Two-dimensional flows in finite-width channels partially filled with porous medium

Abstract

A stationary two-dimensional liquid flow in a channel partially filled with a porous non-deformable homogeneous medium is investigated numerically based on the proposed mathematical model. The homogeneous liquid is bounded by a solid or free rigid boundary, while the channel bottom is solid. The model is constructed based on the Berman transformation. The free flow is described by the Navier-Stokes equations and the filtration flow is described by the Darcy-Brinkman model. The interfacial boundary conditions include the velocity continuity and the balance of tangential and normal viscous stresses; the tangential stresses may have a discontinuity at the boundary. The numerical solution is found by the finite-difference relaxation method. It has been shown that the flow of the liquid into porous medium takes place for a wide parameter range. However in most cases, the transversal velocity is about 10-7of the maximum longitudinal velocity. The dependence of the transversal velocity on the permeability of porous medium and its thickness is examined. The velocity profiles are plotted for different sets of parameters. The effect of the relative thickness of porous medium on the transversal velocity maximum and its position is also considered. It has been found that the effect of the transversal flow becomes significant if the width of the free portion of the channel is less than 10% of the width of the entire system. In spite of the smallness of the transversal velocity, the total flow rate through the interface is about 0.1% of the total flow rate through the channel provided that the channel length is 105÷106of the channel width.