On the Stability of the System of Thomson’s Vortex $n$-Gon and a Moving Circular Cylinder

    Received 19 August 2022; accepted 09 November 2022; published 27 December 2022

    2022, Vol. 18, no. 5, pp.  915-926

    Author(s): Kurakin L. G., Ostrovskaya I. V.

    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 $q=\frac{R^2}{R_0^2}$.
    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 $a\to\infty$ agree with the known results for the model of point vortices outside the circular domain.
    Keywords: point vortices, Hamiltonian equation, Thomson’s polygon, stability
    Citation: Kurakin L. G., Ostrovskaya I. V., On the Stability of the System of Thomson’s Vortex $n$-Gon and a Moving Circular Cylinder, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 5, pp.  915-926

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