Axisymmetric motion of a porous sphere through a spherical envelope subject to a stress jump condition

Abstract

The flow problem of an incompressible axisymmetrical quasisteady translation and steady rotation of a porous sphere in an eccentric spherical container is discussed using a combined analytical–numerical technique. A continuity of velocity components and normal stress together with the stress jump condition for the tangential stress are used at the interface between porous and clear-fluid regions. The fluid flow outside the particle is governed by the classical Stokes equations while the fluid flow inside the porous region is treated by Brinkman model. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based on both the porous sphere and spherical envelope. Solutions for translational and rotational motion of porous eccentric spherical particle in a spherical envelope are obtained using the boundary collocation technique. The hydrodynamic drag force and couple exerted by the surrounding fluid on the porous particle which is proportional to the translational and angular velocities, respectively, are calculated with good convergence for various values of the ratio of porous-to-container radii, the relative distance between the centers of the porous and container, the stress jump coefficient, and a coefficient that is proportional to the permeability. In the limits of the motions of a porous sphere in a concentric container and near a container surface with a small curvature, the numerical values of the normalized drag force and the normalized coupling coefficient are in good agreement with the available values in the literature.