On the Stability of a System of Two Identical Point Vortices and a Cylinder
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the Stability of a System of Two Identical Point Vortices and a Cylinder A. V. Borisov a and L. G. Kurakin b,c,d Received March 2, 2020; revised March 2, 2020; accepted April 27, 2020
Abstract—We consider the stability problem for a system of two identical point vortices and a circular cylinder located between them. The circulation around the cylinder is zero. There are two parameters in the problem: the added mass a of the cylinder and q = R2 /R02 , where R is the radius of the cylinder and 2R0 is the distance between vortices. We study the linearization matrix and the quadratic part of the Hamiltonian of the problem, find conditions of orbital stability and instability in nonlinear statement, and point out parameter domains in which linear stability holds and nonlinear analysis is required. The results for a → ∞ are in agreement with the classical results for a fixed cylinder. We show that the mobility of the cylinder leads to the expansion of the stability region. DOI: 10.1134/S008154382005003X
1. INTRODUCTION The problem of interaction of a cylinder and a vortex in an ideal fluid was posed by V. V. Kozlov. It was solved in [1, 2] and turned out to be integrable. The problem was further developed in [3, 4], where the authors presented various forms of the equations of motion for a cylinder interacting with a set of point vortices and pointed out particular integrable cases. Nevertheless, the problem of interaction of a cylinder with two point vortices is not integrable. For the general system of equations of motion of point vortices and a cylinder, there are particular Thomson’s configurations where the vortices uniformly distributed over a circle move around the center of the circular cylinder. In the classical case when the cylinder is fixed and the circulation around it is zero, the stability analysis was started by T. H. Havelock [5], who solved this problem in the linear statement. The analysis has been continued in many other works (see the survey papers [6, 11]). However, the question is not completely closed. It turned out that in the nonlinear statement the problem splits into a series of meaningful subproblems that require an individual approach. In particular, an analysis of the Taylor series of the Hamiltonian of the problem up to the forth order inclusive was needed. All resonances arising in the problem have been analyzed [17]. It turned out that two of them lead to instability, although the linear statement exhibits stability [9, 13]. The problem of the stability of the motion of Thomson’s vortex configuration and a moving cylinder is new, and its analysis has not yet been performed. This problem is important because the motion of a cylinder and vortices often arises in problems of the motion of a body in a fluid and is of practical interest in connection with the development of underwater robots. a Udmurt State University, Universitetskaya ul. 1, Izhevsk, 426034 Russia. b Water Problems Institute of the Russian Academy of Sciences, ul. Gubkina 3, Moscow, 119333 Russia. c Southern Mathematical Institute
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