Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations

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

Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations

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

Department of Mathematical Sciences

University of British Columbia

Research output: Working paper / Preprint › Preprint

Abstract

A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid 3-dimensional fluid. Helmholtz (1858) 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
Publisher	arXiv
Number of pages	67
Publication status	Acceptance date - 8 Mar 2024

Bibliographical note

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

Keywords

Euler equation
leapfrogging

Access to Document

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

Concentration phenomena in nonlinear analysis
Musso, M. (PI)
Engineering and Physical Sciences Research Council
27/04/20 → 31/07/24
Project: Research council

Cite this

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title = "Leapfrogging vortex rings for the 3-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 3-dimensional fluid. Helmholtz (1858) 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 - Musso, Monica

AU - Del Pino, Manuel

AU - Wei, Juncheng

N1 - Submitted to ArXiv on 7/7/2022; updated on 11/1/2023 (v.3)

PY - 2024/3/8

Y1 - 2024/3/8

N2 - A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid 3-dimensional fluid. Helmholtz (1858) 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.

AB - A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid 3-dimensional fluid. Helmholtz (1858) 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.

KW - Euler equation

KW - leapfrogging

M3 - Preprint

BT - Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations

PB - arXiv

ER -

Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations

Abstract

Bibliographical note

Keywords

Access to Document

Fingerprint

Projects

Concentration phenomena in nonlinear analysis

Cite this