Axisymmetric motion of a spherical porous particle perpendicular to two parallel plates with slip surfaces

Abstract

A combined analytical–numerical approach to the problem of the low Reynolds number motion of a porous sphere normal to one of two infinite parallel plates at an arbitrary position between them in a viscous fluid is investigated. The clear fluid motion governed by the Stokes equation and the Darcy–Brinkman equation is used to model the flow inside the porous material. The motion in each of the homogeneous regions is coupled with the continuity of the velocity components, the continuity of the normal stress, and the tangential stress jump condition. The fluid is allowed to slip at the surface of the walls. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The collocation solutions for the hydrodynamic interactions between the porous sphere and the walls are calculated with good convergence for various values of the slip coefficient of the walls, the separation between the porous sphere and the walls, the stress jump coefficient, and a coefficient that is proportional to the permeability. For the special cases of a solid sphere, our drag results show excellent agreement with the available solutions in the literature for all relative particle-to-wall spacing.