## Abstract

We show how the ideas of topology and variational principle, opened up by Euler, facilitate the calculation of motion of vortex rings. Kelvin-Benjamin's principle, as generalised to three dimensions, states that a steady distribution of vorticity, relative to a moving frame, is the state that maximizes the total kinetic energy, under the constraint of constant hydrodynamic impulse, on an iso-vortical sheet. By adapting this principle, combined with an asymptotic solution of the Euler equations, we make an extension of Fraenkel-Saffman's formula for the translation velocity of an axisymmetric vortex ring to third order in a small parameter, the ratio of the core radius to the ring radius. Saffman's formula for a viscous vortex ring is also extended to third order.

Original language | English |
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Pages (from-to) | 2210-2217 |

Number of pages | 8 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 237 |

Issue number | 14-17 |

DOIs | |

Publication status | Published - Aug 15 2008 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics