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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.

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)
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