Eur. Phys. J. B 29, 167-171 (2002)
https://doi.org/10.1140/epjb/e2002-00279-5

Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy

Graduate School of Mathematics and Space Environment Research Center, Kyushu University 33, Fukuoka 812-8581, Japan

Corresponding author: ^a yasuhide@math.kyushu-u.ac.jp

Received: 2 October 2001
Revised: 10 May 2002
Published online: 2 October 2002

Abstract

Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated under the localized induction approximation for the Euler equations. Using the method of matched asymptotic expansions in a small parameter ϵ, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to , whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. It is found that there is an integrable line in the core whose evolution obeys a summation of the first and third terms of the localized induction hierarchy.