On the dynamics of point vortices in an annular region Nadezhda N Erdakova 1 and Ivan S Mamaev 1,2 1 Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia 2 Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia E-mail: enn@rcd.ru and mamaev@rcd.ru Received 1 July 2013, revised 14 April 2014 Accepted for publication 17 April 2014 Published 28 May 2014 Communicated by Y Fukumoto Abstract This paper reviews the results of stability analysis for polygonal configurations of a point vortex system in an annular region depending on the ratio of the inner and outer radii of the annulus. Conditions are found for linear stability of Thomsonʼs configurations for the case < N 7. The paper also shows that a system of two vortices between parallel walls is a limiting case of a two-vortex system in an annular region, as the radii of the annulus tend to infinity. (Some figures may appear in colour only in the online journal) 1. Introduction A large body of literature has been devoted to the study of the stability of polygonal vortex configurations, both in the case of a free plane, and inside and outside an annulus—see, e.g., (Thomson 1883, Havelock 1931, Kurakin 2010, 2012, Borisov and Mamaev 2005). The problem of the dynamics of two vortices in an annulus is addressed in Pashaev and Yilmaz (2011), Zueva (1996), Erdakova (2010) and Lakaniemi (2007). In Bolsinov et al (2010) and Vaskin and Erdakova (2010), a topological approach is used to search for relative equilibria and to analyze their stability. This approach has yielded new stationary configurations for the system of tree vortices in a circle and two vortices in an annulus. In this paper, we use a Hamiltonian representation for the vortex system in an annular region in terms of θ 1 -functions and explore the stability of polygonal configurations of the system of N vortices in an annulus depending on the ratio between the inner and outer radii of the annulus. Further, we show that the system of two vortices between parallel walls is a limiting case in an annular region as the radii of the annulus tend to infinity. 0169-5983/14/031420+07$33.00 © 2014 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK 1 | The Japan Society of Fluid Mechanics Fluid Dynamics Research Fluid Dyn. Res. 46 (2014) 031420 (7pp) doi:10.1088/0169-5983/46/3/031420