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

TL;DR

This paper develops a geometric theory of pitch for rigid bodies in low Reynolds number fluids, linking rotational and translational motion, and provides computational tools to analyze various chiral and achiral objects.

Contribution

It introduces a comprehensive pitch matrix framework, decomposes it into principal axes, and offers a boundary element model for calculating it for arbitrary bodies.

Findings

Chiral objects with certain symmetries exhibit predictable pitch matrix patterns.

Some achiral objects have non-zero pitch matrices, explaining observed microswimmer behaviors.

The model applies to a range of objects, including molecules, helices, and isotropic helicoids.

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.