Vortex rings with swirl: Axisymmetric solutions of the Euler equations with nonzero helicity. (English) Zbl 0667.76038

Turkington, Bruce

SIAM J. Math. Anal. 20, No. 2, 57-73 (1989).

This work introduces a new class of steady solutions of the axisymmetric Euler equations for an incompressible inviscid fluid. Each solution represents a three-dimensional vortex flow whose azimuthal components of vorticity and velocity are nonzero inside a toroidal region determined by the solution. The governing free-boundary problem is solved by variational techniques. The underlying variational principle is formulated from the natural invariants associated with the evolution equations for axisymmetric flows, and involves a family of invariants that generalizes the standard angular impulse and helicity integrals. A direct method is employed to prove the existence of steady solutions in a bounded domain and steadily translating solutions in space. Qualitative properties of these vortices are discussed and concentrated vortex rings with large swirl are shown to constitute a desingularization of the classical circular vortex filament.

MSC:

76B47	Vortex flows for incompressible inviscid fluids
49S05	Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)

Keywords:

steady solutions; axisymmetric Euler equations; incompressible inviscid fluid; three-dimensional vortex flow; free-boundary problem; variational techniques; existence of steady solutions; desingularization of the classical circular vortex filament