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

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)
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