Couette-Poiseuille flow in a fluid overlying an anisotropic porous layer

Abstract

In this study, Couette-Poiseuille flow inside a channel partially filled with a porous medium is treated. Considering the applicability to the practical scenarios, we assume that the porous medium is anisotropic in nature. More precisely, the model under steady-state assumption is described by the coupling of Brinkman-Forchheimer and Stokes equations. We develop the existence and uniqueness results corresponding to this two-dimensional Stokes-Brinkman-Forchheimer coupled system using the Browder-Minty theorem. For this we restrict the permeability to a diagonal matrix. The coupled system involves a non-linear term due to the Brinkman-Forchheimer equation. We propose an algorithm based on shooting and searching techniques to develop a numerical solution for this system, however, for an arbitrary anisotropic permeability matrix. The robustness of the presented numerical solution is established by comparing it with asymptotic solutions corresponding to small and large Darcy numbers. For the case of a large Darcy number, the problem of interest is a regular perturbation while the same is a singular perturbation problem for the case of a small Darcy number, which is dealt with using Prandtl's matching principle. With the velocity distribution determined, we have presented a comparative analysis of the shear stress distribution at the liquid-porous interface and at the bottom plate, which is of interest to analyze the stress distribution on the arterial wall. The asymptotic results are found to be in good agreement with those obtained numerically.