Two bubble rings colliding

Not a math nerd, but...

I think it's, in part, Bernoulli's principle. Pressure drops as fluid moves faster. The reduced static pressure surrounding the rings brings them closer together.

But those rings are made of compressed air, and normally you see Bernoulli's principle applied in the context of incompressible fluids, like water. The air forming the rings is expanding in volume as it ascends on account of lower water pressure at shallower depths, just like how any underwater bubbles expand as they ascend. Since the air density inside the rings is decreasing with the growing ring volume, Navier-Stokes would be more accurate.

An actual math nerd could probably also produce the topological expressions to represent the ring merger.

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