Clogging-flowing transition of granular media in a two-dimensional vertical pipe

We experimentally and numerically investigate the clogging behavior of granular materials in a two-dimensional vertical pipe. The nonmonotonicity of clogging probability found in a cylindrical vertical pipe [López et al., Phys. Rev. E 102, 010902 (2020)] is also observed in the two-dimensional case. We numerically demonstrate that the clogging probability strongly correlates with the friction coefficient $μ$ in addition to the pipe-to-particle diameter ratio $D/d$. We thus construct a clogging-flowing $D/d$-$μ$ phase diagram within the $2<D/d<3$ range. Finally, by analyzing the geometrical arrangements of particles and using a simple analysis of forces and torques, we are able to predict the clogging-flowing transition in the $D/d$-$μ$ phase diagram and explain the mechanism of the observed counterintuitive nonmonotonicity in more detail. We demonstrate that for pipe clogging, friction-mediated force and torque equilibrium are essential for arch formation.

💡 Research Summary

In this work the authors investigate the clogging‑flow transition of granular media flowing down a two‑dimensional vertical pipe, combining systematic experiments with discrete element method (DEM) simulations. The study was motivated by a recent report of a non‑monotonic clogging probability in a three‑dimensional cylindrical pipe and seeks to determine whether the same phenomenon occurs in a planar geometry and, if so, what physical mechanisms control it.

The experimental apparatus consists of a transparent rectangular pipe of height 300 mm, inner diameters D ranging from 20 to 36 mm, and a thickness of 7.5 mm, ensuring a monolayer of particles. Stainless‑steel disks of two diameters (d = 10 mm and 12 mm) are introduced one by one from the top until a prescribed number N (typically 40–50) is reached. After the particles settle, a binder clip sealing the bottom is removed, allowing gravity to drive the flow. A run is classified as “non‑clogging” if all particles exit the pipe; otherwise it is a “clogging” event. For each set of parameters the authors perform 100–300 repetitions, defining the clogging probability J = N_c/N_t.

DEM simulations are performed with the LMGC90 contact dynamics code. Particles are modeled as perfectly rigid, inelastic disks interacting via a Coulomb friction law with a single coefficient μ (identical for particle–particle and particle–wall contacts). The time step is 5 × 10⁻⁴ s and each simulation runs for up to 2000 steps (≈1 s). Clogging is identified when the maximum particle velocity falls below 10⁻⁶ m s⁻¹ while particles remain in the pipe. The same definitions of J and bulk packing fraction ϕ_b are applied to the simulated data, enabling a direct comparison with experiments.

Both experimental and numerical results reveal a striking non‑monotonic dependence of J on the diameter ratio D/d. Clogging is most likely for D/d ≈ 2.45, while it is strongly suppressed at D/d ≈ 2 and again at D/d ≈ 3. The authors identify a critical diameter ratio (D/d)_c ≈ 2.45 that separates a high‑clogging regime from a flowing regime for 2 < D/d < 3. Importantly, (D/d)_c shifts systematically with the friction coefficient: increasing μ lowers the critical ratio, meaning that more frictional particles clog at smaller pipe openings.

To rationalize these observations the authors focus on the geometry and mechanics of the most common clogging structure: a three‑particle arch spanning the pipe. By measuring the arch angle θ in experiments and simulations they confirm the geometric relation θ = arccos(0.5 D/d − 0.5), which follows from the non‑overlap condition of three disks in a planar channel. The distribution of θ narrows for smaller D/d and widens as D/d approaches 3, reflecting the decreasing likelihood of forming a stable three‑particle arch.

A force‑ and torque‑balance analysis is then carried out for a particle in contact with the wall and its upper neighbor. Assuming (i) the upper neighbor is the only particle contacting the wall particle and (ii) the wall particle contacts only that neighbor, the equilibrium equations are:

1. Vertical force balance: F₂ sinθ + G = f₁ + f₂ cosθ
2. Horizontal force balance: F₂ cosθ + f₂ sinθ = F₁
3. Coulomb limits: f₁ ≤ μ F₁, f₂ ≤ μ F₂
4. Torque balance about the wall contact point: f₁ = f₂

Solving these yields a friction condition μ ≥ 1 − cosθ sinθ. Substituting the geometric expression for θ gives a compact stability criterion for the arch:

 D/d ≥ max{ (3 − μ²)/(1 + μ²), 2 }.

For the experimentally relevant μ = 0.4 this predicts D/d ≥ 2.45, precisely the value observed for the clogging‑flow transition. The analysis shows that when μ is large enough, the dominant factor is the ability of friction to sustain the necessary tangential forces and torques; geometric constraints become secondary. Conversely, for very large D/d the probability of forming a three‑particle arch drops dramatically, making clogging unlikely regardless of μ.

Additional insight is obtained by counting contacts in the bulk during simulations. For 2 < D/d < 2.73 a wall particle typically contacts only one neighbor, whereas for 2.73 < D/d < 3 it contacts two. The latter situation demands larger tangential forces to balance gravity, explaining the abrupt drop in J near D/d ≈ 2.73 observed in the phase diagram.

The authors compile all results into a D/d–μ phase diagram for 2 < D/d < 3, showing a clear boundary between flowing and clogging regions that is accurately described by the theoretical line derived above. The diagram demonstrates that both geometric confinement and inter‑particle friction jointly control the transition, and that the simple three‑particle arch model captures the essential physics across the investigated parameter space.

In summary, this paper provides a comprehensive experimental and numerical characterization of clogging in a 2D vertical pipe, identifies the critical role of friction in shifting the clogging‑flow transition, and offers a concise analytical framework that predicts the transition line with quantitative accuracy. The findings have practical relevance for the design of narrow conduits in industrial processes (e.g., pneumatic transport, pharmaceutical handling, and nuclear fuel element transfer) where unexpected clogging can cause severe operational hazards.