On the Motion of Two Mass Vortices in Perfect Fluid

Abstract

The system of two interacting dynamically 2D rigid circular cylinders in an infinite volume of perfect fluid was considered in [4, 5], while the pioneering contribution is due to Hicks [1, 2]. An allied problem—the motion of two spheres in perfect fluid—was studied by Stokes, Hicks, Carl and Vilhelm Bjerknes, Kirhhoff, and Joukowski (the references can be found in [3] and [7]). Assuming the circulations around the cylinders to be constant and making the radii of the cylinders infinitely small result in new 2D hydrodynamic objects called mass vortices [5]. The equations of motion for mass vortices expand upon the classical Kirhhoff equations governing the motion of ordinary point vortices. In this paper the motion of two mass vortices is examined in greater detail (some results have been obtained already in [5]). A reduction of order is performed; using the Poincaré surface-of-section technique the system is shown to be generally non-integrable. Some integrable cases are indicated. In conclusion the motion of a single mass vortex and the motion of cylinder in a half-plane are briefly investigated.