An approximate model for slow free-surface flows of a fluid through a highly porous medium accounting for the effect of a capillary fringe

Abstract

The paper proposes a model for unconfined creeping flows of a thin fluid layer through a highly porous medium accounting for the effect of a capillary fringe at the free surface. To derive the model, we used the thin-layer approximation for the Brinkman equation with the Navier slip conditions at both the impermeable underlying surface and the free surface. The equation for the fluid layer thickness appears to be a special type of the non-linear heat conduction equation. We present similarity solutions that describe fluid propagation over a horizontal substrate.