Transient rotation of a spherical particle in a concentric cavity with slip surfaces

Abstract

The start-up of slow rotation of a spherical particle due to the sudden application of a constant torque about a diameter in a viscous fluid inside a concentric spherical cavity with slip surfaces is studied. The unsteady Stokes equations governing the fluid velocity are solved by means of the Laplace transform, and an analytical formula for the dimensionless transient angular velocity of the particle is obtained as a function of the relevant dimensionless parameters. The influence of the confining slip cavity on the starting rotation of the slip sphere is significant and interesting. The angular velocity develops continuously with the elapsed time from zero at the beginning to its terminal value and the angular acceleration decreases monotonically with the time. The transient angular velocity increases with an increase in the particle-to-cavity radius ratio as the cavity is sufficiently slippery to enhance the particle rotation, decreases with an increase in this radius ratio as the cavity is sufficiently sticky to retard the rotation, and is not a sensitive function of the radius ratio as the slippage or stickiness of the cavity wall is in between. A particle with a higher relative density, smaller radius ratio, or less stickiness (or stickiness of the cavity) trails behind the reference particle in the development of the angular velocity with respect to individual terminal values, even if the transient angular velocity increases with a decrease in the stickiness of the particle or cavity.