The general motion of a circular disk in a Brinkman medium

Abstract

Solutions of the Brinkman equation for the arbitrary motion of a circular disk are obtained which examine for the first time the effect of particle orientation on the particle drag and torque. Four elementary motions are studied analytically: broadside translation, edgewise translation, rotation about the axis of symmetry, and rotation about the diameter. These motions are closely related to the analogous unsteady oscillation of a disk in Stokes flow [Zhang and Stone, J. Fluid Mech. (1998)]. However, our solution procedure differs in that the problems are formulated using a general solution of the Brinkman equation and are solved by reducing the dual integral equations arising from the mixed boundary conditions in the plane of the disk to a Fredholm integral equation of the second type. Asymptotic results for the drag and torque are derived for both small and large values of the permeability parameter α defined by a/Kp, where a is the radius of the disk and Kp the Darcy permeability. In contrast to the Stokesian motion of a disk, where the drag differs by only a factor of 1.5 for broadside and edgewise translational motion, and is isotropic for rotation about any axis through its center, there is a large difference in the drag and torque with increasing α. In a Brinkman medium, the drag on the disk is proportional to α for edgewise motion and to α2 for broadside motion and the torque is proportional to α2 for out-of-plane rotation and to α for in-plane rotation. For intermediate values of α, the integral equations are solved numerically for the drag and torque exerted by the porous medium on the disk. These results are of importance in probing the microstructure of the porous medium and thus provide a way to test the validity of the effective medium approach.