Parking 3-sphere swimmer. I. Energy minimizing strokes

TL;DR

This paper analyzes the energy-efficient periodic strokes of a low-Reynolds number three-ball swimmer, revealing that optimal strokes are elliptical in shape and providing explicit analytic expressions for these strokes.

Contribution

It introduces the geometric symmetries of the sPr_3 swimmer and characterizes optimal small-amplitude strokes as elliptical curves with explicit formulas.

Findings

Optimal small strokes are elliptical in 3D space.

Explicit analytic expressions for stroke vectors u and v are derived.

Results set the stage for analyzing long-arm dynamics in a subsequent study.

Abstract

The paper is about the parking 3-sphere swimmer ($\text{sPr}_3$). This is a low-Reynolds number model swimmer composed of three balls of equal radii. The three balls can move along three horizontal axes (supported in the same plane) that mutually meet at the center of $\text{sPr}_3$ with angles of $120^{\circ}$ . The governing dynamical system is introduced and the implications of its geometric symmetries revealed. It is then shown that, in the first order range of small strokes, optimal periodic strokes are ellipses embedded in 3d space, i.e. closed curves of the form $t\in [0,2\pi] \mapsto (\cos t)u + (\sin t)v$ for suitable orthogonal vectors $u$ and $v$ of $\mathbb{R}^3$. A simple analytic expression for the vectors $u$ and $v$ is derived. The results of the paper are used in a second article where the real physical dynamics of $\text{sPr}_3$ is analyzed in the asymptotic range of very long arms.