Creeping motions of a porous spherical shell in a concentric spherical cavity

The quasisteady translation and steady rotation of a spherically symmetric porous shell located at the center of a spherical cavity filled with an incompressible Newtonian fluid are investigated analytically. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field of the system, and the hydrodynamic drag force and torque exerted by the fluid on the porous shell which are proportional to the translational and angular velocities, respectively, are obtained in closed form. For a given geometry, the normalized wall-corrected translational and rotational mobilities of the porous shell decrease monotonically with a decrease in its permeability. The boundary effects of the cavity wall on the creeping motions of a porous shell can be quite significant in appropriate situations. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a porous spherical shell in the cavity reduce to those for a solid sphere and for a porous sphere. The hydrodynamic behavior for a porous spherical shell may be approximated by that for a permeable sphere when the porous shell is sufficiently thick, depending on its permeability.