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 in [22] 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 γ * > 1 for the ratio γ of the absolute value of the circulations such that when γ ∈ [1, γ * ], two approaching rings will collide, and when γ ∈ (γ * , ∞), the ring with the larger circulation passes through the other and then separate indefinitely.