Dynamics of two interacting circular cylinders in perfect fluid
A. V. Borisov I.S. Mamaev S. M. Ramodanov
Discrete & Continuous Dynamical Systems - A 2007, 19(2): 235-253 doi: 10.3934/dcds.2007.19.235
In this paper we consider the system of two 2D rigid circular cylinders immersed in an unbounded volume of inviscid perfect fluid. The circulations around the cylinders are assumed to be equal in magnitude and opposite in sign. We also explore some special cases of this system assuming that the cylinders move along the line through their centers and the circulation around each cylinder is zero. A similar system of two interacting spheres was originally considered in the classical works of Carl and Vilhelm Bjerknes, H. Lamb and N. E. Joukowski.
    By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for $n$ point vortices.
keywords: multifractal analysis. Poincaré recurrences Dimension theory
Dynamics of a circular cylinder interacting with point vortices
A. V. Borisov I. S. Mamaev S. M. Ramodanov
Discrete & Continuous Dynamical Systems - B 2005, 5(1): 35-50 doi: 10.3934/dcdsb.2005.5.35
The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.
keywords: Point vortex integrable system Hamiltonian system qualitative and topological analysis. Poisson bracket structure first integral advection

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