Chaos and Integrability in Ideal Body-Fluid Interactions

We consider interaction of a rigid body with a surrounding ideal uid containing a number of point vortices. The uid is assumed to be planar and unbounded and the body is assumed to be free to move in response to the uid forces. Except that the body should be simply connected and rigid, no assumptions are made on the body shape or on its internal mass distribution. There may also be an arbitrary and constant circulation around the body. The governing equations reduce to an autonomous set of coupled ODEs for the vortex positions and the body position and orientation. The form of these equations are derived by combining the classical equations for free body motion in ideal uid of G. Kirchho and Lord Kelvin with C. C. Lin's bounded domain generalisation of H. Helmholtz's celebrated point vortex equations. The Hamiltonian nature of the coupled body-vortex ODEs is demonstrated and the existence of additional conserved quantities is discussed. A survey of the integrable motions of the system is given. Integrability is demonstrated explicitly by exploiting conservation laws to devise reduced phase space coordinates in which the orbits of the system are the contours of an energy landscape. The existence of relative equilibria, their stability, and the qualitatively dierent kinds of motion is studied analytically and numerically. We then perform small parametric perturbations destroying the symmetry or conservation law that makes the system integrable. The emergence of chaos in the system is diagnosed by generating Poincare sections from numerically obtained solutions. By identifying the chaotic solutions and studying the body and vortex orbits, we obtain a better mechanistic understanding of the causes of chaotic behavior. As is well-known from dynamical system theory, the chaos can often be traced back to unstable relative equilibria of the perturbed integrable system. By this methodology we demonstrate that, even when there are no vortices in the uid, a freely moving elongated body, whose motion is dominated by rotation, may have an atmosphere of uid particles following it through the uid. This atmosphere contains both regular and chaotic regions, and may be understood from KAM theory. We also discover two separate chaotic regimes in the interaction of a body and one point vortex when the body is either noncircular or has asymmetric internal mass distribution. For one of these chaotic regimes the eect of chaos seems to be largest on the vortex, while for the other, the chaos primarily expresses itself in the body motion. Finally we brie y demonstrate the occurrence of chaos in the interaction of a body with two point vortices.