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

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