No net motion for oscillating near-spheres

We investigate the hydrodynamics of oscillating nearly-spherical particles -- defined by their radius $r=1+\epsilon f(\theta,\phi)$ -- at low, yet non-vanishing, Reynolds numbers. In contrast with the results of [1], we analytically demonstrate that no net motion can arise up to order one in Re and order one in the asphericity parameter $\epsilon$, regardless of the shape function $f$.
[1] F. Nadal and E. Lauga, Physics of Fluids 26(8) 082001 (2014)