Slow motion of a paraboloid of revolution in a rotating fluid

Abstract

The slow uniform motion, after an impulsive start from relative rest, of a paraboloid of revolution along the axis of a rotating fluid is investigated by using a perturbation method. The principal purpose of the note is to illustrate the mechanism by which the fluid is not subjected to any substantial radial displacement, which is a direct consequence of the requirement that the circulation round material circuits should be constant when the perturbation velocities due to the motion of the paraboloid remain small. It appears that the mechanism is an oscillatory one in which the distance between any fluid particle and the axis of rotation oscillates sinusoidally in time with small amplitude. As time progresses, the amplitude of the oscillation decays to zero everywhere except on the paraboloid. The ultimate motion is then a rigid body rotation everywhere except on the paraboloid and the axis of rotation, where the perturbation velocities continue to oscillate indefinitely with small amplitude.