Falling motion of a circular cylinder interacting dynamically with a vortex pair in a perfect fluid

We consider a system which consists of a circular cylinder subject to gravity interacting with $N$ vortices in a perfect fluid. Generically, the circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit evident integrals of motion: the horizontal and vertical components of the momentum; the latter is obviously non-autonomous. We then focus on the study of a configuration of the Föppl type: a falling cylinder is accompanied with a vortex pair ($N=2$). Now the circulation about the cylinder is assumed to be zero and the governing equations are considered on a certain invariant manifold. It is shown that, unlike the Föppl configuration, the vortices cannot be in a relative equilibrium. A restricted problem is considered: the cylinder is assumed to be sufficiently massive and thus its falling motion is not affected by the vortices. Both restricted and general problems are studied numerically and remarkable qualitative similarity between the problems is outlined: in most cases, the vortex pair and the cylinder are shown to exhibit scattering.