A NEW SOLUTION OF EULER'S EQUATION OF MOTION WITH HELICITY

Summary On the basis of the principle of least action, a new representation of solution is derived for the velocity field of rotational flows of a compressible ideal fluid. The velocity field is represented in general by scalar potentials and vector potentials of frozen field, i.e. the latter potentials is convected with the fluid flow under effect of stretching. It is verified that the system of new expressions in fact satisfies the Euler’s equation of motion. The Lagrangian for the action consists of main terms of total kinetic energy and internal energy (with negative sign), together with two terms yielding the equations of continuity and entropy and the third term which yields a new rotational component of velocity field. This solution gives an explicit expression of non-vanishing helicity, and improves the classical Clebsch-type solution in the sense that the Clebsch potentials are simply convected by the flow (without stretching effect).