## Kinematic variational principle for motion of vortex rings

*Yasuhide Fukumoto*, Graduate School of Mathematics, Kyushu University

Vortex rings are prominent coherent structures in a diversity
of fluid flows. I will show how topological ideas work to derive a
theoretical upper bound on translation speed of a vortex
ring. According to Kelvin-Benjamin's principle, a steady distribution
of vorticity, relative to a moving frame, is realized as the state
that maximizes the total kinetic energy, under the constraint of
constant hydrodynamic impulse, with respect to variations preserving
the vorticity-field topology. Combined with an asymptotic solution of
the Euler equations for a family of vortex rings, we can skip the
detailed solution for the flow field to obtain the translation
velocity of a vortex ring valid to third order in a small parameter,
the ratio of the core radius to the ring radius. Including small
viscosity, Saffman's velocity formula of a viscous vortex ring is
extended to third order, which gives an improved upper bound on the
translation speed. Similarity of this principle is found with
Rasetti-Regge's theory for three-dimensional motion of a vortex
filament.