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

We reconsider fluid dynamics of a self-propulsive swimmer in Stokes flow. With an exact definition of deformation of a swimmer, a proof is given to the scallop theorem including the body rotation of the swimmer. Introducing a virtual swimmer, which has the same shape as the real swimmer but has no ambient fluid, we give an exact definition of the surface deformation of the generally translating and rotating swimmer, and then prove the scallop theorem rigorously for the massless swimmer. We also discuss the breakdown of the scallop theorem due to a finite mass (finite Stokes number) of the swimmer by using a perturbation expansion method and it is found that the breakdown generally occurs at the first order of Stokes number. In addition, employing the Purcell's ``scallop'' model, we show that the theorem holds up to a higher order of the Stokes number, if the swimmer's stroke has some symmetry. \\[4pt] [1] K. Ishimoto and M. Yamada, ``A rigorous proof of the scallop theorem and a finite mass effect of a microswimmer,'' arXiv:1107.5938v1 [physics.flu-dyn].