A systematic treatment of moving axes in hydrodynamics

Abstract

A method is developed for transforming the equations of hydrodynamics to a system of curvilinear co-ordinates in motion relative to fixed axes using the tensor-calculus and without employing Coriolis’s theorem .The basic entity is the kinetic metric, a four-dimensional quadratic form defined through the kinetic energy of unit mass of fluid.The mechanics of special relativity are used to obtain, by approximation in terms of 1/ c 2 , where c is the velocity of light, the classical formulae.The equations of motion in terms of the energy-tensor, the four-dimensional vorticity-tensor and the˙ velocity-components are successively obtained and the equation of continuity is shown to be independent (in mathematical form) of the motion of the co-ordinate-system . This property holds also for the equation of heat-transfer in a non-viscous fluid. Applications are made to the case of local Cartesian and local cylindrical polar co-ordinates on the Earth ’s surface. Formulae for the rate of change of vorticity due to Helmholtz and to Petterssen, respectively, are obtained as special cases and Sawyer’s theory of tropical cyclones is also discussed.