2024
DOI: 10.1002/cpa.22199
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Leapfrogging vortex rings for the three‐dimensional incompressible Euler equations

Juan Dávila,
Manuel del Pino,
Monica Musso
et al.

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 con… Show more

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Cited by 2 publications
(1 citation statement)
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“…Moreover, beside the convergence is proved for short times only, it is possible to chose initial data for which the convergence to a periodic solution cover several periods. This provide a mathematical justification of the leapfrogging, which was first proved in [13] (but with different techniques and scaling). For α " 0 (see [2] for a vortex ring alone and [8,4] for several rings) the dynamics converges to simple translations parallel to the symmetry axis with constant speed, while the case 0 ă α ă 1 has not been studied explicitly, but we believe that the behavior is analogous to the case α " 0.…”
Section: Introductionmentioning
confidence: 92%
“…Moreover, beside the convergence is proved for short times only, it is possible to chose initial data for which the convergence to a periodic solution cover several periods. This provide a mathematical justification of the leapfrogging, which was first proved in [13] (but with different techniques and scaling). For α " 0 (see [2] for a vortex ring alone and [8,4] for several rings) the dynamics converges to simple translations parallel to the symmetry axis with constant speed, while the case 0 ă α ă 1 has not been studied explicitly, but we believe that the behavior is analogous to the case α " 0.…”
Section: Introductionmentioning
confidence: 92%