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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
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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

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