Vortex-rings of prescribed impulse

Reference:

Burton, G. R., 2003. Vortex-rings of prescribed impulse. Mathematical Proceedings of the Cambridge Philosophical Society, 134, pp. 515-528.

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Abstract

We consider a variational problem for steady axisymmetric vortex-rings, in which kinetic energy is maximised subject to prescribed impulse, with the ratio of vorticity to axial distance belonging to the class of rearrangements of a prescribed function, as proposed by Benjamin. We prove existence of maximisers in an extended constraint set, allowing some loss of vorticity. We then study a particular family of vortex-rings including Hill's spherical vortex, determining the precise range of impulse values for which the maximiser loses vorticity, and show that the maximisers are spherical when this happens.

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