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A constitutive law for dense granular flows

Abstract

A continuum description of granular flows would be of considerable help in predicting natural geophysical hazards or in designing industrial processes. However, the constitutive equations for dry granular flows, which govern how the material moves under shear, are still a matter of debate1,2,3,4,5,6,7,8,9,10. One difficulty is that grains can behave11 like a solid (in a sand pile), a liquid (when poured from a silo) or a gas (when strongly agitated). For the two extreme regimes, constitutive equations have been proposed based on kinetic theory for collisional rapid flows12, and soil mechanics for slow plastic flows13. However, the intermediate dense regime, where the granular material flows like a liquid, still lacks a unified view and has motivated many studies over the past decade14. The main characteristics of granular liquids are: a yield criterion (a critical shear stress below which flow is not possible) and a complex dependence on shear rate when flowing. In this sense, granular matter shares similarities with classical visco-plastic fluids such as Bingham fluids. Here we propose a new constitutive relation for dense granular flows, inspired by this analogy and recent numerical15,16 and experimental work17,18,19. We then test our three-dimensional (3D) model through experiments on granular flows on a pile between rough sidewalls, in which a complex 3D flow pattern develops. We show that, without any fitting parameter, the model gives quantitative predictions for the flow shape and velocity profiles. Our results support the idea that a simple visco-plastic approach can quantitatively capture granular flow properties, and could serve as a basic tool for modelling more complex flows in geophysical or industrial applications.

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Figure 1: Friction coefficient µ as a function of the dimensionless parameter I (µs = tan(20.9), µ2 = tan(32.76) and I0 = 0.279).
Figure 2: Experimental set-up of granular flows on a pile between rough sidewalls.
Figure 3: Typical 3D velocity profile predicted by the rheology ( W = 142 d, θ = 22.6°, Q/d3/2g1/2 = 15.2).
Figure 4: Comparison of 3D simulations (lines) and experimental results (symbols) for different flow rates ( Q* = Q/d3/2g1/2).
Figure 5: Scaling laws for the experimental measurements (symbols) compared to the predictions of the model (lines).

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Acknowledgements

This work was supported by grants from CEFIPRA and from the French ANR (PIGE project). Author Contributions P.J. performed all experimental work and numerical simulations; P.J., Y.F. and O.P analysed results and co-wrote the paper.

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Correspondence to Pierre Jop.

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Supplementary information

Supplementary Notes

This section includes the derivation of the scaling laws from the 3D rheology, and a Supplementary Discussion about the transition to a collisional regime observed for high inclination (see also supplementary movies). (PDF 385 kb)

Supplementary Video 1

Side-view video of the flowing layer taken at 2500 frames s-1 and played back at 10 frames s-1. The inclination of the free surface is 32 degrees, below the critical angle θ2. (MOV 1294 kb)

Supplementary Video 2

Side-view video of the flowing layer taken at 2500 frames s-1 and played back at 10 frames s-1. The inclination of the free surface is 34 degrees, above the critical angle θ2. (MOV 1271 kb)

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Jop, P., Forterre, Y. & Pouliquen, O. A constitutive law for dense granular flows. Nature 441, 727–730 (2006). https://doi.org/10.1038/nature04801

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