  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.
