A rigorous proof of the scallop theorem and a finite mass effect of a microswimmer

We reconsider fluid dynamics for a self-propulsive swimmer in Stokes flow. With an exact definition of deformation of a swimmer, a proof is given to Purcells scallop theorem including the body rotation. The breakdown of the theorem due to a finite Stokes number is discussed by using a perturbation expansion method and it is found that the breakdown generally occurs at the first order of the Stokes number. In addition, employing the Purcells scallop model, we show that the theorem holds up to a higher order if the strokes of the swimmer has some symmetry.