Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

Krishnamurthy, Vikas S.; Wheeler, Miles H.; Crowdy, Darren; Constantin, Adrian

doi:10.1017/jfm.2021.285

Cited by 10 publications

(4 citation statements)

References 57 publications

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“…The Kimura conjecture and other matters related to dipole motion has been discussed also in [37,38,41,30,32], and from slightly different points of view in [7,33,27,6]. As mentioned, the analysis in the present paper leads to a new confirmation of Kimura's conjecture, although in a rather weak form: expanding all quantities in Taylor series and keeping only the leading terms, the dynamical equations for a vortex pair reduce to the geodesic equation in the dipole limit.…”

Section: Introductionsupporting

confidence: 66%

Vortex pairs and dipoles on closed surfaces

Gustafsson

2022

Preprint

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We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account.Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S. Boatto and J. Koiller by using Gaussian geodesic coordinates. In the present paper we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable.In a final section we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces, and in two appendices we give some necessary background on affine and projective connections.

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Section: Introductionsupporting

confidence: 66%

Vortex pairs and dipoles on closed surfaces

Gustafsson

2022

Preprint

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“…See Theorem 8.1. The Kimura conjecture and other matters related to dipole motion have been discussed also in Smith (2011), Smith and Nagem (2013), Rodrigues et al (2018), Koiller (2020), Krishnamurthy et al (2021), and from slightly different points of view in Chorin (1973), Kulik et al (2010), Jacobs (2017), andCawte et al (2019). It may be remarked that Kimura's conjecture is counterintuitive.…”

Section: Introductionmentioning

confidence: 89%

Vortex Pairs and Dipoles on Closed Surfaces

Gustafsson

2022

J Nonlinear Sci

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We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S. Boatto and J. Koiller by using Gaussian geodesic coordinates. In the present paper, we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section, we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces.

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“…In this regard, there are many known special solutions. (See the above mentioned references and also recently [2,4,11,13].) In this paper, we adopt a different approach from the mentioned works above as follows: we do not specify f (u) in the beginning, but instead specify the form of solution which is related to the domain of flow or shape of level curves (i.e.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions of steady Euler equations in two dimension by functional separation

Kim

2024

J. Phys. A: Math. Theor.

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Exact solutions of two dimensional Euler equations for incompressible fluid flows arefound by the functional separation of variables. More specifically, two independent variablesare considered in the Cartesian coordinates, polar coordinates and also general orthogonalcurvilinear coordinates which include elliptic, parabolic and hyperbolic systems. Several newsolutions are obtained and other classical solutions are rediscovered.

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Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

Abstract: Abstract

Cited by 10 publications

References 57 publications

Vortex pairs and dipoles on closed surfaces

Vortex pairs and dipoles on closed surfaces

Vortex Pairs and Dipoles on Closed Surfaces

Exact solutions of steady Euler equations in two dimension by functional separation

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