Motion through a viscous fluid of a spherical particle touching a plane wall: Slip boundary conditions

摘要

Understanding the hydrodynamic forces acting upon immersed particles touching surfaces, is of central importance in clean room technology and a variety of rheological and biological applications. This paper addresses the translation and rotation of a sphere translating and rotating parallel to a nearby plane wall bounding an otherwise quiescent semi-infinite viscous fluid, allowing for slip on the wall and/or the sphere. The motivation for disregarding the classical, no-slip boundary condition on solid surfaces aries from an embarrassing discrepancy between theoretical and observed predictions of the translational velocity of a sphere 'rolling' under the influence of gravity down an inclined plane bounding an effectively semi-infinite viscous fluid. According to theory the force and torque on a translating and/or rotating sphere moving parallel to the plane wall become logarithmically infinite with the gap width as the gap between the sphere and well goes to zero. As such, the theoretical conclusion is that the sphere cannot translate down the plane, despite the gravity force that acts to animate it. Experiments, however, reveal that the sphere does, in fact, roll down the plane - at a reproducible mean terminal velocity. In the noninertial, small Reynolds number limit, the experimentally observed drag coefficient was found to be about 8.9 times that given by Stokes law for the unbounded case - thereby suggesting a conventional hydrodynamic wall effect, rather than the logarithmically singular behavior predicted by the theory. It was in an attempt to resolve this glaring contradiction that we have elected here to examine the possible effects of slip.