The hydrodynamics of an oscillating porous sphere

Abstract

We determine the hydrodynamics of a rigid, weakly permeable sphere undergoing translational oscillations in an incompressible Newtonian fluid. We check using homogenization and scaling arguments that the flow inside the sphere may be modeled by Darcy’s law and that the Beavers–Joseph–Saffman (BJS) boundary condition still applies for oscillatory flows, provided the frequency of oscillation is not too high. The BJS boundary condition introduces a slip velocity and to leading order in ε=k/a, where k is the particle permeability and a is the radius, the particle may be regarded as impermeable with a slip length independent of frequency. Under these circumstances we solve for the flow field, pressure distribution and drag explicitly and show their behavior for 0⩽ε⩽0.05 and frequencies relevant to electroacoustics (1–10 MHz). From the drag we find the leading order corrections due to particle permeability of the pseudo-steady drag, Basset force and added mass.