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Bulldozing an immersed granular material in a confined channel

This paper presents and validates a reduced-order continuum model, supported by numerical simulations and experiments, to describe the bulldozing of an immersed granular pile in a confined channel by treating the system as coupled thin films of visco-plastic grains and fluid.

Original authors: Liam C. Morrow, Oliver W. Paulin, Matthew G. Hennessy, Duncan R. Hewitt, Miles L. Morgan, Bjørnar Sandnes, Christopher W. MacMinn

Published 2026-08-13

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CC BY 4.0

Original authors: Liam C. Morrow, Oliver W. Paulin, Matthew G. Hennessy, Duncan R. Hewitt, Miles L. Morgan, Bjørnar Sandnes, Christopher W. MacMinn

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 a world where sand doesn't just sit still; it can flow like a liquid, pile up like a solid, or even act like a gas, depending on how much you squeeze it. This is the strange and wonderful realm of granular materials—think of everything from the sugar in your kitchen to the sand on a beach. Scientists have long studied how these materials move, especially when they are "dry," sliding against each other like a crowd of people shoving through a door. But what happens when you add water? When sand is soaked, the liquid between the grains acts like a lubricant, but it also creates resistance, like trying to run through a pool. This mix of solid grains and fluid is everywhere in nature, from underwater landslides that threaten coastlines to the way blood flows through tiny vessels in our bodies. Understanding how this "wet sand" moves is crucial for predicting natural disasters and designing better industrial machines.

Now, picture a giant, invisible bulldozer pushing through a narrow hallway filled with this wet sand. As the bulldozer (or a piston, in scientific terms) shoves forward, the sand doesn't just slide away; it piles up, gets squished, and sometimes even gets stuck, bridging the gap between the floor and the ceiling of the hallway. This is the puzzle that Liam Morrow and his team at Oxford, Bristol, Cambridge, and Swansea set out to solve. They wanted to know: How does the water trapped in the sand change the way the pile forms, and how much force does it take to keep pushing?

The team built a clever computer model to act as a virtual laboratory. Instead of just guessing, they treated the sand and the water as two separate but connected layers of a thin film, like a sandwich where the bottom slice is the squishy, wet sand and the top slice is the free-flowing water. They used a set of rules called "rheology" (which is just a fancy word for how materials flow) to describe how the sand grains rub against each other and the walls. They tested two main scenarios: one where the sand is so fast-moving that the water barely matters (the "dry" limit), and another where the water is thick and sticky, dominating the movement (the "wet" limit).

Here is what they found, and it's a bit like a rollercoaster ride for the sand pile.

First, in the "dry" or fast-moving world, the sand behaves a bit like a stubborn crowd. When the piston pushes, the sand piles up into a wedge shape. If the friction between the sand and the walls is low, the whole pile slides easily. But if the walls are rough, the sand gets stuck in the middle, forming a "plug" that moves as a solid block while the grains near the walls slide past it. The most surprising thing here is that the force required to push the piston grows exponentially the longer the pile gets. It's like trying to push a snowplow: the longer the snowbank gets, the harder it becomes to move, not just a little bit harder, but much harder, very quickly.

Then, they turned up the water. In the "wet" or slow-moving world, the water becomes the boss. At first, when the water is thin or the piston moves slowly, the sand behaves much like the dry version. But as the water gets stickier (or the piston moves faster), something weird happens. The pile of sand doesn't just get shorter or steeper; it starts to stretch out. The water trapped above the sand acts like a heavy blanket, resisting the sand's attempt to pile up high. This creates a long, flat transition zone where the sand slowly rises.

The team discovered a "Goldilocks" zone for the length of this pile. As they increased the stickiness of the water (represented by a number called $\theta$ ), the pile first got slightly shorter, but then, suddenly, it started to grow longer and longer. If the water was too sticky or the piston too fast, the sand simply couldn't pile up high enough to touch the ceiling. The water layer would just stay on top, and the sand would never "bridge" the gap. It's like trying to build a sandcastle in a tsunami; no matter how hard you try, the water keeps flattening your tower.

To make sure their computer model wasn't just playing pretend, the team went into the lab. They built a narrow, clear tube (3 mm wide and 3 mm tall) and pushed wet glass beads through it using a piston. They used water and a thick glycerol-water mixture to change the stickiness. The results matched their simulations beautifully. When they used thin water, the sand piled up in a neat, triangular wedge. When they used the thick glycerol, the pile stretched out, just as the model predicted. They even found that the sides of the tube played a sneaky role; the friction against the side walls made the pile shorter in real life than in the simple computer model, which only looked at the front and back.

So, what's the big takeaway? The paper suggests that the behavior of wet sand isn't just a simple mix of sand and water rules. It's a complex dance where the water can either help the sand slide or hold it back, creating a pile that can suddenly stretch out or collapse depending on how fast you push and how thick the fluid is. While the model is a strong simulation that matches real-world experiments, the authors note that in the real world, things like the sand getting squished tighter or the water swirling in circles might add even more complexity. But for now, we have a much clearer picture of how to bulldoze a wet, sandy world without getting stuck.

Technical Summary: Bulldozing an Immersed Granular Material in a Confined Channel

Problem Statement
This study investigates the mechanics of an immersed granular material being "bulldozed" by a piston within a confined, horizontal channel. The system consists of a dense, fluid-saturated granular layer initially resting on the bottom plate of a channel with a gap height $b$ . A rigid piston moves at a constant velocity, displacing both the grains and the interstitial fluid. The problem is characterized by complex interactions between grain-grain friction, grain-fluid drag, and grain-wall friction. The specific focus is on the formation of a "bridged" region where grains pile up to fill the channel gap, a transition zone where the granular layer height increases from its initial value to the gap height, and the undisturbed far-field. This work generalizes previous studies on dry granular bulldozing (which ignored fluid) and capillary-driven granular displacement (which often relied on rule-based simulations) by developing a continuum model for the coupled fluid-grain system.

Methodology
The authors develop a reduced-order continuum model based on a depth-averaged, two-phase, two-layer approach. The system is treated as a coupled set of thin films: an upper layer of incompressible Newtonian fluid and a lower layer of fluid-saturated granular material.

Governing Equations: Conservation of mass and momentum are applied under a lubrication approximation (assuming the channel is thin and piston motion is slow, such that macroscopic inertia is negligible). The granular phase is modeled as a dense, porous, visco-plastic material.
Rheology: The internal mechanics of the granular phase are described using $\mu(I)$ $μ (I)$ -type rheologies. The authors consider two regimes distinguished by the Stokes number ($St$):
- Dry Limit ( $St \gg 1$ ): Inertial effects dominate; the granular viscosity is described by the inertial number $I$ .
- Wet Limit ( $St \ll 1$ ): Viscous effects dominate; the granular viscosity is described by the viscous number $I_v$ .
Boundary Conditions: The model incorporates Coulomb-like friction at the top and bottom walls. Grains slide only when the local shear stress exceeds a threshold defined by the local normal stress and a wall friction coefficient ( $\mu_w$ ). A key challenge addressed is the transition between unbridged (fluid-shear dominated) and bridged (wall-friction dominated) states.
Regularization: To handle the discontinuous nature of yield-stress fluids and wall slip (where regions of sliding and sticking coexist), the authors introduce a novel regularization scheme using hyperbolic tangent functions. This allows for a single, continuous numerical solution without explicitly tracking free boundaries between yielded and unyielded regions.
Numerical and Experimental Validation: The model is solved numerically in a frame moving with the piston. Results are compared against a series of laboratory experiments conducted in a square glass tube using glass beads and interstitial fluids (water and glycerol solutions) at varying piston speeds.

Key Contributions and Results

Dry Regime ( $St \gg 1$ ):
- In the bridged region, the granular velocity profile is non-monotonic, featuring a central "plugged" region (where grains move as a rigid body) sandwiched between yielded regions near the walls.
- The normal stress at the piston grows exponentially with the length of the bridged region, consistent with the classical Janssen effect and previous discrete-element method (DEM) simulations.
- The shape of the transition zone (the wedge of rising grains) depends on the Froude number ($Fr$). As $Fr$ increases, the transition zone shortens, and the slope of the granular pile steepens, approaching the wall friction angle.
Wet Regime ( $St \ll 1$ ):
- The presence of the fluid introduces a non-monotonic behavior in the length of the transition zone ( $\lambda$ ) as a function of the modified Shields parameter ( $\theta$ ), which measures fluid shear relative to grain weight.
- At low $\theta$ , the fluid has negligible impact, and the system behaves similarly to the dry case.
- As $\theta$ increases, viscous resistance from the overlying fluid layer elongates the transition zone.
- Beyond a critical $\theta$ , the model predicts that a steady travelling-wave solution for the transition zone ceases to exist; the zone lengthens continually, and for sufficiently high $\theta$ , the grains fail to bridge the gap entirely.
Experimental Comparison:
- The model shows broad qualitative agreement with experiments regarding the shape of the granular pile and the non-monotonic variation of the transition zone length with $\theta$ .
- Quantitative discrepancies in the transition zone length were observed. The authors attribute this to the omission of side-wall friction in the base model. When side-wall friction is included in the model, agreement with experimental data improves significantly.
- Velocity profiles measured via Particle Image Velocimetry (PIV) generally match the model's prediction of monotonically increasing velocity with height in the transition zone, though the model predicts a convex curvature ( $\partial^2_z u_s < 0$ ) not fully captured in the experiments.

Significance and Claims
The paper claims to provide the first quasi-1D/2D continuum model derived from fundamental mechanics for the bulldozing of immersed granular materials in confined geometries. It successfully bridges the gap between dry granular flow theories and fluid-grain interaction problems.

The authors modestly state that their model captures the essential physics of the interactions between wall friction, internal viscous-like stresses, and fluid flow. They highlight that the regularization technique allows for the efficient simulation of complex frictional flow patterns without the need for free-boundary tracking. While the model reproduces key experimental trends, the authors acknowledge limitations, including the assumption of constant solid fraction (ignoring compaction/dilation), the neglect of macroscopic inertia, and the simplified treatment of side-wall effects. They suggest that future work could extend this framework to incorporate non-wetting fluids to better model capillary-driven granular displacement in Hele-Shaw cells and capillary tubes.

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