American Institute of Mathematical Sciences

May  2007, 19(2): 235-253. doi: 10.3934/dcds.2007.19.235

Dynamics of two interacting circular cylinders in perfect fluid

 1 Institute of Computer Science, Udmurt State University, Universitetskaya ul. 1, 426034 Izhevsk, Russian Federation, Russian Federation 2 Institute of Computer Science, Udmurt State University, 426034 Izhevsk

Received  January 2006 Revised  August 2006 Published  July 2007

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.
Citation: A. V. Borisov, I.S. Mamaev, S. M. Ramodanov. Dynamics of two interacting circular cylinders in perfect fluid. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 235-253. doi: 10.3934/dcds.2007.19.235
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