Flow of Viscous Fluid at Small Reynolds Numbers past a Porous Sphere

Abstract

The problem of a porous and permeable sphere in a uniform flow at small Reynolds number is considered. It is assumed that the Stokes equation holds outside the sphere and generalized Darcy’s law holds inside the sphere. A simple closed solution which depends on permeability of the sphere is derived. A formula for the drag acting on the sphere is also obtained.
It is found that for small permeability the flow in the sphere is governed essentially by Darcy’s law except at a boundary layer near the surface of the sphere. A proper boundary condition at the surface for the Stokes equation and Darcy’s law is given when the latter is assumed to hold all over the sphere by neglecting the boundary layer.