Mathematical results for slipping flows past a cylinder or a sphere embedded in a porous medium

Abstract

Flow past a circular cylinder or a sphere embedded in a porous medium is investigated mathematically by treating porous matrix as an incompressible fluid. Closed form analytic solutions for the fourth order scalar boundary-value problems for Ψ (r, θ) - known as the Stokes stream function - are obtained by using Navier-slip conditions. Our exact results for Ψ (r, θ) capture flow fields prevailing in the vicinity of a cylinder/sphere suspended in a uniform or a shear flow field. All the physical quantities computed from our solutions depend on two key parameters, namely, δ (the effective permeability) and ζ (slip coefficient). Flow separation and velocity overshoot behavior are found for certain values of δ and ζ. The force acting on the cylinder/sphere is calculated in each case. It is observed that the presence of slip decreases the force on the boundary. Our results show that in the limit δ -> 0, there is no solution to the two-dimensional boundary-value problem, confirming the familiar Stokes paradox