Dynamics of an Elliptic Foil with an Attached Vortex in an Ideal Fluid: The Integrable Case

Abstract

This paper is concerned with the plane-parallel motion of an elliptic foil with an attached vortex of constant strength in an ideal fluid. Special attention is given to the case in which the vortex lies on the continuation of one of the semiaxes of the ellipse. It is shown that in this case there exist no attracting solutions and the system is integrable by the Euler – Jacobi theorem. A complete qualitative analysis of the equations of motion is carried out for cases where the vortex lies on the continuation of the large or the small semiaxis of the ellipse. Possible types of trajectories of an elliptic foil with an attached vortex are established: quasi-periodic, unbounded (going to infinity) and periodic trajectories.