^{1,a)}

### Abstract

The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a circle inside an annulus is presented. The problem is reduced to one of the equilibrium stability of the Hamiltonian system with a cyclic variable. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The stability of the stationary motion is interpreted as a Routh stability. The exponential instability is shown always to take place if N ≥ 7. For N = 2, 4, and 6, the parameter space is divided in two: a Routh stability domain in an exact nonlinear setting and an exponential instability domain. For N = 3 and 5, the parameter space consists of three domains. The stability of the third in an exact nonlinear setting is sandwiched between the Routh and exponential domains. Its analysis remains an open problem with its solution requiring nonlinear analysis.

*N*-gon) was posed in the 19th century and solved in an exact nonlinear setting in the 21st century. The solution of the problem for vortices inside and outside a circle started by Havelock

^{8}requires the use of Kolmogorov-Arnold-Moser (KAM)-theory methods. In this paper, a complete linear stability analysis of Thomson's vortex polygon inside an annulus is presented.

The research is supported by the base part of the project No 213.01-11/2014-1, Southern Federal University.

I. INTRODUCTION II. STATEMENT OF THE PROBLEM III. STABILITY AND INSTABILITY OF THOMSON'S VORTEX POLYGON INSIDE AN ANNULUS IV. CONSTRUCTION OF THE REDUCED SYSTEM V. PROOF OF STATEMENTS IN SECTION III ON THE STABILITY OF STATIONARY ROTATION OF THOMSON's VORTEX POLYGON VI. CONCLUSION

### Key Topics

- Rotating flows
- 59.0
- Vortex stability
- 49.0
- Eigenvalues
- 30.0
- Polynomials
- 13.0