mécaniques intrinsèques des hématies par viscosimétrie et filtration. PhD thesis, Université de Technologie de Compiègne.
Drochon, A., Barthes-Biesel, D., Bucherer, C, Lacombe, C. & Lelievre, J. C. 1993 Viscous filtration of red blood cell suspensions. Biorheology 30, 1–8.Google Scholar
Drochon, A., Barthes-Biesel, D., Lacombe, C. & Lelievre, J. C. 1990 Determination of the Red Blood Cell apparent membrane elastic modulus from viscometric measurements. J. Biomech. Engng 112, 241–249.Google Scholar
Evans, E. A. 1973 A new material concept for the red cell membrane. Biophys. J. 13, 926–940.Google Scholar
Fischer, T. C., Wenby, R. B. & Meiselman, H. J. 1992 Pulse shape analysis of RBC micropore flow via new software for the cell transit analyser (CTA). Biorheology 29, 185–201.Google Scholar
Green, A. E. & Adkins, J. C. 1960 Large Elastic Deformation and Non-linear Continuum Mechanics. Oxford University Press.
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Helmy, A. & Barthes-Biesel, D. 1982 Migration of a spherical capsule freely suspended in an unbounded parabolic flow. J. Méc. Théor. Appl. 1, 859–880.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.Google Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.
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, 1051–1056.Google Scholar
Li, X. Z., Barthes-Biesel, D. & Helmy, A. 1988 Large deformations and burst of a capsule freely suspended in an elongational flow. J. Fluid Mech. 187, 179–196.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water. I. A numerical method of computation. Proc. R. Soc. Lond. A 350, 1–26.Google Scholar
Martinez, M. J. & Udell, K. S. 1990 Axisymmetric creeping motion of drops through circular tubes. J. Fluid Mech. 210, 565–591.Google Scholar
Pozrikidis, C. 1990 The axisymmetric deformation of a red blood cell in uniaxial straining Stokes flow. J. Fluid Mech. 216, 231–254.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
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.Google Scholar
Secomb, T. W., Skalak, R., Ozkaya, N. & Gross, J. F. 1986 Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.Google Scholar
Skalak, R., Tozeren, A., Zarda, R. P. & Chien, S. 1973 Strain energy function of red blood cell membranes. Biophys. J. 13, 245–264.Google Scholar
Sutera, S. P., Pierre, P. R. & Zahalak, G. I. 1989 Deduction of intrinsic mechanical properties of the erythrocyte membrane from observations of tank-treading in the rheoscope. Biorheology 26, 177–197.Google Scholar
Tözeren, H. & Skalak, R. 1979 The flow of closely fitting particles in tapered tubes. Intl J. Multiphase Flow 5, 395–412.Google Scholar
Tran-Son-Tay, R., Sutera, S. P. & Rao, P. R. 1984 Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion. Biophys. J. 46, 65–72.Google Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69, 377–403.Google Scholar
Youngren, G. K. & Acrivos, A. 1976 On the shape of a gas bubble in a viscous extensional flow. J. Fluid Mech. 76, 433–442.Google Scholar