Chiral propulsion: the method of effective boundary conditions

TL;DR

This paper introduces an effective boundary condition method to analyze chiral propulsion of helices in low Reynolds number fluids, simplifying complex geometries to cylinders with partial slip conditions, and derives universal propulsion characteristics.

Contribution

It develops a novel effective boundary condition approach for modeling chiral propulsion, applicable to various helical shapes with a universal dependence on the helical angle.

Findings

The method accurately reproduces known results for spirals.

Chiral propulsion depends universally on the helical angle as $ heta$ varies.

Maximum propulsion occurs at approximately 35.26 degrees.

Abstract

We propose to apply an "effective boundary condition" method to the problem of chiral propulsion. For the case of a rotating helix moving through a fluid at a low Reynolds number, the method amounts to replacing the original helix (in the limit of small pitch) by a cylinder, but with a special kind of partial slip boundary conditions replacing the non-slip boundary conditions on the original helix. These boundary conditions are constructed to reproduce far-field velocities of the original problem, and are defined by a few parameters (slipping lengths) that can be extracted from a problem in planar rather than cylindrical geometry. We derive the chiral propulsion coefficients for spirals, helicoids, helically modulated cylinders, and some of their generalizations using the introduced method. In the case of spirals, we compare our results with the ones derived by Lighthill and find a very good agreement. The proposed method is general and can be applied to any helical shape in the limit of a small pitch. We have established that for a broad class of helical surfaces the dependence of the chiral propulsion on the helical angle $\theta$ is universal, $\chi\sim \cos\theta\sin 2\theta$ with the maximal propulsion achieved at the universal angle $\theta_m = \tan^{-1}(1/\sqrt{2})\approx 35.26^\circ$.