An exact solution for Stokes flow in a channel with arbitrarily large wall permeability

TL;DR

This paper presents an exact solution for Stokes flow in a channel with walls of arbitrary permeability, extending previous models to regimes of high permeability where transverse velocities become significant.

Contribution

It derives an exact solution for Stokes flow with highly permeable walls, relaxing the small permeability assumption common in prior studies.

Findings

Transverse velocity can be significant at high wall permeability.

Existing asymptotic theories break down in the large permeability limit.

Provides a framework for analyzing flows with highly permeable boundaries.

Abstract

We derive an exact solution for Stokes flow in an in a channel with permeable walls. We assume that at the channel walls, the normal component of the fluid velocity is described by Darcy's law and the tangential component of the fluid velocity is described by the no slip condition. The pressure exterior to the channel is assumed to be constant. Although this problem has been well studied, typical studies assume that the permeability of the wall is small relative to other non-dimensional parameters; this work relaxes this assumption and explores a regime in parameter space that has not yet been well studied. A consequence of this relaxation is that transverse velocity is no longer necessarily small when compared with the axial velocity. We use our result to explore how existing asymptotic theories break down in the limit of large permeability.