Going in circles: Slender body analysis of a self-propelling bent rod

TL;DR

This paper analyzes the self-propelling motion of a bent, asymmetric slender rod in two dimensions, revealing that geometry and surface activity induce circular trajectories, with specific configurations maximizing propulsion.

Contribution

It provides a mathematical framework for understanding how geometric asymmetry and surface activity drive self-propulsion in slender, bent bodies using slender body theory and the Lorentz reciprocal theorem.

Findings

Particle trajectories are circular and determined by rotational speed.

Uniform surface activity with asymmetry causes self-propulsion.

Maximum propulsion occurs when one arm is four times longer with a  angle.

Abstract

We study the two-dimensional motion of a self-propelling asymmetric bent rod. By employing slender body theory and the Lorentz reciprocal theorem, we determine particle trajectories for different geometric configurations and arbitrary surface activities. Our analysis reveals that all particle trajectories can be mathematically expressed through the equation for a circle. The rotational speed of the particle dictates the frequency of the circular motion and the ratio of translational and rotational speeds describes the radius of the circular trajectory. We find that even for uniform surface activity, geometric asymmetry is sufficient to induce a self-propelling motion. Specifically, for uniform surface activity, we observe (i) when bent rod arm lengths are equal, the particle only translates, (ii) when the length of one arm is approximately four times the length of the other arm and the angle between the arms is approximately {\pi}/2, the rotational and translational speeds are at their maximum. We explain these trends by comparing the impact of geometry on the hydrodynamic resistance tensor and the active driving force. Overall, the results presented here quantify self-propulsion in composite-slender bodies and motivate future research into self-propulsion of highly asymmetric particles.