Leapfrogging vortex rings for the three-dimensional incompressible Euler equations

Juan Davila Bonczos; Monica Musso; Manuel Del Pino; Juncheng Wei

doi:10.1002/cpa.22199

Leapfrogging vortex rings for the three-dimensional incompressible Euler equations

Juan Davila Bonczos, Monica Musso, Manuel Del Pino, Juncheng Wei

Department of Mathematical Sciences

University of British Columbia

Research output: Contribution to journal › Article › peer-review

3 Citations (SciVal)

Abstract

A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid three-dimensional fluid. In 1858, Helmholtz observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. We provide a mathematical justification for this phenomenon by constructing a smooth solution of the 3D Euler equations exhibiting this motion pattern.

Original language	English
Pages (from-to)	3843-3957
Number of pages	115
Journal	Communications on Pure and Applied Mathematics
Volume	77
Issue number	10
Early online date	6 May 2024
DOIs	https://doi.org/10.1002/cpa.22199
Publication status	Published - 31 Oct 2024

Bibliographical note

Submitted to ArXiv on 7/7/2022; updated on 11/1/2023 (v.3)

Funding

J. D\u00E1vila has been supported by a Royal Society Wolfson Fellowship, UK and Fondecyt Grant 1170224, Chile. M. del Pino has been supported by a Royal Society Research Professorship, UK. M. Musso has been supported by EPSRC research Grant EP/T008458/1.

Funders	Funder number
Royal Society
Fondo Nacional de Desarrollo Científico y Tecnológico	1170224
Fondo Nacional de Desarrollo Científico y Tecnológico
Engineering and Physical Sciences Research Council	EP/T008458/1
Engineering and Physical Sciences Research Council

Keywords

Euler equation
leapfrogging

Access to Document

10.1002/cpa.22199Licence: CC BY

https://arxiv.org/abs/2207.03263Licence: CC0

Cite this

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title = "Leapfrogging vortex rings for the three-dimensional incompressible Euler equations",

abstract = "A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid three-dimensional fluid. In 1858, Helmholtz observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. We provide a mathematical justification for this phenomenon by constructing a smooth solution of the 3D Euler equations exhibiting this motion pattern.",

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AU - Del Pino, Manuel

AU - Wei, Juncheng

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N2 - A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid three-dimensional fluid. In 1858, Helmholtz observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. We provide a mathematical justification for this phenomenon by constructing a smooth solution of the 3D Euler equations exhibiting this motion pattern.

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Leapfrogging vortex rings for the three-dimensional incompressible Euler equations

Abstract

Bibliographical note

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Keywords

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Concentration phenomena in nonlinear analysis

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Leapfrogging vortex rings for the three-dimensional incompressible Euler equations

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

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Concentration phenomena in nonlinear analysis

Cite this