Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid

Abstract

The motion of a bounded shallow liquid, initially of arbitrary shape, in an arbitrary state of motion and lying on a paraboloid of revolution (including a level surface as a special case) can always be separated into three parts: the motion of the centre of gravity, which is entirely independent of the other motions and is governed by a pair of simple ordinary linear differential equations;an isotropic two-dimensional dilatation and rotation which are also governed by a simple linear differential equation;the motions that remain after removal of the velocity fields associated with the preceding motions; these will be called additional motions.The additional motions exert a ‘pressure’, determined by their total energy, which tends to increase the spread of the liquid. If the spread does increase then the additional motions lose energy which then appears as energy associated with the dilatation.The effect of the dilatation and rotation on the additional motions can be described by transformation into a co-ordinate system that rotates and dilates with the liquid. In these co-ordinates, with a properly adjusted time scale, the additional motions satisfy equations that are isomorphic with the original equations of motion; however, the liquid now appears to be lying on a parabola that is always concave upwards.