On the Theoretical Studies about the Vortex Motion of Perfect Fluid I

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The investigation about the theory of vortex motion with vorticity continuously distributed in the perfect fluid, was mainly carried out about 90 years ago, by Clebsch and Helmholtz etc., and soon after Kelvin established his famous circulation theorem. So it is considered that in these periods of 19th century the theoretical study of the vortex motion reached the vertex of its studies. In the following periods it was even said that there were few contributions about theoretical studies of vortex motion except Bjerknes' circulation theorem of baroclinic fluid. Besides the only useful method about analytical expression of vortex motion in incompressible fluid, is Stokes' stream function in the cases of plane motion and axial symmetric one. Even in these cases soluble ones are constrained to linear or some special types of partial differential equations. Many people have considered that the theory of vortex motion in perfect fluid reaches the limit of its analytical method. Causes of the above-mentioned incorrect considerations may consist in the following circumstances. (i) Vector potential method introduced by Helmholtz has been prevailed in the theoretical view on account of his fame. But the application of this method to vortex motion is incorrect as shown below. (ii) In the treatment about the vortex motion of perfect fluid by the method of his expression, Clebsch commited a gross mistake, as pointed out in 5 of this paper. This fact prevented the theoretical development of vortex motion by Clebsch's expression. It is one of the objects of this paper to rectify the above mistake. (iii) To solve generally the differential equations for vortex motion is very difficult, because they have non-linear form. (iv) The interest of hydrodynamical researches has been directed recently to the turbulent problems on hydraurics and aerodynamics and the high speed problems on aerodynamics. However in the domain of meteorology the effects of rotation of the earth to motion of the atmosphere are essential, by which effects its motion may be transformed necessarily into vortex motion. So to solve the fundamental problem of meteorological dynamics, in the first place, we must treat the problem how to solve the vortex motion of perfect fluid from the pure theoretical point of view, especially the problem of solution of non-linear type from the analytical point of view (cf. 7), While the author consulted the classical literature about vortex motion such as Helmholtz and Clebsch, he has found the hope to solve and develop these problems by amending their errors in the use of Clebsch's expression and applying a method of contact transformation. In this paper, differing from the usual treatments of vortex motion, by employing exclusively the method of Clebsch's expression for velocity we shall develop the theory of fluid motion with spread vorticity. In Part I its general theory is developed. In 1-3 a preliminary note for further discussions is delivered. In 4-5 the extension of Kelvin's circulation theorem and its relation with Bernoulli's equation are also discussed. By using Clebsch's in 6-7 the partial differential equation for stream function is transformed from the second order to the first order, but the general solutions of the latter will be discussed in the next paper. Considering our following discussions this Clebsch's expression seems to be the most powerful method to accomplish the theoretical study of vortex motion either in perfect fluid or in viscous fluid.
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On the Theoretical Studies about the Vortex Motion of Perfect Fluid I

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