Reciprocal swimming in viscoelastic granular hydrogels

TL;DR

This work tackles reciprocal locomotion in viscoelastic granular media by driving a scallop-like swimmer with reciprocal wing flaps inside a bed of hydrogel spheres, comparing to polystyrene beads. The authors combine force measurements, swimmer displacement, and X-ray radiography to show that propulsion occurs only at intermediate flapping frequencies, with backward motion arising from an interplay of inertia and viscoelastic relaxation that creates density voids around the wings. The key findings include a resonance-like enhancement of net displacement near $1\,\text{Hz}$, two relaxation times $\tau_1$ and $\tau_2$, and a hysteresis in density due to void formation and refilling. The study provides a minimal dynamical framework linking wing actuation to swimmer motion via an equation of motion that couples inertia, drag, elasticity, and time-varying density, offering insights for control of soft robots in cohesive granular environments.

Abstract

We experimentally study a scallop-like swimmer with reciprocally flapping wings in a nearly frictionless, cohesive granular medium consisting of hydrogel spheres. Significant locomotion is found when the swimmer's flapping frequency matches the inverse relaxation time of the material. Remarkably, the swimmer moves in the opposite direction compared to its motion in a cohesion-free granular material of hard plastic spheres. At higher or lower frequencies, we observe no motion of the swimmer, apart from a short initial transient phase. X-ray radiograms reveal that the wing motions create low-density zones, which in turn give rise to a hysteresis in drag and propulsion forces. This time-dependent effect, combined with the swimmer's inertia, accounts for locomotion at intermediate frequencies.

Figures (8)

• Figure 1: Experimental setup for swimming in granular media. The apparatus consists of a pair of wings operating in a reservoir filled with granular particles. The inset shows a photograph of the hydrogel particles.
• Figure 2: Swimmer displacement as a function of time (dashed lines) for several flapping periods: $T=2.0\,\text{s}$ (blue), $T=1.0\,\text{s}$ (purple), $T=0.4\,\text{s}$ (red). For comparison, the black line shows data for polystyrene granulate at $T=2.0\,\text{s}$. Symbols and solid lines indicate the displacement at the end of each cycle, averaged over two repetitions.
• Figure 3: Net swimmer displacement per cycle as a function of the flapping frequency. The inset shows the data for a swimmer moving in a polystyrene granulate; here, the error bars are very small.
• Figure 4: Phase-resolved swimmer velocity (black dashed line, right axis) and driving force (red solid line, left axis) at a flapping frequency of $1/T=1\,\text{Hz}$. The vertical dotted line separates the opening and closing half-cycles, sketched in the insets.
• Figure 5: Averaged relative displacement of the swimmer as a function of the phase, $t/T$, for various flapping frequencies. The vertical dotted line separates the opening and closing half-cycles. The figure also defines the (positive) peak location, $\tau$, and the peak-to-peak amplitude, $\Delta y$, that depend on the flapping frequency, see \ref{['fig:inertiaB']}.