Flow Past a Pair of Separated Porous Spheres using Brinkman model

Abstract

In this work, we obtained an analytical solution to the problem of viscous fluid flow past a pair of porous separated spheres. Stokesian approximation of the Navier-Stokes equation for the viscous fluid governs the flow in the region outside the two spheres, whereas Brinkman's model describes the flow in the porous region (within the spheres). Since the bipolar coordinate system is the most convenient system to represent separated spheres' geometry, we formulated this problem in the bipolar coordinate system. We then eliminated the pressure term from the equations governing the flow in the region outside the spheres, and they got reduced to a separable equation in terms of the stream function. Further, the flow governing equations inside the porous spheres gave rise to the Helmholtz equation.  As the Helmholtz equation is not separable in the bipolar system, we used the spherical coordinate system to describe the fluid flow within each separated spheres and solved the resulting problem in the spherical coordinate system. Because of this, the flow variables on either side of the interface (spheres) are in different coordinate systems.  To match the values of the field variables at the boundary, we used the transformation equations between the bipolar and spherical systems and transformed all the variables into the bipolar coordinate system. We then solved the governing equations with appropriate boundary conditions for the arbitrary constants and derived expressions for the stream and pressure functions. We plotted the respective functions for various values of the mathematical model's parameters to understand the flow pattern and pressure distribution in the flow domain and noted our observations. This study revealed an intuiting insight that the pressure distribution inside the porous spheres is independent of the non-dimensional parameter related to the medium's permeability and the fluid's viscosity.