Free motion of a sphere in a rotating liquid parallel to the axis of rotation

Abstract

1. The motion is supposed to be a small disturbance from one of uniform rotation like a rigid body, small motion being defined to be such that the squares and products of velocity and vorticity components may be neglected in the expressions for acceleration. The system is supposed free from bodily forces, and is initially disturbed from relative rest by a motion suddenly communicated to the sphere. The pressure intensity of the liquid consists of two parts,one depending only on the distance from the axis, and the other on the disturbed motion. If at any instant the sphere is moving parallel to the axis of rotation, and the disturbed motion of the liquid is symmetrical with respect to a line through the centre of the sphere parallel to the axis of rotation, the motional part of the pressure intensity will also be symmetrical with respect to this line, and its resultant effect on the sphere will be to produce an acceleration parallel to the axis of rotation. Now the positional part of the pressure intensity would maintain any portion of the liquid in relative rest, and will consequently maintain the sphere in relative rest, providing the density of the sphere is equal to that of the liquid, which we shall suppose to be the case.