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

A general numerical method using the boundary integral equation technique o f Pozrikidis (1994) for Stokes how in an axisymmetric domain is used to obt ain the first solutions to the Brinkman equation for the motion of a partic le in the presence of planar confining boundaries. The method is first appl ied to study the perpendicular and parallel motion of a sphere in a fibre-f illed 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 effici ently and accurately treat flow problems with high alpha defined by r(s)/K- p(1/2) in which r(s) is the radius of the sphere and K-p is the Darcy perme ability. Convergence and accuracy of the new technique are tested by compar ing results for the drag with the solutions of Kim & Russell (1985a) for th e motion of two spheres perpendicular to their line of centres in a Brinkma n 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 epsilon greater than or equal to 0.1, in which epsilon r(s) is the gap between the particle and the boundary. When the gap width is muc h smaller than r(s), a local analysis using stretched variables for motion of a sphere indicates that the leading singular term for both drag and torq ue is independent of alpha provided that alpha = O(1). These results are of interest in modelling the penetration of the endothelial surface glycocaly x 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 inves tigate the motion of a sphere translating in a channel. The drag and torque exerted on the sphere are obtained for various values of alpha, the channe l height H and particle position b. These numerical results are used to des cribe the diffusion of a spherical solute molecule in a parallel walled cha nnel filled with a periodic array of cylindrical fibres and to assess the a ccuracy of a simple multiplicative formula proposed in Weinbaum et al. (199 2) for diffusion of a solute in the interendothelial cleft.