Relative equilibrium and collapse configurations of heterogeneous vortex triple rings

O’Neil, Kevin A.

doi:10.1016/j.physd.2007.07.015

Cited by 6 publications

(1 citation statement)

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“…The more general problem with three polygons was also solved in the particular case where the vorticities of the three polygons are equal (Aref and van Buren, 2005). We also know that, for three polygons, the number of relative equilibria corresponding to a generic set of vorticities (identical on a same polygon) is finite (O'Neil, 2007). The purpose of this section is to provide an exhaustive classification of all the relative equilibria formed of two polygons, which will be valid for every Γ 1 and Γ 2 .…”

Section: Two Polygons With the Same Vorticity On Each Polygonmentioning

confidence: 99%

On polygonal relative equilibria in the N-vortex problem

Celli¹,

Lacomba²,

Pérez‐Chavela³

2011

Journal of Mathematical Physics

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Helmholtz's equations provide the motion of a system of N vortices which describes a planar incompressible fluid with zero viscosity. A relative equilibrium is a particular solution of these equations for which the distances between the vortices are invariant during the motion. In this article, we are interested in relative equilibria formed of concentric regular polygons of vortices. We show that in the case of one regular polygon (and a possible vortex at the center) with more than three vertices (two if there is a vortex at the center), a relative equilibrium requires equal vorticities (on the polygon). We also determine all the relative equilibria with two concentric regular n-gons and the same vorticity on each n-gon. This result completes the classical studies for two regular n-gons when all the vortices have the same vorticity or when the total vorticity vanishes.

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Section: Two Polygons With the Same Vorticity On Each Polygonmentioning

confidence: 99%