### 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 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Physica D: Nonlinear Phenomena*,

*237*(14-17), 2210-2217. https://doi.org/10.1016/j.physd.2008.02.003

**Kinematic variational principle for motion of vortex rings.** / Fukumoto, Yasuhide; Moffatt, H. K.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 237, no. 14-17, pp. 2210-2217. https://doi.org/10.1016/j.physd.2008.02.003

}

TY - JOUR

T1 - Kinematic variational principle for motion of vortex rings

AU - Fukumoto, Yasuhide

AU - Moffatt, H. K.

PY - 2008/8/15

Y1 - 2008/8/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=47049101777&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=47049101777&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2008.02.003

DO - 10.1016/j.physd.2008.02.003

M3 - Article

AN - SCOPUS:47049101777

VL - 237

SP - 2210

EP - 2217

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 14-17

ER -