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.

Bibliographie

[1] Thomson, W., Floating Magnets, “Illustrating Vortex Systems”, Nature, 18 (1878), 13–14

[2] Thomson, J. J., Treatise on the Motion of Vortex Rings, Macmillan, London, 1883, 156 pp.

[3] Havelock, T. H., “The Stability of Motion of Rectilinear Vortices in Ring Formation”, Philos. Mag. (7), 11:70 (1931), 617–633

[4] Kurakin, L. G. and Yudovich, V. I., “The Stability of Stationary Rotation of a Regular Vortex Polygon”, Chaos, 12:3 (2002), 574–595

[5] Borisov, A. V. and Mamaev, I. S., “The Kelvin Problem and Its Solutions”, Mathematical Methods in the Dynamics of Vortex Structures, R Dynamics, Institute of Computer Science, Izhevsk, 2005, 261–265 (Russian)

[6] Kilin, A. A., Borisov, A. V., and Mamaev, I. S., “The Dynamics of Point Vortices Inside and Outside a Circular Domain”, Basic and Applied Problems of the Theory of Vortices, eds. A. V. Borisov, I. S. Mamaev, M. A. Sokolovskiy, R Dynamics, Institute of Computer Science, Izhevsk, 2003, 414–440 (Russian)

[7] Kurakin, L. G., Melekhov, A. P., and Ostrovskaya, I. V., “A Survey of the Stability Criteria of Thomson's Vortex Polygons outside a Circular Domain”, Bol. Soc. Mat. Mex., 22:2 (2016), 733–744

[8] Kurakin, L. and Ostrovskaya, I., “On the Effects of Circulation around a Circle on the Stability of a Thomson Vortex $N$-Gon”, Mathematics, 8 (2020), 1033, 19 pp.

[9] Ramodanov, S. M., “Motion of a Circular Cylinder and $N$ Point Vortices in a Perfect Fluid”, Regul. Chaotic Dyn., 7:3 (2002), 291–298

[10] Shashikanth, B. N., Marsden, J. E., Burdick, J. W., and Kelly, S. D., “The Hamiltonian Structure of a $2$D Rigid Circular Cylinder Interacting Dynamically with $N$ Point Vortices”, Phys. Fluids, 14 (2002), 1214–1227

[11] Borisov, A. V., Mamaev, I. S., and Ramodanov, S. M., “Motion of a Circular Cylinder and $n$ Point Vortices in a Perfect Fluid”, Regul. Chaotic Dyn., 8:4 (2003), 449–462

[12] Borisov, A. V. and Mamaev, I. S., “An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid”, Regul. Chaotic Dyn., 8:2 (2003), 163–166

[13] Borisov, A. V., Mamaev, I. S., and Ramodanov, S. M.?????????????, “Dynamics of a Cylinder Interacting with Point Vortices, in Borisov, A. V. and Mamaev, I. S.”, Mathematical Methods in the Dynamics of Vortex Structures, R Dynamics, Institute of Computer Science, Izhevsk, 2005, 286–307 (Russian)

[14] Mamaev, I. S. and Bizyaev, I. A., “Dynamics of an Unbalanced Circular Foil and Point Vortices in an Ideal Fluid”, Phys. Fluids, 33:8 (2021), 087119, 18 pp.

[15] Tr. Mat. Inst. Steklova, 310 (2020), 33–39 (Russian)

[16] Ramodanov, S. M. and Sokolov, S. V., “Dynamics of a Circular Cylinder and Two Point Vortices in a Perfect Fluid”, Regular Chaotic Dyn., 26:6 (2021), 675–691