Head-On Collision of a Pair of Coaxial Circular Vortex Filament

Abstract

We consider the head-on collision of two coaxial vortex rings described as the motion of two circular vortex filaments under the localized induction approximation. We prove the existence of solutions to a system of nonlinear partial differential equations proposed by the author which exhibit head--on collision. We also give a necessary and sufficient condition for the initial configuration and parameters of the filaments for head-on collision to occur. Our results suggest that there exists a critical value ${\gamma }_{\ast }>1$ for the ratio $\gamma$ of the absolute value of the circulations such that when $\gamma \in [1,{\gamma }_{\ast }]$, two approaching rings will collide, and when $\gamma \in ({\gamma }_{\ast },\infty )$, the ring with the larger circulation passes through the other and then separate indefinitely.

18 pages, 0 figures