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

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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 ratioB[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 whenB> 5, but breaks down whenB[les ]O(1). A new interpolation formula is proposed for this last regime. Numerical results for the detailed velocity profiles, the drag coefficientf, and the Darcy permeabilityKpare 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 heightB′. One finds that when the aspect ratioB> 5, the neglect of the vertical velocity componentvzcan 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 whichvzand 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 thanB′. 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).