Lie Symmetry Analysis of a cross-flow induced porous media flow in a weakly permeable channel

This investigation focuses on the symmetry analysis and solution of a viscous and incompressible flow in a channel filled with isotropic porous material. There is a uniform cross-flow through the upper and lower walls of the channels that generates a flow inside the porous media. The flow dynamics inside the channel is governed by the full Darcy-Brinkman equations. The tangential velocity slip along the walls is modeled by Navier slip boundary condition. The Lie symmetry analysis based on invariance principle are adopted to reduce the number of independent variables of the system of governing equations. Consequently, a single fourth order non-linear ordinary differential equation is obtained, which is solved analytically by the perturbation method in the small parameter range. However, it is solved for arbitrary parameter range analytically with variation of Iteration method (VIM) as well as numerically using a fourth order Runge–Kutta solver. A stronger cross-velocity enhances the flow rate and advances the mean flow inside the channel. Consequently, the maximum velocity and total shear rate become higher. A relatively larger Darcy number allows a faster flow through the porous medium and the axial velocity profile becomes fuller. The obtained results and outcomes are usefull for biomedical applications.