Chaotic scattering of the point vortex by falling circular cylinder

S. V. Sokolov ; I. S. Koltsov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 184-196

Résumé

We consider a system which consists of a circular cylinder subject to gravity interacting with a point vortex in a perfect fluid. In contrast to previous works, in this paper the circulation about the cylinder is assumed to be 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. Using autonomous integral we reduce the order of the system by one degree of freedom in a case of zero circulation which early was not considered. Unlike nonzero circulation in the absence of point vortices when the cylinder moves inside a certain horizontal stripe it is shown that in the presence of vortices and with circulation equal to zero a vertical coordinate of the cylinder is unbounded decreasing. We then focus on the numerical study of dynamics of our system. In a case of zero circulation trajectories are noncompact. The different kinds of the scattering function of the vortex by cylinder were obtained. The form of these functions argues to chaotic behavior of the scattering which means that an additional analytical integral is absent.

Détail

Keywords: point vortices, rigid body, chaotic scattering, Hamiltonian systems, reduction.