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.

Interactions between stably rolling leukocytes in vivo

“…[8][9][10]12 Many of these previous studies have demonstrated the importance particle surface roughness in relieving the lubrication singularity of spheres at contact. 9,11 The flattening of a liquid drop adherent on a wall in shear flow 14 is relevant to leukocyte adhesion at much higher shear stresses than studied here ͑ ϳ 10 dyn/ cm 2 ͒, when significant deformation of the cell occurs. In a work closely related to the present study, Chapman and Cokelet 15 calculated the drag force on multiple spheres adherent on the wall of a tube; however, they did not consider any motion of the spheres.…”

“…Several recent fluid mechanical studies have examined the interaction of small particles with a planar wall, [8][9][10][11][12][13] however, none of these have focused on the adhesive rolling motion of spheres along the wall of a circular tube. In our system, characteristic Reynolds numbers based on the leukocyte radius are O͑10 −3 ͒, so that inertial effects are expected to be negligible.…”

Dynamic adhesion behavior of micrometer-scale particles flowing over patchy surfaces with nanoscale electrostatic heterogeneity

“…Furthermore, surface contact (D = δ) introduces a non-hydrodynamic frictional resistance to particle motion. Problems involving the rolling of a sphere down an inclined plane immersed in a viscous liquid have been addressed in earlier studies [33,34] in which the normal force between the particle and the surface was due to gravity, and the contact between surface roughness elements present on the surface gave rise to a non-hydrodynamic resistive force. The present analysis extends this work to a neutrally buoyant particle experiencing a spatially varying normal force due to the physicochemical interactions between the particle and the nanotextured wall.…”

How to elutriate particles according to their density

“…Although subjected to a lift force, the particles will still remain in virtual contact with the incline. The torque balance and rotation of the particles (Galvin et al, 2001) are neglected given that most "real" particles have an irregular geometry and therefore resist any tendency to rotate, preferring to slide.…”