Kinematic variational principle for motion of vortex rings

Y. Fukumoto, H. K. Moffatt

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)


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 languageEnglish
Pages (from-to)2210-2217
Number of pages8
JournalPhysica D: Nonlinear Phenomena
Issue number14-17
Publication statusPublished - Aug 15 2008

All Science Journal Classification (ASJC) codes

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


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