Reciprocal swimming in granular media: the role of jamming and swimmer inertia

TL;DR

Problem: How can a scallop-like swimmer achieve net locomotion in granular media using reciprocal wing flapping, challenging time-reversal constraints in a non-Newtonian substrate? Approach: discrete element method (DEM) simulations of a winged swimmer in a dense granular reservoir, with analysis of jamming via the strong-contact count $N_c$, coordination number $Z$, and a coasting-time metric $T_c$ to identify inertia-driven effects. Key contributions: (i) a jamming-based mechanism yielding $\Delta y/L = \alpha\,\Delta N_c$ with $\alpha \approx 8.63\times10^{-6}$; (ii) an inertia-driven mechanism governed by $\Delta y/L \approx \alpha\,\Delta N_c - \beta (T_c/T)^n$ with $\beta \approx 0.24$ and $n \approx 0.44$; (iii) validation against experiments and CT imaging confirming asymmetric force chains and stagnation zones. Significance: offers physics-based design principles for granular propulsion and reveals how microstructure and inertia enable locomotion under reciprocal actuation.

Abstract

We use particle simulations to reveal two distinct propulsion mechanisms for a scallop-like swimmer to locomote itself in granular media by reciprocally flapping its wings. Based on the discrete element method, we examine the kinematics and contact forces of particles near the swimmer to identify jamming effects induced by the swimmer in a frictional granular medium, which are less intense during the opening stroke than the closing. This broken symmetry is quantified by the difference in the number of strong particle contact forces formed during opening and closing, which shows a linear relation with the swimmer's net displacement across various swimmer and medium configurations, all favoring the opening stroke. We identify a secondary propulsion mechanism in a dynamic regime with significant swimmer inertia, as the flapping period approaches the coasting time for a moving swimmer to come to rest under the medium resistance. In this case, the swimmer's net displacement is correlated to the ratio between these two time scales, and the swimming direction favors the closing stroke due to the smaller medium resistance as the swimmer coasts with closed wings.

Figures (6)

• Figure 1: Schematics of the simulation setup and particle contact models. (a) 3D view of the setup. (b) Top-down view of the setup. (c) Normal and tangential contact forces between particles $i$ and $j$.
• Figure 2: Swimmer and particle kinematics from the experiment xiao2024locomotion and from DEM simulations, see configurations in Table \ref{['tab:1']}. (a) Swimmer displacement vs time in the experiment (dashed, orange), the DEM result with simulated mounting rods (yellow), the simplified simulation base case (blue), and a frictionless simulation (red). (b) and (c) Highlighting particles with a displacement larger than five times the average displacement in the DEM results (left panels) and the experimental X-ray CT results (right panels), at $\theta=75^\circ$ during opening and $\theta=25^\circ$ during closing, respectively. (d) and (e) Comparison of the particle displacement field at the mid-plane of the swimmer between DEM results (left panels) and X-ray CT results (right panels), at $\theta=75^\circ$ during opening and $\theta=25^\circ$ during closing, respectively.
• Figure 3: Force transmission and structural analyses for the base simulation case. (a) Particle velocity and inter-particle contact forces visualized during one swimming cycle. (b) Sketch for spatial decomposition of the swimmer's surroundings. (c) and (d) Average coordination number $Z$ over $\theta$ for zone I and II, respectively. (e) Total force exerted on different lateral surfaces of the wings.
• Figure 4: The influence of jamming on net locomotion. (a) Cycle-averaged swimmer net displacement $\Delta y$ for cases with different gap widths $S$, with error bars indicating cycle-cycle fluctuations, along with the case with inert mid-body (purple) and the case with $\mu=0$ (black). The black dashed line marks the value of $\Delta y$ for the one-wing case. (b) Cycle-averaged $\Delta y$ vs. the contact number difference $\Delta N_c$. Inset shows the $N_c$ in the base case in a swimming cycle. The purple diamond represents the case with inert mid-body, the orange square represents the one-wing case, and the green triangle represents the case with $\mu=0$. The red dashed line is a linear fit, $\Delta y/L = \alpha \ \Delta N_c$, where $\alpha=8.63\times10^{-6}$.
• Figure 5: Locomotion in the dynamic regime. (a) The normalized swimmer displacement $\Delta y$ vs time in quasi-static (dashed blue curve) and dynamic (solid green curve) regimes. The quasi-static case is with $m_s=0.60 \text{ g}$, $T=2.0 \text{ s}$, and $T_c/T=5.16\times10^{-5}$, and the dynamic case has $m_s=2.38 \text{ g}$, $T=0.05 \text{ s}$, and $T_c/T=0.33$. (b) Normalized $\Delta y$ vs $\theta$ in both quasi-static (dashed curve) and dynamic (solid curve) regimes. Blue represents opening, and red represents closing. (c) The difference in retardation distance vs the coasting time $T_c$ in frictional (blue circle) and frictionless (red triangle) media. The inset shows the difference in the retardation time between the two strokes vs $T_c$.