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Viscous flow in a channel with periodic cross-bridging fibres: exact solutions and Brinkman approximation

Published online by Cambridge University Press:  26 April 2006

Ruey-Yug Tsay  and
Sheldon Weinbaum
Ruey-Yug Tsay
Affiliation:
Department of Chemical Engineering, The City College of the City University of New York, New York, NY 10031, USA
Sheldon Weinbaum
Affiliation:
Department of Mechanical Engineering, The City College of the City University of New York, New York, NY 10031, USA
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Abstract

A general solution of the three-dimensional Stokes equations is developed for the viscous flow past a square array of circular cylindrical fibres confined between two parallel walls. This doubly periodic solution, which is an extension of the theory developed by Lee & Fung (1969) for flow around a single fibre, successfully describes the transition in behaviour from the Hele-Shaw potential flow limit (aspect ratio B [Lt ] 1) to the viscous two-dimensional limiting case (B [Gt ] 1, Sangani & Acrivos 1982) for the hydrodynamic interaction between the fibres. These results are also compared with the solution of the Brinkman equation for the flow through a porous medium in a channel. This comparison shows that the Brinkman approximation is very good when B > 5, but breaks down when B [les ] O(1). A new interpolation formula is proposed for this last regime. Numerical results for the detailed velocity profiles, the drag coefficient f, and the Darcy permeability Kp are presented. It is shown that the velocity component perpendicular to the parallel walls is only significant within the viscous layers surrounding the fibres, whose thickness is of the order of half the channel height B′. One finds that when the aspect ratio B > 5, the neglect of the vertical velocity component vz can lead to large errors in the satisfaction of the no-slip boundary conditions on the surfaces of the fibres and large deviations from the approximate solution in Lee (1969), in which vz and the normal pressure field are neglected. The numerical results show that the drag coefficient of the fibrous bed increases dramatically when the open gap between adjacent fibres Δ′ becomes smaller than B′. The predictions of the new theory are used to examine the possibility that a cross-bridging slender fibre matrix can exist in the intercellular cleft of capillary endothelium as proposed by Curry & Michel (1980).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 226 , May 1991 , pp. 125 - 148
Copyright
© 1991 Cambridge University Press

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