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

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).

Darcy’s Law for Flow in a Periodic Thin Porous Medium Confined Between Two Parallel Plates

“…Flow past an array of circular cylindrical fibres confined between two parallel walls has been studied by Tsay and Weinbaum (1991), who extended a result obtained by Lee and Fung (1969). Their analysis is based on an approximate series solution of the Stokes equation.…”

Permeability calculations in three-dimensional isotropic and oriented fiber networks

“…28,29 In three dimensions, rigorous solution of creeping flow past a fibrous system is more difficult. For spatially periodic media, Tsay and Weinbaum 30 solved the Stokes equations for flow past a square array of fibers confined between two parallel walls. Higdon and Ford 12 used a spectral boundary element method to calculate the permeability in simple cubic, body-centered cubic, and face-centered cubic structures, and Palassini and Remuzzi 31 employed a finite element method to solve the Stokes equations for a tetrahedral periodic array of cylinders to model the glomerular basement membrane.…”

Effect of flow and stretch on the [Ca2+]iresponse of principal and intercalated cells in cortical collecting duct

“…2 to determine the average velocity field and the forces and torques on the cilia has been examined in a hydrodynamically equivalent problem, the Stokes flow past a periodic array of slender vertical fibers in a parallel walled channel (39). Comparison with exact Stokes solutions for the average flow past each fiber and the resulting drag show excellent agreement for fiber solid fractions up to 0.7.…”