🔬 condensed matter

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

Using particle simulations, this study reveals that a scallop-like swimmer propels itself in granular media through two distinct mechanisms: a primary jamming-induced asymmetry favoring the opening stroke due to stronger contact forces, and a secondary inertia-driven mechanism favoring the closing stroke when the flapping period approaches the swimmer's coasting time.

Original authors: Amir Nazemi, Hongyi Xiao

Published 2026-07-09

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CC BY 4.0

Original authors: Amir Nazemi, Hongyi Xiao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to swim through a giant bathtub filled not with water, but with a mountain's worth of sand. This is the challenge faced by the "scallop swimmer" in this study—a robot with two wings that flap open and shut, just like a real scallop.

In normal water, a famous rule called the "Scallop Theorem" says that if you just open and close your wings symmetrically (reciprocally), you won't go anywhere. You'd just wiggle in place. But in sand, this rule doesn't apply. The researchers used powerful computer simulations to figure out why this robot can actually move through sand, and they discovered it uses two different "engines" depending on how fast it flaps.

Here is how those two engines work, explained with everyday analogies:

Engine 1: The "Traffic Jam" Effect (Slow Swimming)

When the robot flaps slowly, it moves forward because of jamming.

Think of the sand particles like a crowd of people at a concert.

The Opening Stroke: When the robot's wings open, they push the "crowd" (sand) aside. The sand flows relatively easily, like people stepping back to make room.
The Closing Stroke: When the wings snap shut, they squeeze the sand together. Because sand is frictional, the particles get stuck on each other, forming a solid "traffic jam" or a rigid block right in front of the closing wings.

The Analogy: Imagine trying to walk through a hallway.

Opening: You push a group of people aside, and they scatter easily. You move forward a bit.
Closing: You try to pull your hands back, but you've accidentally created a solid wall of people holding hands in front of you. Because this wall is so solid, it's actually harder to pull your body backward through it than it was to push forward.

The robot ends up moving forward because it's easier to push the sand away (open) than to pull the solid "sand-wall" back with it (close). The researchers found that the more "stuck" the sand gets during the closing stroke, the more the robot is forced to move forward.

Engine 2: The "Coasting" Effect (Fast Swimming)

When the robot flaps very quickly, a second engine kicks in, driven by inertia (the robot's own weight and momentum).

Think of this like driving a car on a rough, sticky road.

The Physics: When you stop pressing the gas, a heavy car doesn't stop instantly; it "coasts" for a while. The time it takes to stop depends on how heavy the car is and how much resistance the road offers.
The Asymmetry: The robot's wings are shaped differently when they are open versus when they are closed.
- Open: The wings are spread wide, presenting a big surface area to the sand. It's like a parachute. If you stop flapping, the sand resistance is huge, and the robot stops almost immediately.
- Closed: The wings are folded tight, presenting a tiny surface area. It's like a sleek arrow. If you stop flapping here, the sand resistance is much lower, and the robot "coasts" for a longer time.

The Result: Because the robot coasts much longer after the wings are closed than after they are open, it ends up drifting backward. Even though the flapping motion is the same, the "braking" is different. The robot gets a free ride backward during the closed phase that it doesn't get during the open phase.

The Grand Finale: A Unified Picture

The researchers found that these two mechanisms are like two people pulling on a rope in opposite directions.

The "Jamming" person pulls the robot forward.
The "Coasting" person pulls the robot backward.

If the robot flaps slowly, the "Jamming" person is stronger, and the robot moves forward. If the robot flaps very fast (and is heavy enough), the "Coasting" person takes over, and the robot moves backward.

By combining these two effects into a single mathematical formula, the researchers created a "master map" that predicts exactly how far the robot will move based on how fast it flaps and how much it weighs.

In short: The robot moves because sand isn't just a fluid; it's a tricky material that can jam into a solid. By exploiting the difference between how the sand flows when the wings open versus how it jams when they close, and by using the robot's own weight to "coast" differently in each direction, the robot breaks the rules of normal swimming and finds a way to travel through the sand.

Technical Summary: Reciprocal Swimming in Granular Media: The Role of Jamming and Swimmer Inertia

Problem Statement
Locomotion in granular media presents a unique challenge because the medium can exhibit both fluid-like flow and solid-like jamming. While many biological and robotic swimmers utilize complex, non-reciprocal gaits (e.g., undulation) to navigate these environments, simple reciprocal motions—such as the flapping of wings in a scallop-like manner—are theoretically predicted to produce zero net displacement in low-Reynolds-number Newtonian fluids (the "Scallop Theorem"). However, in granular media, time-reversal symmetry is generally broken due to the medium's non-Newtonian rheology and history-dependent response. This paper investigates how a scallop-like swimmer achieves net propulsion through reciprocal wing flapping in granular media, specifically isolating the roles of jamming-induced hysteresis and swimmer inertia.

Methodology
The authors employ Discrete Element Method (DEM) simulations using the MercuryDPM package to model a scallop-like swimmer submerged in a granular reservoir.

Swimmer Geometry: The swimmer consists of two square wings rotating around hubs with a gap width $S$ . The wings oscillate reciprocally between a closed angle ( $\theta_c = 20^\circ$ ) and an open angle ( $\theta_o = 80^\circ$ ) with a period $T$ .
Medium Properties: The granular medium consists of $\approx 510,000$ particles with a density of $1050 \text{ kg/m}^3$ and a mean diameter of $1.0 \text{ mm}$ . Inter-particle interactions include Hertzian elastic contact, viscous dissipation, and Coulomb friction.
Parametric Study: The study systematically varies the swimmer mass ( $m_s$ ), flapping period ( $T$ ), wing gap width ( $S$ ), and particle friction coefficient ( $\mu$ ).
Validation: Simulation results are validated against prior experimental data (Xiao et al., 2025) regarding swimmer displacement and particle kinematics (verified via X-ray CT imaging).
Analysis Tools: The authors analyze particle kinematics (velocity fields, strain rates), structural metrics (coordination number $Z$ ), and force transmission (contact forces, strong contact networks) to quantify stagnant zones and resistance.

Key Contributions and Results

1. Jamming-Induced Propulsion (Quasi-Static Regime)
In the quasi-static regime (low swimmer inertia, slow flapping), net propulsion is driven by the history-dependent formation of stagnant zones (regions of jammed particles that move with the swimmer).

Asymmetric Stagnant Zones: During the opening stroke, "propulsive stagnant zones" (PSZ) form near the wing tips. During the closing stroke, "resistive stagnant zones" (RSZ) form near the wing hubs.
Force Hysteresis: The number of strong particle contacts ( $N_c$ ) differs significantly between the fully open and fully closed states. Specifically, the closing stroke generates a higher density of strong force chains (higher $N_c$ ) near the hubs, creating greater resistance to backward motion than the resistance encountered during the forward opening stroke.
Linear Correlation: The net displacement per cycle ( $\Delta y$ ) exhibits a linear relationship with the difference in strong contact numbers between the closed and open states ( $\Delta N_c = N_c(\theta_c) - N_c(\theta_o)$ ). This confirms that the symmetry breaking is caused by the hysteretic evolution of the jammed contact network.
Friction Dependence: In frictionless simulations ( $\mu=0$ ), stagnant zones do not form, and net locomotion vanishes, confirming that inter-particle friction is essential for this mechanism.

2. Inertia-Induced Propulsion (Dynamic Regime)
In the dynamic regime (high swimmer mass, fast flapping), a second mechanism emerges driven by the swimmer's inertia and the difference in coasting times ( $T_c$ ) between strokes.

Coasting Time Definition: $T_c$ is defined as the characteristic time for a moving swimmer to come to rest under medium drag. It is proportional to the ratio of swimmer mass to the drag force ( $T_c \propto m_s / P_s L T$ ).
Retardation Asymmetry: When the flapping period $T$ approaches the coasting time $T_c$ , the swimmer does not stop instantaneously when the wings reverse direction. Instead, it undergoes a "retardation" process.
Directional Reversal: The resistive drag force is larger during the opening stroke (due to a larger projected area) than during the closing stroke. Consequently, the swimmer coasts for a longer duration and distance after the closing stroke than after the opening stroke. This asymmetry in retardation leads to a net displacement in the backward direction (opposite to the jamming-driven forward direction).
Frictionless Confirmation: This inertial mechanism persists even in frictionless media, distinguishing it from the jamming mechanism.

3. Unified Scaling Law
The paper demonstrates that these two mechanisms coexist and compete.

Unified Model: The net displacement can be described by a unified scaling law:
$\Delta y/L = \alpha \Delta N_c - \beta (T_c/T)^n$
where the first term represents the jamming contribution (favoring forward motion) and the second term represents the inertial contribution (favoring backward motion).
Regime Transition: At low $T_c/T$ , the jamming mechanism dominates (forward motion). As $T_c/T$ increases, the inertial mechanism grows, eventually overcoming the jamming effect and causing a transition to backward locomotion.

Significance
The paper claims to reveal two distinct physical mechanisms that break time-reversal symmetry in granular locomotion, allowing reciprocal swimmers to move.

Mechanism 1 (Jamming): Establishes that the history-dependent reorganization of the contact network in a frictional, jammed medium creates asymmetric resistance, enabling forward propulsion. This is quantified by the difference in strong contact forces.
Mechanism 2 (Inertia): Identifies that swimmer inertia, when comparable to the medium's drag response time, creates a second symmetry-breaking mechanism based on asymmetric coasting times, enabling backward propulsion.
Theoretical Implication: The work extends the understanding of the Scallop Theorem to granular media, showing that while the theorem holds for Newtonian fluids, granular complexity (elasto-plasticity and inertia) provides multiple pathways for net displacement.
Design Insight: The results suggest that swimmer geometry (e.g., adding an inert mid-body) and mass can be tuned to optimize locomotion direction and efficiency in granular environments.

Engine 1: The "Traffic Jam" Effect (Slow Swimming)

Engine 2: The "Coasting" Effect (Fast Swimming)

The Grand Finale: A Unified Picture

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