| Abstract: |
| We consider the motion of a pair of coaxial vortex rings immersed in an incompressible and inviscid fluid. The motion is formulated as the motion of a pair of coaxial circular vortex filaments. The motion of such vortex rings have been studied for a very long time, dating back to the work by Helmholtz in 1858, and many motion patterns are known to exist for interacting coaxial vortex rings. One such pattern that attracted the attention of many is leapfrogging. Leapfrogging was suggested to occur by Helmholtz, but it took until 1971 for leapfrogging motion to be experimentally confirmed in the work by Yamada and Matsui. Since then, more laboratory experiments and numerical experiments of leapfrogging motion has been done, but the mathematical analysis of the leapfrogging motion is scarce.
To this end, we propose a system of partial differential equations describing such motion and prove that solutions corresponding to leapfrogging exist. We further give necessary and sufficient conditions for leapfrogging to occur. |
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