Slip at the surface of a translating–rotating sphere bisected by a free surface bounding a semi-infinite viscous fluid: Removal of the contact-line singularity

A linear slip, Basset-type, boundary condition having an experimentally adjustable phenomenological slip coefficient is used to remove the contact-line singularity that would otherwise prevent the movement of a partially penetrating sphere normal to a planar free surface F bounding a semi-infinite viscous fluid. Stokes flow calculations are presented for the quasistatic hydrodynamic force and torque resistance matrix for a half-submerged sphere that is instantaneously translating and rotating with vector velocities that are arbitrarily oriented relative to the free-surface unit normal vector. The singular components of this material matrix (arising either during translational motion normal to F or rotational motion about an axis lying within F) are shown to be finite for finite slip coefficients β, and to become logarithmically infinite in the traditional nonslip limit β→∞. The relative weakness of this logarithmic singularity suggests that a degree of slip as small as, say, 0.01%—which would presumably be kinematically indistinguishable from the no-slip case—could easily masquerade as a conventional ‘‘wall effect’’ on the Stokes drag. A small degree of slip is thus hypothesized as a mechanism that would permit the observed transport of Brownian corpuscles across interfacial regions.