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On stability of a regular vortex polygon in the circular domain. (English) Zbl 1097.76031

The paper deals with Kelvin problem for the case in which the vortex \(n\)-gon is located within a circular domain with the common center of symmetry. According to the well-known Lyapunov theorem, the equilibrium of a complete system is unstable when a linearized system is exponentially unstable. The power-law instability is insufficient to draw this conclusion; therefore, nonlinear terms should be involved in the analysis. The paper presents necessary and sufficient conditions for the stability and instability of a regular \(n\)-gon of point vortices located at the circle. For a vortex pentagon, the answer to the question concerning the instability remains unclear for the null set of a governing parameter \(p\). The author also formulates two theorems concerning the stability in the Routh sense.

MSC:

76E30 Nonlinear effects in hydrodynamic stability
76B47 Vortex flows for incompressible inviscid fluids
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