Gravitational and zero-drag motion of a sphere of arbitrary size in an inclined channel at low Reynolds number

Abstract

The strong-interaction theory developed in Ganatos, Weinbaum & Pfeffer (1 9804 and Ganatos, Pfeffer & Weinbaum (1980b) for the normal and parallel creeping motion of a sphere of arbitrary size between two infinite plane-parallel walls is applied to several particle-boundary interaction problems of long-standing interest. The first highly accurate solutions are presented for the slip and angular velocity of a neutrally buoyant sphere carried by the fluid in a Couette or Poiseuille channel flow and the gravitational settling of a non-neutrally buoyant sphere in an inclined channel. The latter problem clearly illustrates the non-isotropy of the frictional resistance tensor on the particle motion. The solutions for the fluid velocity field exhibit an induced circulation extending to infinity fore and aft of the sphere for a neutrally buoyant sphere in Couette flow and an induced back-flow ahead of the sphere for the Poiseuille flow geometry. Approximate but highly accurate solutions are presented for small gap widths between a sphere and the neighbouring boundary, which take account of the influence of the second wall.