Effects of inertia on the slow rotation of a slip spherical particle

Abstract

The steady rotational motion of a spherical particle with frictionally slip surface about a diameter in a quiescent unbounded fluid is studied analytically for a small but finite Reynolds number Re. The Navier–Stokes equation governing the axisymmetric fluid flow around the particle is solved by using a regular perturbation method. The expansion expression for the retarding torque exerted by the fluid on the particle good to O(Re3) is obtained as a function of the scaled slip coefficient of the particle in closed form. This perturbation analysis shows that the perturbed fluid velocity field is of O(Re), the same as that for the translation of the particle, but the first correction to the hydrodynamic torque occurs at O(Re2), much weaker than that to the hydrodynamic drag force acting on a translating particle which is still of O(Re). The hydrodynamic torque is found to be a monotonic increasing function of the Reynolds number and a monotonic decreasing function of the scaled slip coefficient, similar to the effect of fluid inertia on the hydrodynamic force on a slip particle undergoing translation. The inertial effect on the hydrodynamic torque, which can be a sensitive function of the slip coefficient and vanishes as the particle surface is fully slip, is generally negligible for the case of Re<3 but significant as Re is greater than about 10.