Motion of three vortices

Abstract

A qualitative analysis of the motion of three point vortices with arbitrary strengths is given. This simplifies and extends recent work by Novikov on the motion of three identical vortices. Using a phase diagram technique, the possible regimes of motion are classified according to the signs of the arithmetic, geometric, and harmonic means of the three vortex strengths. For the special case where the vortex strengths (κ₁,κ₂,κ₃) take the values (+κ,+κ,−κ), the diagram has an interpretation in terms of the scattering of a neutral pair by a single vortex. Quantitative details are presented for this case. If the harmonic mean of the three vortex strengths is zero, the triangle of vortices can collapse to a point in a finite time for certain initial conditions.