A global theory of steady vortex rings in an ideal fluid

Abstract

The question of whether the equations governing the motion of an inviscid, incompressible fluid admit solutions representing steady vortex rings has not been studied widely, despite the central place of such rings in the theory of vortex motion initiated by Helmholtz [1] By a steady vortex ring we mean a figure of revolution si that is expected to be homeomorphic to a solid torus in most cases, and is associated with a continuous, axi-symmetric, solenoidal vector field q (the fluid velocity) defined in a cylinder V or in the whole of R 3 , and having the following properties when we take axes fixed in the ring si. (a) Both si and q do not vary with time; (b) the vorticity <o = curl g has positive magnitude in si, vanishes in V -si or R 3 -si, and satisfies a nonlinear equation of motion which, among other things, determines the boundary of st\ (c) q has a prescribed normal component on dV, or tends to a constant value at infinity in R 3 . For the case of a cylinder F these requirements lead to equations (1) and (2) below, and we solve the resulting problem by means of the calculus of variations in the large [7], [8]. We also use Steiner symmetrization [9], the generalized maximum principle [10] and certain a priori estimates to describe the solution and to deal with two limiting cases : (i) that when the nonlinear term in the differential equation (2a) is discontinuous, and (ii) that when the domain of q is the whole of R 3 (so that the usual compactness theorems, needed for the solution of variational problems, do not hold).

Vortex Rings: Existence and Asymptotic Estimates

“…Remark 3. The approach of Fraenkel and Berger [13] is to pose a variational problem for the function u defined in (1.17); the propagation speed W, the flux constant y, and the vorticity function /(f) (suitably normalized) are prescribed in this approach. The method is more complicated technically than a variational principle based on f mainly due to the fact that u does not have compact support.…”

“…Using an analysis based on (1.18), Caffarelli and Friedman [9] have proved that the number of components of Q is finite, and that the free boundary 3fl (as given by z = Z(r)) is real analytic. Benjamin [13] asserts that there is just one component, arguing that a positive second variation for E(£) is obtained for (infinitesimal) relative displacements of two distinct components.…”

“…A general existence theorem for vortex rings was first established by Fraenkel and Berger [13] (see also the very recent work [5], [20] with a similar approach); this paper also contains an excellent survey of the subject. The approach in [13] is based on a variational principle for the stream function.…”

Nested axi-symmetric vortex rings

“…They are nested in the sense that two consecutive rings Ri and are such that Ri+l C co(Ri) (the convex hull of Ri) but ~ Ri = 0. In a referential frame attached to the vortices, the equations of motion are given in cylindrical coordinates (r, 8, z) on the domain n == {(r, 6~) : r > 0, -7r ~ 71-, -oo ~ +00} by where L1/J = + 1/Jzz (see [18,21,7,8]). The so called Stokes stream function ~ is in C1 (fi) n 788 B. BUFFONI finite union of non-degenerate level sets of In cylindrical coordinates, the velocity field q is then simply given by and the amplitude of the vorticity by curl q = As the 03B8 component does not appear in Equation (1), the vortices described in this way are without swirl [20].…”

“…The choice of the functional space is crucial. We can mention C1 spaces [18] and W1,2 Sobolev spaces, preferred in the variational approach [21,2]. A comment on the respective advantages of these spaces may be found in [4].…”