Granular Matter

, Volume 14, Issue 2, pp 229–234 | Cite as

Surface curvature of steady granular flows

  • Jim N. McElwaineEmail author
  • Daisuke Takagi
  • Herbert E. Huppert
Original Paper


Laboratory experiments have shown that the steady flow of granular material down a rough inclined plane has a surface that is not parallel to the plane, but has a curvature across the slope with the height increasing toward the middle of the flow. We study this observation by postulating a new granular rheology, similar to that of a second order fluid. This model is applied to the experiments using a shallow water approximation, given that the depth of the flow is much smaller than the width. The model predicts that a second normal stress difference allows cross-slope height variations to develop in regions with considerable cross-slope velocity shear, consistent with the experiments. The model also predicts the development of lateral eddies, which are yet to be observed.


Shallow granular flow Normal stress difference Levée formation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Felix G., Thomas N.: Relation between dry granular flow regimes and morphology of deposits: formation of levees in pyroclastic deposits. Earth Planet Sci. Lett. 221, 197 (2004)ADSCrossRefGoogle Scholar
  2. 2.
    Deboeuf S., Lajeunesse E., Dauchot O., Andreotti B.: Flow rule, self-channelization, and levees in unconfined granular flows. Phys. Rev. Lett. 97, 158303 (2006)ADSCrossRefGoogle Scholar
  3. 3.
    Takagi D., McElwaine J.N., Huppert H.H.: Shallow granular flows. Phys. Rev. E 83, 031306 (2011)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Tanner R.I.: Some methods for estimating the normal stress functions in viscometric flows. J. Rheol. 14, 483–507 (1970)ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Bird R.B., Armstrong R.C., Hassager O.: Dynamics of Polymeric Liquids. 2 edn. Volume 1. Wiley, (1987)Google Scholar
  6. 6.
    Singh A., R.Nott P.: Experimental measurements of the normal stresses in sheared stokesian suspensions. J. Fluid Mech. 490, 293–320 (2003)ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Zarraga I.E., Hill D.A., Leighton D.T.: The characterization of the total stress of concentrated suspensions of noncolloidal spheres in newtonian fluids. J. Rheol. 44, 185 (2000)ADSCrossRefGoogle Scholar
  8. 8.
    Bagnold R.A.: Experiments on a gravity-free dispersion of large solid spheres in a newtonian fluid under shear. Proc. R. Soc. Lond. A Math. Phys. Sci. 225, 49–63 (1954)ADSCrossRefGoogle Scholar
  9. 9.
    Boyer F., Pouliquen O., Guazzelli E.: Dense suspensions in rotating-rod flows: normal stresses and particle migration. J. Fluid Mech. 686, 5–25 (2011)ADSCrossRefGoogle Scholar
  10. 10.
    Campbell C.S.: Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57–92 (1990)ADSCrossRefGoogle Scholar
  11. 11.
    Silbert L.E., Ertaz D., Grest G.S., Halsey T.C., Levine D., Plimpton S.J.: Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 051302 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    Börzsönyi T., Ecke R.E., McElwaine J.N.: Patterns in flowing sand: understanding the physics of granular flow. Phys. Rev. Lett. 103, 178302 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    Goldhirsch I., Sela N.: Origin of normal stress differences in rapid granular flows. Phys. Rev. E 54, 4458–4461 (1996)ADSCrossRefGoogle Scholar
  14. 14.
    Goldhirsch I.: Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293 (2003)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    McElwaine J.N., Nishimura K.: Ping-pong ball avalanche experiments. Ann. Glaciol. 32, 241–250 (2001)ADSCrossRefGoogle Scholar
  16. 16.
    Depken M., Lechman J.B., van Hecke M., van Saarloos W., Grest G.S.: Stresses in smooth flows of dense granular media. EPL (Europhys. Lett.) 78, 58001 (2007)ADSCrossRefGoogle Scholar
  17. 17.
    Jop P., Forterre Y., Pouliquen O.: A constitutive law for dense granular flows. Nature 441, 727 (2006)ADSCrossRefGoogle Scholar
  18. 18.
    Leal L.G.: The slow motion of slender rod-like particles in a second-order fluid. J. Fluid Mech. 69, 305–337 (1975)ADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Truesdell C.A.: A First Course in Rational Continuum Mechanics. Academic Press, New York (1977)zbMATHGoogle Scholar
  20. 20.
    Rajchenbach J.: Dense, rapid flows of inelastic grains under gravity. Phys. Rev. Lett. 90, 144302 (2003)ADSCrossRefGoogle Scholar
  21. 21.
    Pouliquen O., Forterre Y.: A non-local rheology for dense granular flows. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 367, 5091 (2009)ADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Pouliquen O.: Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11, 542 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Dalziel, S.B.: Digiflow, dl research partners, version 0.7 (2003)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Jim N. McElwaine
    • 1
    Email author
  • Daisuke Takagi
    • 1
  • Herbert E. Huppert
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsInstitute of Theoretical Geophysics, Centre for Mathematical Sciences, University of CambridgeCambridgeUK

Personalised recommendations