Symmetry analysis of a cross-velocity induced flow in a channel filled with porous media

The study comprehensively analyses the symmetries and similarity solutions for a viscous, incompressible flow in a channel filled with isotropic porous material. One end of the channel is closed, and a uniform cross-flow through the upper and lower porous walls generates the flow inside the channel. The dynamics of the flow are governed by the full Darcy-Brinkman equations with Navier slip boundary conditions at the walls. The Classical and non-classical Lie symmetry analysis, based on the principle of invariance, are applied to reduce the number of independent variables in the governing equations. This reduction leads to a single fourth-order nonlinear ordinary differential equation, which is solved analytically using the perturbation method for small parameter ranges and through the Variational Iteration Method (VIM) for arbitrary parameters. A fourth-order Runge-Kutta method is also employed for numerical solutions. The results show that stronger cross-flow enhances the flow rate and increases the mean velocity in the channel, leading to higher maximum velocities and total shear rates. Wall slip conditions raise the velocity near the walls but reduce it near the centerline to maintain a constant volumetric flow rate. Additionally, significant wall slip strengthens wall shear, and a larger Darcy number promotes faster flow through the porous medium, resulting in a fuller axial velocity profile.