Time-averaged hydrodynamic roughness of a noncolloidal sphere in low Reynolds number motion down an inclined plane

Abstract

A system consisting of a spherical particle in motion down an inclined planar surface in a viscous liquid was investigated theoretically and experimentally to examine the effects of surface roughness on the interactions between the sphere and the plane. Two characteristic roughness scales were used to describe the microscopic surface roughness of the sphere. The smallest roughness elements are assumed to dominate the surface, and the largest roughness elements are more sparse. The time-averaged nominal separation between the sphere and the plane was found to increase as the planar surface was made steeper. This apparent hydrodynamic roughness is governed by the heights of the smallest roughness when the sphere resides on a horizontal plane, whereas the largest roughness elements govern the apparent hydrodynamic roughness when the plane is inclined at a steep angle. On a steep incline, the normal component of the gravitational force that drives the sphere toward the plane is relatively weak. Hence, as the sphere migrates toward the plane after contact with a large asperity ends, its rotation may result in another large asperity forcing the sphere away from the plane before contact with the smaller asperities occurs. The time-averaged separation at intermediate angles increases with increasing surface coverage by the largest roughness elements. The method of Smart and Leighton [Phys. Fluids A 1, 526 (1989)] was modified to determine the hydrodynamic separation between the sphere and the plane during its motion down the incline. The apparent hydrodynamic roughness values obtained in the experiments increase as the angle of inclination of the plane was increased, and provide a satisfactory validation of the model. The relatively large but sparse roughness elements have a disproportionate effect on the time-averaged hydrodynamic roughness, especially at high angles of inclination. These findings may be important in the interaction of pairs of spherical particles in viscous suspensions, where the effective angle of inclination varies significantly. For example, the presence of a low concentration of relatively large roughness elements should result in significantly higher levels of hydrodynamic diffusion.