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

A system consisting of a spherical particle in motion down an inclined plan ar surface in a viscous liquid was investigated theoretically and experimen tally to examine the effects of surface roughness on the interactions betwe en the sphere and the plane. Two characteristic roughness scales were used to describe the microscopic surface roughness of the sphere. The smallest r oughness elements are assumed to dominate the surface, and the largest roug hness elements are more sparse. The time-averaged nominal separation betwee n the sphere and the plane was found to increase as the planar surface was made steeper. This apparent hydrodynamic roughness is governed by the heigh ts of the smallest roughness when the sphere resides on a horizontal plane, whereas the largest roughness elements govern the apparent hydrodynamic ro ughness when the plane is inclined at a steep angle. On a steep incline, th e normal component of the gravitational force that drives the sphere toward the plane is relatively weak. Hence, as the sphere migrates toward the pla ne after contact with a large asperity ends, its rotation may result in ano ther large asperity forcing the sphere away from the plane before contact w ith the smaller asperities occurs. The time-averaged separation at intermed iate angles increases with increasing surface coverage by the largest rough ness elements. The method of Smart and Leighton [Phys. Fluids A 1, 526 (198 9)] was modified to determine the hydrodynamic separation between the spher e and the plane during its motion down the incline. The apparent hydrodynam ic roughness values obtained in the experiments increase as the angle of in clination of the plane was increased, and provide a satisfactory validation of the model. The relatively large but sparse roughness elements have a di sproportionate effect on the time-averaged hydrodynamic roughness, especial ly at high angles of inclination. These findings may be important in the in teraction of pairs of spherical particles in viscous suspensions, where the effective angle of inclination varies significantly. For example, the pres ence of a low concentration of relatively large roughness elements should r esult in significantly higher levels of hydrodynamic diffusion. (C) 2001 Am erican Institute of Physics.