Scaling laws for velocity profile of granular flow in rotating drums

This paper uses discrete element method simulations and $\mu(I)$-rheology continuum modeling to demonstrate that the velocity profile and surface flow layer thickness in rotating drums follow specific scaling laws derived through dimensional analysis.

The Gist

Imagine you are looking at a giant, rotating drum filled with sand or marbles—the kind you might see in a factory mixing ingredients or a construction site moving gravel.

If you watch closely, you’ll notice something interesting: the sand doesn't just move as one big clump. Instead, it behaves like two different worlds living in one drum. Near the top, there is a thin, fast-moving "river" of sand sliding down the slope. But underneath that river, the sand is actually "frozen" in place, moving in a slow, heavy circle along with the drum itself.

Scientists have long struggled to predict exactly how thick that "river" of sand will be. Does it get thicker if the drum is bigger? Does it change if you spin it faster? For a long time, different experiments gave different answers, leaving everyone a bit confused.

The "Recipe" for the Sand River

In this paper, researchers Hiroki Oba and Michio Otsuki set out to find the "universal recipe" for this flow. To do this, they used two different methods:

1. The "Micro" View (DEM): They used supercomputers to simulate every single tiny grain of sand, watching how they bump and grind against each other.
2. The "Macro" View (Continuum Model): Instead of looking at grains, they treated the sand like a thick, gooey liquid (kind of like honey or lava) and used math to describe how it flows.

The Big Discovery: The Scaling Law

The researchers discovered that if you have a large enough drum, the thickness of that top "river" follows a very predictable rule.

Think of it like scaling a recipe for a cake. If you want to make a cake for 10 people versus 100 people, you don't just add more flour; you scale everything up proportionally so the texture stays the same.

The researchers found that:

• The size matters most: If you double the diameter of the drum, the thickness of the sand river also roughly doubles. It scales linearly with the size of the container.
• Speed is a minor player: While spinning the drum faster does change things slightly, it doesn't have a massive impact on the thickness compared to the size of the drum itself.

Why does this matter?

Imagine you are an engineer designing a massive industrial mixer for a pharmaceutical company. If you don't know how thick that "river" of powder is, you might not mix the medicine correctly, leading to a bad batch.

Before this paper, engineers had to rely on "rules of thumb" or expensive, time-consuming experiments. This study provides a mathematical "cheat sheet." It tells them: "If you know how big your drum is and how fast it's spinning, you can use this formula to predict exactly how the material will behave."

In short: The researchers turned a messy, chaotic pile of moving sand into a predictable, mathematical dance, proving that even in a world of billions of tiny grains, there is a beautiful, simple order underneath.

Technical Summary

Technical Summary: Scaling Laws for Velocity Profile of Granular Flow in Rotating Drums

Authors: Hiroki Oba and Michio Otsuki
Date: February 10, 2026 (as per manuscript)

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1. Problem Statement

Granular flows in rotating drums are critical to various industrial processes (mixing, drying, granulation). In the "rolling state" (low Froude number, $10^{-4} < \text{Fr} < 10^{-2}$), the flow is characterized by two distinct regions: a thin surface flow layer where particles cascade down a slope, and a static flow regime where particles undergo rigid-body rotation with the drum.

Despite extensive experimental and numerical studies, a consistent theoretical explanation for how the thickness of the surface flow layer ($h$) scales with the drum diameter ($D$) and angular velocity ($\Omega$) has been lacking. Previous studies reported conflicting results—some suggesting $h$ scales linearly with $D$ and is independent of $\Omega$, while others suggested power-law dependencies on $\Omega$—often due to variations in system geometry, sidewall effects, or inertial regimes.

2. Methodology

The authors employ a dual-approach strategy to bridge the gap between particle-level dynamics and macroscopic continuum behavior:

• Discrete Element Method (DEM): Used to provide high-fidelity, particle-scale benchmark data. The simulations involve polydisperse particles in a 2D rotating drum with diameters $D = 150d, 300d,$ and $450d$.
• Continuum Modeling ($\mu(I)$-rheology): The granular bed is treated as an incompressible, non-Newtonian fluid governed by mass and momentum conservation equations. The constitutive relation is the $\mu(I)$-rheology, where the effective friction coefficient $\mu$ depends on the inertial number $I$.
• Computational Fluid Dynamics (CFD): The continuum equations are solved numerically using the Volume of Fluid (VoF) method and the Simplified Marker and Cell (SMAC) method to capture the free surface and velocity fields.
• Dimensional Analysis: The authors use the continuum equations to perform a scaling analysis, seeking to identify dimensionless parameters that govern the velocity profile and layer thickness.

3. Key Contributions

• Validation of the Continuum Model: The study demonstrates that the $\mu(I)$-rheology-based continuum model quantitatively reproduces the velocity fields obtained from DEM simulations across a wide range of $D$ and $\Omega$.
• Analytical Derivation of Scaling Laws: By non-dimensionalizing the governing equations in the limit of large $D/d$, the authors analytically derive that the velocity profile and the surface layer thickness are functions of the Froude number ($\text{Fr}$) and the particle-to-drum ratio ($d/D$).
• Resolution of Scaling Discrepancies: The paper provides a theoretical basis for why $h$ appears to scale with $D$ and remain nearly independent of $\Omega$ in large, 2D systems, while explaining that deviations in other studies likely stem from sidewall friction, axial inhomogeneities, or non-local rheological effects.

4. Results

• Velocity Profile Scaling: The normalized velocity $u/\Omega D$ collapses onto a single curve when plotted against $z/D$ for different drum diameters, confirming the scaling law:
  $$\frac{u(z)}{\Omega D} = U\left(\frac{z}{D}; \text{Fr}\right)$$
• Surface Layer Thickness Scaling: The dimensionless thickness $h/D$ is found to be a function solely of the Froude number:
  $$\frac{h}{D} = H(\text{Fr})$$
• Dependence on $\Omega$ and $D$: For low Froude numbers ($\text{Fr} \ll 1$), the scaling function $H(\text{Fr})$ is nearly constant. This implies that $h$ is proportional to $D$ and nearly independent of $\Omega$.
• Numerical Confirmation: Both DEM and CFD results show excellent agreement with the derived scaling laws, with $h/D$ plotted against $\text{Fr}$ showing a nearly horizontal trend (weak dependence on $\Omega$).

5. Significance

This work provides a much-needed theoretical framework for understanding steady granular flows in rotating drums. By moving beyond purely numerical observations (DEM) and providing analytical scaling laws via continuum mechanics, the authors offer a predictive tool for industrial applications.

The study's significance lies in its ability to rationalize the discrepancies in existing literature: it suggests that the "ideal" scaling behavior is a feature of large-scale, 2D-like flows, and that the power-law dependencies observed in other experiments are likely "corrections" caused by finite-size effects (sidewalls) or non-local granular physics. This provides a systematic way to interpret experimental deviations from ideal scaling.