Helical beating of an actuated elastic filament

We investigate the propulsive force resulting from the rotation of a flexible filament in the low Reynolds number regime. Using a simple linear model, we establish the nonlinear torque-force relations for two torque-driven actuation modes. When the rotation of the filament is induced by two perpendicular transverse oscillating torques, the propulsive force increases monotonically with the torque amplitude. Conversely, when a constant axial torque is applied, the torque-force characteristics displays an unstable branch, related to a discontinuous transition in the shape of the filament. We characterize this shape transition using two geometrical parameters, quantifying the wrapping around and the collapse on the axis of the filament. The proposed theoretical description correctly accounts for our experimental observations and reveals a strong dependence of the filament dynamics on the anchoring conditions.