The drag on a sphere moving axially in a long rotating container

Abstract

A container of viscous incompressible liquid is bounded by rigid parallel planes and rotates steadily about an axis normal to these planes. A rigid sphere moves steadily parallel to the rotation axis and the Rossby and Ekman numbers characterizing the motion are both small. The drag on the sphere is calculated in the case when the length of the Taylor column is comparable to the axial dimension of the container. Viscous effects are allowed for in the boundary of the Taylor column, but the Ekman layers on the sphere and on the bounding planes are shown not to affect the drag to leading order. The determination of the drag involves solving dual integral equations. This is done numerically and, for the limiting cases of long and short containers, analytically. The interaction of the Taylor column and the ends of the container leads to an increase in the drag over its value in an unbounded fluid, but the increase is smaller than that measured by Maxworthy (1970).