Axisymmetric creeping motion of a slip spherical particle in a nonconcentric spherical cavity

A semianalytical study of the creeping flow caused by a spherical fluid or solid particle with a slip surface translating in a viscous fluid within a spherical cavity along the line connecting their centers is presented in the quasisteady limit of small Reynolds number. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the fundamental solutions in the two spherical coordinate systems based on both the particle and cavity. The boundary conditions on the particle surface and cavity wall are satisfied by a collocation technique. Numerical results for the hydrodynamic drag force exerted on the particle are obtained with good convergence for various values of the ratio of particle-to-cavity radii, the relative distance between the centers of the particle and cavity, the relative viscosity or slip coefficient of the particle, and the slip coefficient of the cavity wall. In the limits of the motions of a spherical particle in a concentric cavity and near a cavity wall with a small curvature, our drag results are in good agreement with the available solutions in the literature. As expected, the boundary-corrected drag force exerted on the particle for all cases is a monotonic increasing function of the ratio of particle-to-cavity radii, and becomes infinite in the touching limit. For a specified ratio of particle-to-cavity radii, the drag force is minimal when the particle is situated at the cavity center and increases monotonically with its relative distance from the cavity center to infinity in the limit as it is located extremely away from the cavity center. The drag force acting on the particle, in general, increases with an increase in its relative viscosity or with a decrease in its slip coefficient for a given configuration, but surprisingly, there are exceptions when the ratio of particle-to-cavity radii is large.