Slow rotation of a spherical particle in an eccentric spherical cavity with slip surfaces

Abstract

A semi-analytical study of the steady flow around a spherical particle rotating in an incompressible Newtonian fluid inside an eccentric spherical cavity with slip surfaces about their common diameter is presented at low Reynolds numbers. To solve the Stokes equation, a solution consisting of the general solutions in two systems of spherical coordinates is employed and the boundary conditions are fulfilled by a collocation method. Accurate results of the torque exerted by the fluid on the particle are obtained as a function of the dimensionless parameters a∕b, d∕(b−a), βa∕η, and βwb∕η, where a and b are the radii and β−1 and βw−1 are the Navier slip coefficients of the particle and cavity, respectively, d is the distance between the particle and cavity centers, and η is the fluid viscosity. The boundary effect of the cavity with a slip wall on the rotation of the slip particle is quite significant and interesting. The torque normalized by that on the particle in an unbounded identical fluid vanishes as βwb∕η=0 (the cavity wall is fully slip), equals unity for a value of βwb∕η very close to 3, and in general increases with an increase in βwb∕η (or the stickiness of the cavity wall). When βwb∕η>3, the normalized torque is in general greater than unity, an increasing/decreasing function of the eccentricity parameter d∕(b−a) if the value of βa∕η (or the stickiness of the particle surface) is large/small, and an increasing function of βa∕η and a∕b. When βwb∕η<3, conversely, the normalized torque is in general less than unity, a decreasing/increasing function of d∕(b−a) if the value of βa∕η is large/small, and a decreasing function of βa∕η and a∕b. The cavity wall exerts less torque on the particle when it rotates about their common diameter than about an axis normal to it.