UC San Diego

A growing body of work aims at designing and testing micron-scale synthetic swim- mers. One method, inspired by the locomotion of flagellated bacteria, consists in applying a rotating magnetic field to a rigid, helically- shaped, propeller attached to a magnetic head. When the resulting device, termed an artificial bacteria flagellum, is aligned perpendicularly to the applied field, the helix rotates and the swimmer moves forward. Experimental investigation of artificial bacteria flagella stability shows that at low frequency of the applied field, the helix axis does not align perpendicularly to the field but wobbles around that direction, with an angle which increases as the inverse of the field frequency. We use numerical computations and an asymptotic analysis to provide a theoretical explanation for this wobbling behavior. We first build a dynamical model for the locomotion of an artificial bacteria flagellum based on the mechanical balance of forces and moments and on resistive-force theory for the hydrodynamics. A numerical solution of the dynamical system demonstrates the wobbling -to-swimming transition as a function of the helix geometry and the dimensionless Mason number quantifying the ratio of viscous to magnetic torques. We then employ an asymptotic expansion for near-straight helices to derive an analytical estimate for the wobbling angle allowing to rationalize our computations and past experimental results. These results can help guide future design of artificial helical swimmers