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Nelin. Dinam., 2009, Volume 5, Number 3, Pages 295–317 (Mi nd95)  

This article is cited in 9 scientific papers (total in 9 papers)

The stability of Thomson's configurations of vortices in a circular domain

L. G. Kurakinab

a Southern Federal University, Faculty of Mathematics, Mechanics and Computer Sciences
b South Mathematical Institute of VSC RAS

Abstract: The paper is devoted to stability of the stationary rotation of a system of $n$ equal point vortices located at vertices of a regular $n$-gon of radius $R_0$ inside a circular domain of radius $R$. T. H. Havelock stated (1931) that the corresponding linearized system has an exponentially growing solution for $n\ge 7$, and in the case $2\le n \le 6$ — only if parameter $p={R_0^2}/{R^2}$ is greater than a certain critical value: $p_{*n}<p<1$. In the present paper the problem on stability is studied in exact nonlinear formulation for all other cases $0<p\le p_{*n},$ $n=2,…,6$. We formulate the necessary and sufficient conditions for $n\neq 5 $. We give full proof only for the case of three vortices. A part of stability conditions is substantiated by the fact that the relative Hamiltonian of the system attains a minimum on the trajectory of a stationary motion of the vortex $n$-gon. The case when its sign is alternating, arising for $n=3$, did require a special study. This has been analyzed by the KAM theory methods. Besides, here are listed and investigated all resonances encountered up to forth order. It turned out that one of them lead to instability.

Keywords: point vortices; stationary motion; stability; resonance

Full text: PDF file (535 kB)

Document Type: Article
UDC: 532.517
MSC: 76B47, 34D20, 70K30
Received: 11.03.2009

Citation: L. G. Kurakin, “The stability of Thomson's configurations of vortices in a circular domain”, Nelin. Dinam., 5:3 (2009), 295–317

Citation in format AMSBIB
\Bibitem{Kur09}
\by L.~G.~Kurakin
\paper The stability of Thomson's configurations of vortices in a circular domain
\jour Nelin. Dinam.
\yr 2009
\vol 5
\issue 3
\pages 295--317
\mathnet{http://mi.mathnet.ru/nd95}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. L. G. Kurakin, I. V. Ostrovskaya, “Stability of the Thomson vortex polygon with evenly many vortices outside a circular domain”, Siberian Math. J., 51:3 (2010), 463–474  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. A. V. Vaskina, “Novye statsionarnye konfiguratsii v sisteme trekh tochechnykh vikhrei v krugovoi oblasti i ikh ustoichivost”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 4, 61–70  mathnet  elib
    3. A. V. Borisov, I. S. Mamaev, A. V. Vaskina, “Novye otnositelnye ravnovesiya v sisteme trekh tochechnykh vikhrei v krugovoi oblasti i ikh ustoichivost”, Nelineinaya dinam., 7:1 (2011), 119–138  mathnet  elib
    4. L. G. Kurakin, “Ob ustoichivosti tomsonovskogo vikhrevogo pyatiugolnika vnutri kruga”, Nelineinaya dinam., 7:3 (2011), 465–488  mathnet
    5. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Bifurkatsionnyi analiz i indeks Konli v mekhanike”, Nelineinaya dinam., 7:3 (2011), 649–681  mathnet
    6. L. G. Kurakin, I. V. Ostrovskaya, “Kriterii ustoichivosti pravilnogo vikhrevogo pyatiugolnika vne kruga”, Nelineinaya dinam., 8:2 (2012), 355–368  mathnet
    7. Kurakin L.G. Ostrovskaya I.V., “Nonlinear Stability Analysis of a Regular Vortex Pentagon Outside a Circle”, Regul. Chaotic Dyn., 17:5 (2012), 385–396  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Kurakin L.G., “On the Stability of Thomson's Vortex Pentagon Inside a Circular Domain”, Regul. Chaotic Dyn., 17:2 (2012), 150–169  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. A. Yu. Pogosov, O. V. Derevyanko, “Teoreticheskie osnovy inzhenernogo konstruirovaniya busternykh friktsionno-vikhrevykh turbin privodov nasosov avariinoi podpitki dlya AES”, Energosberezhenie. Energetika. Energoaudit, 2014, no. 10 (129), 18–26  elib
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