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Swimming at small Reynolds number of an elliptical disk propelled by an elliptically polarized surface wave

The swimming of an elliptical disk at small Reynolds number is studied on the basis of a perturbative solution of the Navier-Stokes equations for fluid flow near a deformable infinite sheet. A stroke involving an elliptically polarized plane surface wave is studied, in extension of work by Taylor and Tuck. In general the elliptic polarization of the stroke leads to an asymmetry of the flow in the upper and lower half-space. On the basis of results for an infinite sheet expressions for the mean translational and rotational swimming velocity of an elliptical disk of size much larger than the wavelength of the stroke are deduced. In addition expressions are derived for the mean power and the efficiency of swimming.