On the Motion of Vortices in Two Dimensions

Lin, Chen-Yun

doi:10.1073/pnas.27.12.575

Cited by 94 publications

(56 citation statements)

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“…According to [11], introducing the modified stream function on the Riemannian surface, called Kirchhoff-Routh path function as in the planar case by Lin [22,23],…”

Section: Formulation Of Vortex Dynamics On 2d Riemannian Manifoldsmentioning

confidence: 99%

Point vortex interactions on a toroidal surface

Sakajo¹,

Shimizu²

2016

Proc. R. Soc. A.

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Owing to non-constant curvature and a handle structure, it is not easy to imagine intuitively how flows with vortex structures evolve on a toroidal surface compared with those in a plane, on a sphere and a flat torus. In order to cultivate an insight into vortex interactions on this manifold, we derive the evolution equation for N-point vortices from Green's function associated with the LaplaceBeltrami operator there, and we then formulate it as a Hamiltonian dynamical system with the help of the symplectic geometry and the uniformization theorem. Based on this Hamiltonian formulation, we show that the 2-vortex problem is integrable. We also investigate the point vortex equilibria and the motion of two-point vortices with the strengths of the same magnitude as one of the fundamental vortex interactions. As a result, we find some characteristic interactions between point vortices on the torus. In particular, two identical point vortices can be locally repulsive under a certain circumstance.

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“…According to [11], introducing the modified stream function on the Riemannian surface, called Kirchhoff-Routh path function as in the planar case by Lin [22,23],…”

Section: Formulation Of Vortex Dynamics On 2d Riemannian Manifoldsmentioning

confidence: 99%

Point vortex interactions on a toroidal surface

Sakajo¹,

Shimizu²

2016

Proc. R. Soc. A.

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“…͑1͒ to bounded domains or to surfaces other than the plane ͑the sphere being the surface of particular interest͒, although we shall briefly consider vortices in periodic domains. The extension of the basic dynamical equations of point vortices to bounded domains was considered in two short papers by Lin,32,33 later published as a monograph. 34 This theory has recently been revisited by Crowdy and Marshall 20,21 who show that the Hamiltonian may be given explicitly for a multiconnected domain in terms of the Riemann mapping function of that domain onto a topologically equivalent domain with all boundaries being circles and the Schottky-Klein prime function associated with this set of circles.…”

Section: The Basic Dynamical Equationsmentioning

confidence: 99%

Point vortex dynamics: A classical mathematics playground

Aref¹

2007

Journal of Mathematical Physics

199

213

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The idealization of a two-dimensional, ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics. Several examples are given including the integrability of the three-vortex problem, the interplay of relative equilibria of identical vortices and the roots of certain polynomials, addition formulas for the cotangent and the Weierstraß function, projective geometry, and other topics. The hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community.

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“…In the Hamiltonian (123) we collect to the separated sum all terms describing a vortex interaction with its own images, then we get the Hamiltonian function in the form of the Kirchhoff-Routh function [31,32],…”

Section: The Kirchhoff-routh Functionmentioning

confidence: 99%

Vortex images andq-elementary functions

Pashaev¹,

Yılmaz²

2008

J. Phys. A: Math. Theor.

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In the present paper, the problem of vortex images in the annular domain between two coaxial cylinders is solved by the q-elementary functions. We show that all images are determined completely as poles of the q-logarithmic function, and are located at sites of the q-lattice, where a dimensionless parameter q = r 2 2 r 2 1 is given by the square ratio of the cylinder radii. The resulting solution for the complex potential is represented in terms of the Jackson q-exponential function. Our approach in this paper provides an efficient path to rediscover known solutions for the vortex-cylinder pair problem and yields new solutions as well. By composing pairs of q-exponents to the first Jacobi theta function and conformal mapping to a rectangular domain we show that our solution coincides with the known one, obtained before by elliptic functions. The Schottky-Klein prime function for the annular domain is factorized explicitly in terms of q-exponents. The Hamiltonian, the KirchhoffRouth and the Green functions are constructed. As a new application of our approach, the uniformly rotating exact N-vortex polygon solutions with the rotation frequency expressed in terms of q-logarithms at Nth roots of unity are found. In particular, we show that a single vortex orbits the cylinders with constant angular velocity, given as the q-harmonic series. Vortex images in two particular geometries with only one cylinder as the q → ∞ limit are studied in detail.

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On the Motion of Vortices in Two Dimensions

Cited by 94 publications

References 0 publications

Point vortex interactions on a toroidal surface

Point vortex interactions on a toroidal surface

Point vortex dynamics: A classical mathematics playground

Vortex images andq-elementary functions

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