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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.

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Solid-solid contacts due to microscopic surface roughness in viscous fluids were examined by observing the translational and rotational behaviours of a suspended sphere falling past a lighter sphere or down an inclined surface. In both cases, a roll-slip behaviour was observed, with the gravitational forces balanced by not only hydrodynamic forces but also normal and tangential solid-solid contact forces. Moreover, the nominal separation between the surfaces due to microscopic surface roughness elements is not constant but instead varies due to multiple roughness scales. By inverting the system, so that the heavy sphere fell away from the lighter sphere or the plane, it was found that the average nominal separation increases with increasing angle of inclination of the plane or the surface of the lighter sphere from horizontal; the larger asperities lift the sphere up from the opposing surface and then gravity at large angles of inclination is too weak to pull the sphere back down to the opposing surface before another large asperity is encountered. The existence of microscopic surface roughness and solid-solid contacts is shown to modify the rheological properties of suspensions. For example, the presence of compressive, but not tensile, contact forces removes the reversibility of sphere-sphere interactions and breaks the symmetry of the particle trajectories. As a result, suspensions of rough spheres exhibit normal stress differences that are absent for smooth spheres. For the conditions studied, surface roughness reduces the effective viscosity of a suspension by limiting the lubrication resistance during near-contact motion, and it also modifies the suspension microstructure and hydrodynamic diffusivity.