Start-up flow in a porous medium channel

Abstract

The transient fluid motion in a porous medium channel is considered. Frictional resistance induced by the solid matrix and the channel walls is accounted for by a Darcian body force and a viscous shear stress, respectively. The adopted mathematical model leads to a one-parameter problem, in which the channel half-widthh, the porosity e and the permeabilityK combine into a shape parameterA=(eh2/K)1/2. Exact analytical solutions in terms of infinite series expansions are provided both for the start-up flow following the sudden imposition of a constant pressure gradient and for the transient motion induced by an instantaneously imposed flow rate. Time histories of the centerline velocity and the wall friction are presented, together with time-varying velocity profiles. It is observed that the start-up time required to reach a steady state is significantly reduced in the less porous channels, and this reduction is more pronounced when the start-up flow is driven by a pressure gradient than if the transient motion is forced by an imposed flow rate.