Coupled Parallel Flows in a Channel and a Bounding Porous Medium of Finite Thickness

The steady laminar flow in a parallel plate channel bounded below by a porous layer of finite thickness and above by a rigid impermeable plate moving with a uniform velocity is studied. The two cases, viz., the porous medium bounded below (i) by a static fluid and (ii) by a rigid impermeable stationary wall, are considered separately. The modified slip condition involving the thickness of the porous layer is derived using the variation of velocity in the porous medium with the proper matching conditions based on the physical considerations. It is shown that when the thickness of the porous layer tends to infinity our modified slip condition tends to the slip condition postulated by Beavers and Joseph [13]. Methods to estimate the viscosity factor λ and relative permeability are discussed. The velocity profiles in the porous layer are shown to exhibit the boundary-layer type very near the porous surface; they increase with increase in depth of the porous medium and decrease with increases in λ. We find that the effect of the finite thickness of the porous medium is significant only for large values of λ and small values of the porous parameter σ.