# How can these fluid dynamical smoke-ring phenomena be explained?

The Navier-Stokes fluid dynamics equations, said that, as Sir William Thomson (or Lord Kelvin) predicted:

1. When two smoke-rings are moving in the same direction, with the same speed, one behind the other, the 'leading' ring will slow down and enlarge, while the 'following' one will get smaller and speed up, passing through the ring in front, and this will keep on happening until they fade away. It was experimentally proven by Kelvin.

2. When two smoke-rings move towards each other, rather than colliding and annihilating in a smokey mess, they actually slow down and enlarge, never meeting, just getting larger until they fade away. It was experimentally proven by Kelvin.

My questions:

1. How does phenomena number 1 happen?
2. How does phenomena number 2 happen?
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Smoke rings are very resilient things, because they are topologically protected from disappearing: small disturbances don't usually break them because they'd have to fundamentally alter the flow's topology. I suspect this is behind, at least, the second phenomenon you ask about. – Emilio Pisanty Dec 10 '12 at 23:55

Vortex Ring Interactions (by Wakelin-Riley)
http://qjmam.oxfordjournals.org/content/49/2/287.abstract

Here is my interpretation: Consider two rings moving co-axially in the positive x-direction with their rings in the y-planes (ring A in front of ring B). The particles in each circle of the (fattened) ring circulate and force spreading of the air in front of it (see http://en.wikipedia.org/wiki/File:Vortex_ring.gif). Then
1) A's circulations will force B to spread and enlarge (visually look at the vectors in the vicinity of A). Then A moves through and this repeats, because B's circulation will now spread A.
2) Here ring A is moving in positive x-direction and B in negative x-direction. Their respective circulations will stretch each other and slow them down until they meet at rest and fade.

Here are more direct papers:
A Note on the Leapfroggeing Between Coaxial Vortex Rings at Low Reynolds Numbers (by Lim) http://pof.aip.org/resource/1/phfle6/v9/i1/p239_s1
Interaction of Two Vortex Rings Moving along a Common Axis of Symmetry (by Oshima)