Motion of a sphere near planar confining boundaries in a Brinkman medium

Abstract

A general numerical method using the boundary integral equation technique of Pozrikidis (1994) for Stokes flow in an axisymmetric domain is used to obtain the first solutions to the Brinkman equation for the motion of a particle in the presence of planar confining boundaries. The method is first applied to study the perpendicular and parallel motion of a sphere in a fibre-filled medium bounded by either a solid wall or a planar free surface which remains undeformed. By accurately evaluating the singular integrals arising from the discretization of the resulting integral equation, one can efficiently and accurately treat flow problems with high α defined by rs/K1/2p in which rs is the radius of the sphere and Kp is the Darcy permeability. Convergence and accuracy of the new technique are tested by comparing results for the drag with the solutions of Kim & Russell (1985a) for the motion of two spheres perpendicular to their line of centres in a Brinkman medium. Numerical results for the drag and torque exerted on the particle moving either perpendicular or parallel to a confining planar boundary are presented for ε[ges ]0.1, in which εrs is the gap between the particle and the boundary. When the gap width is much smaller than rs, a local analysis using stretched variables for motion of a sphere indicates that the leading singular term for both drag and torque is independent of α provided that α = O(1). These results are of interest in modelling the penetration of the endothelial surface glycocalyx by microvilli on rolling neutrophils and the motion of colloidal gold and latex particles when they are attached to membrane receptors and observed in nanovid (video enhanced) microscopy. The method is then applied to investigate the motion of a sphere translating in a channel. The drag and torque exerted on the sphere are obtained for various values of α, the channel height H and particle position b. These numerical results are used to describe the diffusion of a spherical solute molecule in a parallel walled channel filled with a periodic array of cylindrical fibres and to assess the accuracy of a simple multiplicative formula proposed in Weinbaum et al. (1992) for diffusion of a solute in the interendothelial cleft.