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Fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He

Published online by Cambridge University Press:  24 July 2012

Robert A. Van Gorder
Robert A. Van Gorder*
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
*
†Email address for correspondence: rav@knights.ucf.edu
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Abstract

We obtain the fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He. As the relevant friction parameters are small, we linearize terms involving such parameters, while keeping the remaining nonlinearities, which accurately describe the curvature of the vortex filament, intact. The resulting equation is a type of nonlinear Schrödinger equation, and, under an appropriate change of variables, this equation is shown to have a first integral. This is in direct analogy with the simpler equation studied previously in the literature; indeed, in the limit where the superfluid parameters are taken to zero, we recover the results of Van Gorder. While this first integral is mathematically interesting, it is not particularly useful for computing solutions to the nonlinear partial differential equation which governs the vortex filament. As such, we introduce a new change of dependent variable, which results in a nonlinear four-dimensional system that can be numerically integrated. Integrating this system, we recover solutions to the fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He. We find that the qualitative features of the solutions depend not only on the superfluid friction parameters, but also strongly on the initial conditions taken, the curvature and the normal fluid velocity.

JFM classification

Mathematical Foundations: Computational methods Vortex Flows: Vortex dynamics Complex Fluids: Quantum fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 707 , 25 September 2012 , pp. 585 - 594
Copyright
Copyright © Cambridge University Press 2012

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