Two counter-rotating vortices of circulation G and separation d translate together at U = G d / (2 pi (d^2 + a^2)). The check integrates the mutually induced motion with RK2 and compares displacement/time with this formula.

A smoke or vapor ring is a vortex ring; slicing it through its axis shows two cores of opposite circulation. Each induces on the other a velocity G/(2 pi d) (Biot-Savart in 2D), so the pair self-propels. A regularised Lamb-Oseen-type core of radius a replaces the singular point vortex: u_theta = G r / (2 pi (r^2 + a^2)).

For a true 3D ring of radius R the self-induced speed is the Kelvin-Saffman result U = G/(4 pi R) [ln(8R/a) - 1/4], shown as a readout using R = d/2; the 2D pair formula is the thin-section limit. Viscosity spreads the core, a^2 = a0^2 + 4 nu t, slowing the ring.

Simplifications: no vortex stretching, no azimuthal waves (Widnall instability) and no entrainment of surrounding air; the 3D formula is quoted, not simulated.