On the motion of a porous sphere in a Stokes flow parallel to a planar confining boundary

Abstract

An analysis is presented of the three-dimensional creeping flow in and around a porous sphere, modelled as a generalized Brinkman medium, near a smooth plane where the sphere (i) translates uniformly without rotating in an otherwise quiescent Newtonian fluid, (ii) rotates uniformly without translating in an otherwise quiescent Newtonian fluid, and (iii) is fixed in a shear field, which is uniform in the far field and has a linearly increasing velocity profile with increasing distance from the plane. The linear superposition of these three flow regimes is also considered for the special case of the free translational and rotational motion of a neutrally buoyant porous sphere in a shear field that is uniform in the far field. Exact series solutions to the momentum equations are derived for the velocity and pressure fields in the Brinkman and Stokes-flow regions. Coefficients in the series solutions for each flow regime are determined using recursion relations derived from the continuity equations in the Brinkman and Stokes-flow regions, from the interfacial boundary conditions on the porous spherical surface, and from the no-slip condition on the plane. Results are presented in terms of the drag force on the porous sphere and torque about the sphere centre as a function of the dimensionless clearance distance between the sphere and the rigid plane for several values of the dimensionless hydraulic permeability of the Brinkman medium. The free motion of the neutrally buoyant sphere is calculated by requiring that the net hydrodynamic drag force and torque acting on the sphere vanish. Results for this case are presented in terms of the dimensionless translational and rotational speeds of the porous sphere relative to the far-field shear rate as a function of the dimensionless clearance distance for several values of the dimensionless hydraulic permeability. The work is motivated by insights it offers into the behaviour of porous agglomerates, and by its potential utility in industrial, biological, biophysical, medicinal and environmental applications wherever gas or liquid suspensions of porous agglomerates might arise.