Vortex Rings

A vortex ring (or toroidal vortex), is a donut-shaped region of circulation that moves through the fluid medium. A vortex ring tends to move in the direction perpendicular to the plane of the ring, possibly extending out a considerable distance. The movement of the fluid can be classified as poloidal (related to the short route around a torus), across the circular axis of the donut, in such a way that the inner part of the ring moves faster toward the front side.
Vortex rings are produced by dolphins, humpback and beluga whales, volcanoes, atomic/hydrogen bombs, fire eaters, and obviously, by smokers. Vortex rings were first analyzed mathematically by H. von Helmholtz in 1858 [1].


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The relevant equations are enumerated in the Initialization Code.
[1] H. von Helmholtz, "On Integrals of the Hydrodynamic Equations that Correspond to Vortex Motions," Journal für die reine und angewandte Mathematik, 55, 1858 pp. 22–55.
[2] Extraordinary Toroidal Vortices. [Video] (Feb 12, 2010) www.youtube.com/watch?v=mHyTOcfF99o.
[3] Wikipedia. "Vortex Ring." (Nov 14, 2012) en.wikipedia.org/wiki/Vortex_ring.
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