Vortex Rings: Existence and Asymptotic Estimates

Abstract

Abstract. The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter X; as A -» oo the vortex ring tends to a torus whose cross-section is an infinitesimal disc. More recently Benjamin [4] developed a new approach based on a variational principle for the vorticity. This approach is more natural since (i) the vorticity has compact support (whereas the stream function does not) and (ii) the quantities involved in the variational principle have direct physical significance.In this paper we establish the existence of vortex rings by a new method. As in [4] we formulate the problem in a variational form for the kinetic energy as a functional of the vorticity. We take the admissible functions to vary in the set S^ of functions f(x) satisfying: f (x) = f (r, z) = f (r, -z) where x = (r, 0, z),i.e., an axisymmetric flow with prescribed impulse (= 1), circulation (< 1) and vortex strength (< X); in [4] f is taken to vary over all rearrangements of a given function f0(r, z). Our approach seems technically simpler; it has the further advantage that it leads to vortex rings with, essentially, any given vorticity function, such as3) fit) = c(t + )p (c>0,B>0).

Planar vortex patch problem in incompressible steady flow

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An integral transform whose kernel is a stream function of a steady Euler flow with axisymmetry

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Existence of solutions for three dimensional stationary incompressible Euler equations with nonvanishing vorticity

“…There exist huge literatures dealing with the stationary incompressible Euler equations, such as exact solutions (see [19,30] and references therein), the existence of solutions (see [2, 3, 5-7, 11, 12, 14, 16, 20-25, 27, 31, 32] and references therein), symmetry of solutions (see [13] and references therein), stability of solutions (see [15,16] and references therein), topological properties of solutions (see [10]) and numerical approximations of solutions (see [8,9,28,35] and references therein). For proving the existence of solutions, there are various methods, such as the variational methods (see [2,3,5,12,14,20,31,32] and references therein), the statistical mechanics methods (see [6,7]), the pseudo-advection method (see [22,24,25]), the magnetohydrodynamic approach (see [21,23]), the fixed points method (see [1]) and some other methods in [29,34]. Most of them can only be used to the two-dimensional or the axisymmetric cases, except for [1,4,23,36].…”