Bending-compression coupling in extensible slender microswimmers

TL;DR

This work develops a geometric framework for bending–compression coupling in extensible slender microswimmers at low Reynolds number, employing a $\mathrm{SE}(2)$ representation and Magnus expansion to capture non-commutative translation–rotation effects. It introduces two minimal models and a systematic small-amplitude perturbation theory to quantify how bending and compression interact, including Lie-bracket contributions, and analyzes various gaits (uniform compression, bending waves, and mixed modes). The key findings show that compression enables motion under isotropic drag when nonuniform, enhances yaw and maneuverability through $O(\epsilon\eta)$ coupling, and that higher-order Lie-bracket terms are essential to predict net locomotion in finite-size swimmers. These results highlight compression as a functional degree of freedom with implications for understanding biology and guiding the design of soft microrobots, while underscoring the importance of non-commutative effects in microswimming dynamics.

Abstract

Undulatory slender objects have been a central theme in the hydrodynamics of swimming at low Reynolds number, where the slender body is usually assumed to be inextensible, although some microorganisms and artificial microrobots largely deform with compression and extension. Here, we theoretically study the coupling between the bending and compression/extension shape modes, using a geometrical formulation of microswimmer hydrodynamics to deal with the non-commutative effects between translation and rotation. By means of a coarse-grained minimal model and systematic perturbation expansions for small bending and compression/extension, we analytically derive the swimming velocities and report three main findings. First, we revisit the role of anisotropy in the drag ratio of the resistive force theory and generally demonstrate that no motion is possible for uniform compression with isotropic drag. We then find that the bending-compression/extension coupling generates lateral and rotational motion, which enhances the swimmer's manoeuvrability, as well as changes in progressive velocity at a higher order of expansion, while the coupling effects depend on the phase difference between the two modes. Finally, we demonstrate the importance of often-overlooked Lie bracket contributions in computing net locomotion from a deformation gait. Our study sheds light on compression as a forgotten degree of freedom in swimmer locomotion, with important implications for microswimmer hydrodynamics, including understanding of biological locomotion mechanisms and design of microrobots.

Figures (10)

• Figure 1: Schematic of a filament in three frames of reference. (a) A filament in a reference state. The arc length $s_0\in [0, L_0]$ is used for parametrisation of the curve. (b) A filament in the body-fixed frame at time $t$. The point labelled by $s_0$ is mapped to $\tilde{\bm{x}}(s_0, t)$, where the distance along the filament is denoted by $s(s_0, t)$ and the local tangent angle is represented as $\tilde{\theta}(s_0, t)$. (c) A filament in the laboratory frame, which is obtained by rigid body transformation, translation with $\bm{X}(t)$ and rotation by $\Theta(t)$, from the filament in (b). The point $\tilde{\bm{x}}(s_0, t)$ is mapped to $\bm{x}(s,t)$.
• Figure 2: Setup and notations for the minimal model.
• Figure 3: Curvature fields for the squeeze-me-bend-you swimmer, as defined in Equation \ref{['eq: curvature field']}. Dotted black lines indicate zero-curvature level set. Suggested gaits are indicated by a continuous black line, and the corresponding deformation sequence of the swimmer is represented on the right of each curvature plot : (a) $y$-displacement for uniform compression, (b) $x$-displacement, (c) $y$-displacement, and (d) $\theta$-displacement with time progress shown by the arrow and colours.
• Figure 4: (Top) Sample trajectories of a swimmer under uniform compression with different phase shifts, (a) $\phi=0$ and (b) $\phi=\pi/2$. The parameters are $\epsilon=\eta=0.4$, $\gamma=2$, $L_0=1, kL_0=2\pi$ and $\omega=1$. With the initial position $(0,0)$ (marked in a red circle) and the initial angle $\theta=0$, we drew the orbits of the leftmost end of the filament from $t=0$ to $t=10$. The configuration at $t=10$ is also shown. (Bottom) Time sequence of swimmer shape from $t=0$ to $t=1$ with the local extension visualised by the colours.
• Figure 5: Schematics for the symmetry arguments of the no-net-rotation. (a) The original shape in the body fixed frame with velocities $\tilde{\bm{U}}$ and $\Omega$. (b) The shape after the time reversal with a shift $c=kL_0/\omega-T/2$ (c) The head-to-tail inversion. (d) The $\pi$-rotation of the system.

Theorems & Definitions (1)

• Proposition 1