Creeping motion of a fluid drop inside a spherical cavity

Abstract

A combined analytical–numerical study for the quasisteady creeping flow caused by a fluid sphere with an arbitrary viscosity translating at an arbitrary position in a second, immiscible fluid within a spherical cavity in the direction perpendicular to the line connecting their centers is presented. To solve the Stokes equations for the fluid velocity fields inside and outside the drop, a general solution is constructed from the fundamental solutions in the two spherical coordinate systems based on both the drop and the cavity. The boundary conditions on the drop surface and cavity wall are satisfied by a multipole collocation technique. Numerical results for the hydrodynamic drag force acting on the drop are obtained with good convergence for various values of the relative viscosity of the drop, the ratio of drop-to-cavity radii, and the relative distance between the centers of the drop and cavity. In the limits of the motions of a fluid sphere 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. The wall-corrected drag force acting on the drop for any case is found to be a monotonic increasing function of the ratio of drop-to-cavity radii. For a fixed ratio of drop-to-cavity radii, the drag force is minimal when the drop is situated at the cavity center and increases monotonically with its relative distance from the cavity center. The drag force exerted on the drop in general increases with an increase in its relative viscosity for a given configuration, but there are exceptions when the ratio of drop-to-cavity radii is large. The boundary effect on the drop motion normal to the line of the drop and cavity centers is found to be significant, but in general weaker than that along the line of the centers.