On the motion of a sphere in a Stokes flow parallel to a Brinkman half-space

Abstract

A three-dimensional analysis is presented of the Stokes flow, adjacent to a Brinkman half-space, that is induced or altered by the presence of a sphere in the flow field that ($a$) translates uniformly without rotating, ($b$) rotates uniformly without translating, or ($c$) is fixed in a shear flow that is uniform in the far field. The linear superposition of these three flow regimes is also considered for the special case of the free motion of a neutrally buoyant sphere. Exact solutions to the momentum equations are obtained in terms of infinite series expansions in the Stokes-flow region and in terms of integral transforms in the Brinkman medium. Attention is focused on the approach to the asymptotic limit as the ratio of Newtonian- to Darcy-drag forces vanishes. From the leading-order asymptotic approximations, implicit recursion relations are derived to determine the coefficients in the series solutions such that those solutions exactly satisfy the boundary and interfacial conditions as well as the continuity equations in both the Stokes-flow and Brinkman regions. For each of the three flow regimes considered, results are presented in terms of the drag force on the sphere and torque about the sphere centre as a function of the dimensionless separation distance between the sphere and the interfacial plane for several small values of the dimensionless hydraulic permeability of the Brinkman medium. Finally, the free motion of a neutrally buoyant sphere is found 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 sphere as a function of the dimensionless separation distance for several small values of the dimensionless hydraulic permeability. The work is motivated by its potential application as an analytical tool in the study of near-wall microfluidics in the vicinity of the glycocalyx surface layer on vascular endothelium and in microelectromechanical systems devices where charged macromolecules may become adsorbed to microchannel walls.