A Theory of Pitch for the Hydrodynamic Properties of Molecules, Helices, and Achiral Swimmers at Low Reynolds Number

Abstract

We present a theory for pitch, a matrix property which is linked to the coupling of rotational and translational motion of rigid bodies at low Reynolds number. The pitch matrix is a geometric property of objects in contact with a surrounding fluid, and it can be decomposed into three principal axes of pitch and their associated \textit{moments of pitch}. The moments of pitch predict the translational motion in a direction parallel to each pitch axis when the object is rotated around that axis, and can be used to explain translational drift, particularly for rotating helices. We also provide a symmetrized boundary element model for blocks of the resistance tensor, allowing calculation of the pitch matrix for arbitrary rigid bodies. We analyze a range of chiral objects, including chiral molecules and helices. Chiral objects with a $C_n$ symmetry axis with $n > 2$ show additional symmetries in their pitch matrices. We also show that some achiral objects have non-vanishing pitch matrices, and use this result to explain recent observations of achiral microswimmers. We also discuss the small, but non-zero pitch of Lord Kelvin's isotropic helicoid.