Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

A constitutive law for dense granular flows


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.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

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).


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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    ADS  MathSciNet  CAS  MATH  Google Scholar 

  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)

    Article  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

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

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Pierre Jop.

Ethics declarations

Competing interests

Reprints and permissions information is available at The authors declare no competing financial interests.

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)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jop, P., Forterre, Y. & Pouliquen, O. A constitutive law for dense granular flows. Nature 441, 727–730 (2006).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing