We formulate a new theoretical model for the swimming of a flexible body in a vortex street. We consider the class of periodic travelling-wave body motions, in the limit of small amplitude. We calculate the output power provided to the body by thrust forces, and the input power done against pressure forces, as functions of the aspect ratio and strength of the vortex street. We then formulate two optimization problems. In the first, we determine the body wave which provides maximum output power for fixed amplitude. We find a closed-form solution with a transition from power law to exponential decay of output power as the vortex street widens. In the second problem, we incorporate internal viscoelasticity to the swimming body and compute its contribution to the input power. We find the body wave which maximizes efficiency for a given output power. The body shape and resulting efficiency are found in closed form and simple approximate formulas are given. We find that efficiency scales as the inverse of the damping parameter. Finally, we compare our results with previous experiments and simulations. We find agreement in some aspects and disagreement in others. We give physical interpretations for agreements and disagreements in terms of the phase between the body wave and vortex street.
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