A non-primitive boundary element technique for modeling flow through non-deformable porous medium using Brinkman equation

Abstract

In this study, we develop a non-primitive boundary integral equation (BIE) method for steady two-dimensional flows of an incompressible Newtonian fluids through porous medium. We assume that the porous medium is isotropic and homogeneous, and use Brinkman equation to model the fluid flow. First, we present BIE method for 2D Brinkman equation in terms of the non-primitive variables namely, stream-function and vorticity variables. Subsequently, a test problem namely, the lid-driven porous cavity over a unit square domain is presented to assert the accuracy of our BEM code. Finally, we discuss an application of our proposed method to flows through porous wavy channel, which is a problem of significant interest in the micro-fluidics, biological domains and groundwater flows. We observe that the rate of convergence ( $$R_{c}$$ ) increases with increasing Darcy number. For low Darcy number streamlines follow the curvature of the wavy-walled channel and no circulation occurs irrespective of the wave–amplitude, while for high Darcy number the flow circulation occurs near the crest of the wavy-walled channel, when the wave–amplitude is large enough.