Slow Translation of a Composite Sphere in an Eccentric Spherical Cavity

Abstract

This semi-analytical study is presented examining the quasi-steady creeping flow caused by a soft (composite) spherical particle, which is a hard (impermeable) sphere core covered by a porous (permeable) layer, translating in an incompressible Newtonian fluid within a non-concentric spherical cavity along the line joining their centers. To solve the Brinkman and Stokes equations for the flow fields inside and outside the porous layer, respectively, general solutions are constructed in two spherical coordinate systems attached to the particle and cavity individually. The boundary conditions at the cavity wall and particle surface are fulfilled through a collocation method. Numerical results of the normalized drag force exerted by the fluid on the particle are obtained for numerous values of the ratios of core-to-particle radii, particle-to-cavity radii, the distance between the centers to the radius difference of the particle and cavity, and the particle radius to porous layer permeation length. For the translation of a soft sphere within a concentric cavity or near a small-curvature cavity wall, our drag results agree with solutions available in the literature. The cavity effect on the drag force of a translating soft sphere is monotonically increasing functions of the ratios of core-to-particle radii and the particle radius to porous layer permeation length. While the drag force generally rises with an increase in the ratio of particle-to-cavity radii, a weak minimum (surprisingly, smaller than that for an unconfined soft sphere) may occur for the case of low ratios of core-to-particle radii and of the particle radius to permeation length. This drag force generally increases with an increase in the eccentricity of the particle position, but in the case of low ratios of core-to-particle radii and particle radius to permeation length, the drag force may decrease slightly with increasing eccentricity.