Asymptotic formula for velocity of a vortex ring and kinematic variational principle

A general formula is established for translation speed of an axisymmetric vortex ring whose core is not necessarily thin. We rely on Lamb-Saffman-Rott-Cantwell's method of calculating the total kinetic energy of fluid in two ways. Combined with the Navier-Stokes equations, we can skip the detailed solution for the flow field to extend Saffman's velocity formula of a viscous vortex ring to third order in the ratio of the core radius to the ring radius, a small parameter, for the entire range of the Reynolds number. At small Reynolds numbers, a solution that describes the whole life of a vortex ring is available. For inviscid motion, a further simplification is achieved by resorting to the variation, under the topological constraints, of the kinetic energy with respect to the hydrodynamic impulse. This principle bears similarity with the variational principle for a vortex ring governed by the Gross-Pitaevskii equation. Similarity is also found with Rasetti-Regge's theory for the three-dimensional motion of a vortex filament.