Viscous/Inertial Flow in Channels with Wavy Permeable Walls Immersed in Porous Medium

The flow of a viscous fluid in a channel with wavy walls immersed in the porous medium has application to many scientific and industrial area. The Poiseuille flow in tubes and cubic law in parallel-plate channels cannot describe this problem, because of the varying cross-section and presences of inflow through the walls. The inflow and the irregular wavy geometry of the walls result in non-negligible inertial and visco-inertial effects along the channel. The asymptotic solutions of Navier-Stokes equations with slip boundary condition are obtained for axisymmetric and parallel-plate wavy channels. Two-scale homogenization technique is used to capture the effect of the corrugations on the flow. We show that the inflow through the walls creates local flow instabilities and forms reverse flow. The averaged pressure drop along the channel includes quadratic and cubic corrections to the linear law. The quadratic correction only exists in case of permeable walls. The cubic correction corresponds to the wavy geometry. In axisymmetric channels, the cubic term decreases the pressure drop as the corrugation amplitude increases, while in parallel-plate channels the cubic term is less dominant. The channel's effective permeability decreases as the amplitude of the corrugations increases.