Asymptotic Analysis of the Laminar Viscous Flow Over a Porous Bed

Abstract

We consider the laminar viscous channel flow over a porous surface. The size of the pores is much smaller than the size of the channel, and it is important to determine the effective boundary conditions at the porous surface. We study the corresponding boundary layers, and, by a rigorous asymptotic expansion, we obtain Saffman's modification of the interface condition observed by Beavers and Joseph. The effective coefficient in the law is determined through an auxiliary boundary-layer type problem, whose computational and modeling aspects are discussed in detail. Furthermore, the approximation errors for the velocity and for the effective mass flow are given as powers of the characteristic pore size $\ep$. Finally, we give the interface condition linking the effective pressure fields in the porous medium and in the channel, and we determine the jump of the effective pressures explicitly.