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Absence of singular stretching of interacting vortex filaments

Published online by Cambridge University Press:  10 August 2012

Sahand Hormoz  and
Michael P. Brenner
Sahand Hormoz*
Affiliation:
School of Engineering and Applied Sciences, and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138, USA
Michael P. Brenner
Affiliation:
School of Engineering and Applied Sciences, and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138, USA
*
†Email address for correspondence: [email protected]
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Abstract

A promising mechanism for generating a finite-time singularity in the incompressible Euler equations is the stretching of vortex filaments. Here, we argue that interacting vortex filaments cannot generate a singularity by analysing the asymptotic dynamics of their collapse. We use the separation of the dynamics of the filament shape, from that of its core, to derive constraints that must be satisfied for a singular solution to remain self-consistent uniformly in time. Our only assumption is that the length scales characterizing filament shape obey scaling laws set by the dimension of circulation as the singularity is approached. The core radius necessarily evolves on a different length scale. We show that a self-similar ansatz for the filament shapes cannot induce singular stretching, due to the logarithmic prefactor in the self-interaction term for the filaments. More generally, there is an antagonistic relationship between the stretching rate of the filaments and the requirement that the radius of curvature of filament shape obeys the dimensional scaling laws. This suggests that it is unlikely that solutions in which the core radii vanish sufficiently fast to maintain the filament approximation exist.

JFM classification

Mathematical Foundations: Mathematical Foundations Vortex Flows: Vortex dynamics Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 707 , 25 September 2012 , pp. 191 - 204
Copyright
Copyright © Cambridge University Press 2012

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