On the Stability of the System of Thomson’s Vortex

L. G. Kurakin ; I. V. Ostrovskaya

Russian journal of nonlinear dynamics, Tome 18 (2022) no. 5, pp. 915-926

Résumé

The stability problem of a moving circular cylinder of radius $R$ and a system of n identical point vortices uniformly distributed on a circle of radius $R_0$, with $n \geqslant 2$, is considered. The center of the vortex polygon coincides with the center of the cylinder. The circulation around the cylinder is zero. There are three parameters in the problem: the number of point vortices n, the added mass of the cylinder a and the parameter $q = \frac{R^2}{R^2_0}$. The linearization matrix and the quadratic part of the Hamiltonian of the problem are studied. As a result, the parameter space of the problem is divided into the instability area and the area of linear stability where nonlinear analysis is required. In the case $n = 2, 3$ two domains of linear stability are found. In the case $n = 4, 5, 6$ there is just one domain. In the case $n \geqslant 7$ the studied solution is unstable for any value of the problem parameters. The obtained results in the limiting case as $a \rightarrow \infty$ agree with the known results for the model of point vortices outside the circular domain.

Détail

Keywords: point vortices, Hamiltonian equation, Thomson’s polygon, stability.