On the Head-on Collision of Coaxial Vortex Rings (Mathematical Analysis of Viscous Incompressible Fluid)

Abstract

We consider the head-on collision of two coaxial vortex rings, which have circulations of opposite sign, described as the motion of two coaxial circular vortex filaments under the localized induction approximation. We prove the existence of solutions to a system of nonlinear partial differential equations modelling the interaction of two vortex filaments proposed by the author [M. Aiki, On the existence of leapfrogging pair of circular vortex filaments, Stud. Appl. Math., 143 (2019), no.3, pp.213-243.] 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 magnitude of the circulations satisfying the following. 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. As far as the author knows, the existence of such threshold γ* is only indirectly suggested via numerical investigations of the head-on collision of coaxial vortex rings. Hence, our paper is the first to obtain the threshold in a way that is possible to numerically calculate γ*, as well as prove that the threshold exists in a framework of a mathematical model.