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Locomotion of a rotating cylinder pair with periodic gaits at low Reynolds numbers

We consider the periodic gaits of a microswimmer formed by two rotating cylinders, placed apart at a fixed width. Through a combination of theoretical arguments and numerical simulations, we derive semi-analytic expressions for the system’s instantaneous translational and rotational velocities, as a function of the rotational speeds of each cylinder. We can then integrate these relations in time to find the speed and efficiency of the swimmer for any imposed gait. Here, we focus particularly on identifying the periodic gaits that lead to the highest efficiency. To do so, we consider three stroke parameterizations in detail: alternating strokes, where only one cylinder rotates at a time; tilted rectangle strokes, which combine co- and counter-rotation phases; and smooth strokes represented through a set of Fourier series coefficients. For each parameterization, we compute maximum efficiency solutions using a numerical optimization approach. We find that the parameters of the global optimum, and the associated efficiency value, depend on the average mechanical input power. The globally optimal efficiency asymptotes toward that of a steadily counter-rotating cylinder pair as the input power increases. Finally, we address a possible three-dimensional (3D) extension of this system by evaluating the efficiency of a counter-rotating 3D cylinder pair with spherical end caps. We conclude that the counter-rotating cylinder pair combines competitive efficiency values and high versatility with simplicity of geometry and actuation, and thus forms a promising basis for engineered microswimmers.