Bending Leaves and Flapping Flight: Transitions in Flow-Body Interaction Problems

The coupled motion of fluids and solids in contact is common in the biological world, and leads to unexpected phenomena in theoretical mechanics. I will address two model problems in this area, both of which arose from experiments. The first problem asks: How can a flexible body reduce its drag by bending in a flow? We specialize to the case of a uniform elastica immersed in a steady planar flow, and find a transition from the quadratic growth of drag with flow speed typical of rigid bodies to a much-reduced 4/3-power law. We find also that the body and wake assume a unified, parabolic form. An asymptotic argument explains the governing phenomenon: the formation of a ``tip region'' on the fiber, which gives rise to global self-similarity. The second problem, initiated by a recent experiment by Vandenberghe, Zhang, and Childress, asks: Under what conditions does a flapping foil spontaneously locomote? We find that, at sufficiently large ``frequency Reynolds number,'' unidirectional locomotion emerges as an attracting state for an initially nonlocomoting body. Locomotion is generated in two stages: first, the fluid field loses symmetry by an instability similar to the classical von Karman instability; and second, precipitous interactions with previously shed vortical structures ``push'' the body into locomotion. Body mass and slenderness play central and unexpected roles in each stage.