Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-18T15:18:42.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deformations of a cylindrical liquid-filled membrane by a viscous shear flow

Published online by Cambridge University Press:  21 April 2006

George I. Zahalak ,
Peddada R. Rao  and
Salvatore P. Sutera
George I. Zahalak
Affiliation:
Department of Mechanical Engineering, Washington University, St Louis, MO 63130, USA
Peddada R. Rao
Affiliation:
Department of Mechanical Engineering, Washington University, St Louis, MO 63130, USA
Salvatore P. Sutera
Affiliation:
Department of Mechanical Engineering, Washington University, St Louis, MO 63130, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper treats the steady flow fields generated inside and outside an initially circular, inextensible, cylindrical membrane filled with an incompressible viscous fluid when the membrane is placed in a two-dimensional shear flow of another viscous fluid. The Reynolds numbers of both the interior and exterior flows were assumed to be zero (‘creeping flow’), but no further approximations were made in the formulation. A series solution of the resulting free boundary-value problem in powers of a dimensionless shear rate parameter was constructed through fifth order. When combined with a conformal coordinate transformation this series gave accurate results for large deformations of the membrane (up to an aspect ratio of 2.5). The rather tedious algebraic manipulations required to obtain the series solution were done by computer with a symbol-manipulation program (reduce), which both formulated the boundary-value problems for each successive order and solved them. Results are presented which show how the shear rate and fluid viscosities influence the internal and external velocity and pressure fields, the membrane deformation and its ‘tank-treading’ frequency, and the membrane tension.

This work demonstrates that classical perturbation techniques combined with computer algebra offer a useful alternative to purely numerical methods for problems of this type.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 179 , June 1987 , pp. 283 - 305
Copyright
© 1987 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

Barthes-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831–853.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
Cox, R. G. 1968 The deformation of a drop in a general time-dependent fluid flow. J. Fluid Mech. 37, 601–623.Google Scholar
Duff, G. D. F. & Naylor, D. 1966 Differential Equations of Applied Mathematics, p. 143. Wiley.
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
Kholief, I. A. & Weymann, H. D. 1974 Motion of a single red blood cell in a plane shear flow. Biorheology 11, 337–348.Google Scholar
Kober, H. 1957 A Dictionary of Conformal Representations, p. 177. Dover.
Langlois, W. E. 1964 Slow Viscous Flows. Macmillan.
Muskhelishvili, N. I. 1953 Some Basic Problems of the Mathematical Theory of Elasticity. Groningen-Holland: P. Noordhoff.
Niimi, H. & Sugihara, M. 1985 Cyclic loading on the red cell membrane in a shear flow: a possible cause of hemolysis. J. Biomech. Engng 107, 91–95.Google Scholar
Rand, R. H. 1984 Computer Algebra in Applied Mathematics: An Introduction to MACSYMA. Boston: Pittman.
Rao, P. R. 1985 Deformation of a fluid-filled cylindrical membrane by a slow viscous shear flow. M.S. thesis, Department of Mechanical Engineering, Washington University, St Louis
Schmid-SchÖnbein, H. & Wells, R. E. 1969 Fluid drop like transition of erythrocytes under shear stress. Science 165, 288–291.Google Scholar
Sugihara, M. & Niimi, H. 1984 Numerical approach to the motion of a red blood cell in Couette flow. Biorheology 21, 735–749.Google Scholar
Sutera, S. P. & Tran-Son-Tay, R. 1983 Mathematical model of the velocity field external to a tank-treading red cell. Biorheology 20, 267–282.Google Scholar
Sutera, S. P., Gardner, R. A., Boylan, C. W., Carroll, G. L., Chang, K. C., Marvel, J. S., Kilo, C., Gonen, B. & Williamson, J. R. 1985 Age-related changes in deformability of human erythrocytes. Blood 65, 275–282.Google Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. A 138, 41–48.Google Scholar