Dynamics and locomotion of flexible foils in a frictional environment.

Over the past few decades, oscillating flexible foils have been used to study the physics of organismal propulsion in different fluid environments. Here, we extend this work to a study of flexible foils in a frictional environment. When the foil is oscillated by heaving at one end but is not free to locomote, the dynamics change from periodic to non-periodic and chaotic as the heaving amplitude increases or the bending rigidity decreases. For friction coefficients lying in a certain range, the transition passes through a sequence of N-periodic and asymmetric states before reaching chaotic dynamics. Resonant peaks are damped and shifted by friction and large heaving amplitudes, leading to bistable states. When the foil is free to locomote, the horizontal motion smoothes the resonant behaviours. For moderate frictional coefficients, steady but slow locomotion is obtained. For large transverse friction and small tangential friction corresponding to wheeled snake robots, faster locomotion is obtained. Travelling wave motions arise spontaneously, and move with horizontal speeds that scale as transverse friction coefficient to the power 1/4 and input power that scales as the transverse friction coefficient to the power 5/12. These scalings are consistent with a boundary layer form of the solutions near the foil's leading edge.