Slow motion of a slip spheroid along its axis of revolution

Abstract

A combined analytical and numerical study of the Stokes flow caused by a rigid spheroidal particle translating along its axis of revolution in a viscous fluid is presented. The fluid is allowed to slip at the surface of the particle. The general solution for the stream function in prolate and oblate spheroidal coordinates can be expressed in an infinite-series form of semi-separation of variables. The slip boundary condition incorporating the shear stress at the particle surface is applied to this general solution to determine its unknown coefficients of the leading orders. The solution of these coefficients can be either numerical results obtained from a boundary-collocation method or explicit formulas derived analytically. The drag force exerted on the spheroidal particle by the fluid is evaluated with good convergence behavior for various values of the slip parameter and aspect ratio of the particle. The agreement between our hydrodynamic drag results and the relevant numerical solutions obtained previously using a singularity method is excellent. Although the drag force acting on the translating spheroid normalized by that on a corresponding sphere with equal equatorial radius increases monotonically with an increase in the axial-to-radial aspect ratio for a no-slip spheroid, it decreases monotonically as this aspect ratio increases for a perfect-slip spheroid. The normalized drag force exerted on a spheroid with a given surface slip coefficient in between the no-slip and perfect-slip limits is not a monotonic function of its aspect ratio. For a spheroid with a fixed aspect ratio, its drag force is a monotonically decreasing function of the slip coefficient of the particle.