Slow flow in channels with porous walls

TL;DR

This paper develops an analytical framework for slow viscous flow in channels with porous walls, reducing complex Navier-Stokes equations to simpler biharmonic problems, enabling solutions for various cross-sectional shapes.

Contribution

It introduces a similarity transformation that simplifies the flow equations, extending Berman flow to three dimensions and providing explicit solutions for different channel geometries.

Findings

Analytic solutions for flow in rectangular and triangular channels.

Reduction of Navier-Stokes to biharmonic equations under specific conditions.

Framework for extending classical flow models to three dimensions.

Abstract

We consider the slow flow of a viscous incompressible liquid in a channel of constant but arbitrary cross section shape, driven by non-uniform suction or injection through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a set of coupled equations for the velocity potential in two dimensions. When the channel aspect ratio and Reynolds number are both small, the problem reduces to solving the biharmonic equation with constant forcing in two dimensions. With the relevant boundary conditions, determining the velocity field in a porous channels is thus equivalent to solving for the vertical displacement of a simply suspended thin plate under uniform load. This allows us to provide analytic solutions for flow in porous channels whose cross-section is e.g. a rectangle or an equilateral triangle, and provides a general framework for the extension of Berman flow (Journal of Applied Physics 24(9), p. 1232, 1953) to three dimensions.