Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation

Abstract

This paper addresses the problem of the self-propulsion of a smooth body in a fluid by periodic oscillations of the internal rotor and circulation. In the case of zero dissipation and constant circulation, it is shown using methods of KAM theory that the kinetic energy of the system is a bounded function of time. In the case of constant nonzero circulation, the trajectories of the center of mass of the system lie in a bounded region of the plane. The method of expansion by a small parameter is used to approximately construct a solution corresponding to directed motion of a circular foil in the presence of dissipation and variable circulation. Analysis of this approximate solution has shown that a speed-up is possible in the system in the presence of variable circulation and in the absence of resistance to translational motion. It is shown that, in the case of an elliptic foil, directed motion is also possible. To explore the dynamics of the system in the general case, bifurcation diagrams, a chart of dynamical regimes and a chart of the largest Lyapunov exponent are plotted. It is shown that the transition to chaos occurs through a cascade of period-doubling bifurcations.