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A constitutive law for dense granular flows

Naturevolume 441pages727730 (2006) | Download Citation



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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  1. 1

    Savage, S. B. Analysis of slow high-concentration flows of granular materials. J. Fluid Mech. 377, 1–26 (1998)

  2. 2

    Mills, P., Loggia, D. & Texier, M. Model for stationary dense granular flow along an inclined wall. Europhys. Lett. 45, 733–738 (1999)

  3. 3

    Aranson, I. S. & Tsimring, L. S. Continuum description of avalanches in granular media. Phys. Rev. E 64, 020301 (2001)

  4. 4

    Pouliquen, O., Forterre, Y. & Le Dizes, S. Slow dense granular flows as a selfinduced process. Adv. Complex Syst. 4, 441–450 (2001)

  5. 5

    Bocquet, L., Losert, W., Schalk, D., Lubensky, T. C. & Gollub, J. P. Granular shear flow dynamics and forces: Experiment and continuous theory. Phys. Rev. E 65, 011307 (2002)

  6. 6

    Ertas, D. & Halsey, T. C. Granular gravitational collapse and chute flow. Europhys. Lett. 60, 931–937 (2002)

  7. 7

    Lemaître, A. Origin of a repose angle: kinetics of rearrangement for granular materials. Phys. Rev. Lett. 89, 064303 (2002)

  8. 8

    Mohan, L. S., Rao, K. K. & Nott, P. R. A frictional cosserat model for the slow shearing of granular materials. J. Fluid Mech. 457, 377–409 (2002)

  9. 9

    Josserand, C., Lagrée, P.-Y. & Lhuillier, D. Stationary shear flows of dense granular materials: a tentative continuum modelling. Eur. Phys. J. E 14, 127–135 (2004)

  10. 10

    Kumaran, V. Constitutive relations and linear stability of a sheared granular flow. J. Fluid Mech. 506, 1–43 (2004)

  11. 11

    Jaeger, H. M., Nagel, S. R. & Behringer, R. P. Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259–1273 (1996)

  12. 12

    Goldhirsch, I. Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293 (2003)

  13. 13

    Nedderman, R. M. Static and kinematics of granular materials. (Cambridge Univ. Press, Cambridge, UK, 1992)

  14. 14

    GDR MiDi group, On dense granular flows. Eur. Phys. J. E 14, 341–365 (2004)

  15. 15

    Da Cruz, F., Emam, S., Prochnow, M., Roux, J.-N. & Chevoir, F. Rheophysics of dense granular materials: Discrete simulation of plane shear flows. Phys. Rev. E 72, 021309 (2005)

  16. 16

    Iordanoff, I. & Khonsari, M. M. Granular lubrication: toward an understanding between kinetic and fluid regime. ASME J. Tribol. 126, 137–145 (2004)

  17. 17

    Pouliquen, O. & Forterre, Y. Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. J. Fluid Mech. 453, 133–151 (2002)

  18. 18

    Forterre, Y. & Pouliquen, O. Long-surface-wave instability in dense granular flows. J. Fluid Mech. 486, 21–50 (2003)

  19. 19

    Jop, P., Forterre, Y. & Pouliquen, O. Crucial role of sidewalls in dense granular flows: consequences for the rheology. J. Fluid Mech. 541, 167–192 (2005)

  20. 20

    Savage, S. B. The mechanics of rapid granular flows. Adv. Appl. Mech. 24, 289–366 (1984)

  21. 21

    Ancey, C., Coussot, P. & Evesque, P. A theoretical framework for very concentrated granular suspensions in a steady simple shear flow. J. Rheol. 43, 1673–1699 (1999)

  22. 22

    Pouliquen, O. Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11, 542–548 (1999)

  23. 23

    Silbert, L. E. et al. Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 051302 (2001)

  24. 24

    Drucker, D. C. & Prager, W. Soil mechanics and plastic analysis of limit design. Q. Appl. Math. 10(2), 157–175 (1952)

  25. 25

    Tanner, R. I. Engineering Rheology (Clarendon Press, Oxford, USA, 1985)

  26. 26

    Mueth, D. M. et al. Signature of granular microstructure in dense shear flows. Nature 406, 385–389 (2000)

  27. 27

    Fenistein, D. & Van Hecke, M. Wide shear zones in granular bulk flow. Nature 425, 256 (2003)

  28. 28

    Howell, D., Behringer, R. P. & Veje, C. Stress fluctuations in a 2D granular couette experiment: a continuous transition. Phys. Rev. Lett. 82, 5241–5244 (1999)

  29. 29

    Daerr, A. & Douady, S. Two types of avalanche behaviour in granular media. Nature 399, 241–243 (1999)

  30. 30

    Börzsönyi, T., Halsey, T. C. & Ecke, R. E. Two scenarios for avalanche dynamics in inclined granular layers. Phys. Rev. Lett. 94, 208001 (2005)

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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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  1. IUSTI, CNRS UMR 6595, Université de Provence, Marseille, 5 rue Enrico Fermi, 13453 cedex 13, France

    • Pierre Jop
    • , Yoël Forterre
    •  & Olivier Pouliquen


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Competing interests

Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.

Corresponding author

Correspondence to Pierre Jop.

Supplementary information

  1. 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)

  2. 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)

  3. 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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