Creeping flow of a viscous fluid past a pair of porous separated spheres^*

In this paper we consider the problem of creeping or the Stokes’ flow of a viscous fluid past a pair of porous separated spheres with the problem formulation done in the bipolar coordinate system. Stokesian approximation of the Navier-Stokes equations for the Newtonian fluid model is taken to describe the fluid flow in the region exterior to the porous spheres, while the classical Darcy's law is for the flow inside the porous spheres. An analytical solution to this problem is found wherein the expressions for stream function, pressure and velocity are derived in terms of the Legendre functions, the hyperbolic trigonometric functions and the Gegenbauer functions. Also, the expression for the drag experienced by each of the spheres is found and we carry out numerical evaluations to compute the values of drag in the cases where the two spheres are of equal radii and the case where they are of unequal radii. The plots of streamlines and pressure contours are presented and discussed.