Granular Matter

, Volume 16, Issue 4, pp 517–530 | Cite as

Subdiffusive behavior in a two-dimensional planar shear granular flow

  • J. M. SalazarEmail author
Original Paper


We study the rheology of a compact ensemble of beads in a two-dimensional planar shear configuration by means of molecular dynamics. The rheology obtained shows a plateau for low shear rates. In this parameter region we observe the presence of a collective behavior, sticking agglomerates of particles, characterized by a velocity lower than the mean flow velocity. The lifetime of this blocking regions can stand several thousands time units. Moreover, our analysis of the mean square displacement shows a growth of the form \(\langle (\Delta r({\tau }))^2\rangle \sim {\tau }^{\beta }\) with \(\beta < 1.0\). This indicates the presence of a subdiffusive flow in the quasi-static regime characterized by the presence of vortices. In this work we detail how the observed transient agglomerates of particles deform the flow and we used a local relaxation time analysis for characterizing the inhomogeneities.


Shear granular Granular flow Vortex Spatial instabilities Molecular dynamics Anomalous diffusion 



The author would like to thanks the comments and remarks of Prof. L. Brenig. Also a special thanks to F. Chevoir for the enlightening discussions held with him during the development of this work. The simulations were performed at the Université de Bourgogne computer facilities.


  1. 1.
    Jenkins, J.T., Savage, S.B.: A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187 (1983)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Wolfson, D., Tsimring, L.S., Aranson, I.S.: Partially fluidized shear granular flows: continuum theory and molecular dynamics simulations. Phys. Rev. E 68, 021301 (2003)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ertaz, D., Halsey, T.C.: Granular gravitational collapse and chute flow. Eurphys. Lett. 60(6), 931–937 (2002)ADSCrossRefGoogle Scholar
  4. 4.
    Iordanoff, I., Khonsari, M.M.: Granular lubrification: toward and understanding between kinetic and dense regime. ASME. J. Tribol. 126, 137–145 (2004)CrossRefGoogle Scholar
  5. 5.
    Barenblatt, G.I.: Scaling, Self Similarity, and Intermediate Asymptotics. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  6. 6.
    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)ADSCrossRefGoogle Scholar
  7. 7.
    Savage, S.B.: The mechanics of rapid granular flows. Adv. Appl. Mech. 24, 289–366 (1984)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bagnold, R.: Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. Ser. A 225, 49 (1954)Google Scholar
  9. 9.
    Midi, G.D.R.: On dense granular flows. Eur Phys J. E 14, 341 (2004)CrossRefGoogle Scholar
  10. 10.
    Johnson, P.C., Jackson, R.: Frictional–collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 67 (1987)ADSCrossRefGoogle Scholar
  11. 11.
    Louge, M.Y.: Model for dense granular flows down bumpy inclines. Phys. Rev. E 67, 061303 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    Pouliquen, O., Forterre, Y., Le Dizes, S.: Slow dense granular flows as a self-induced process. Adv. Complex Syst. 4, 441 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Bocquet, L., Losert, W., Schalk, D., Lubensky, T.C., Gollub, J.P.: Granular shear flow dynamics and forces: experiment and continuum theory. Phys. Rev. E 65, 011307 (2002)ADSCrossRefGoogle Scholar
  14. 14.
    Yop, P., Forterre, Y., Pouliquen, O.: A constitutive law for dense granular flows. Nature 441(8), 727 (2003)Google Scholar
  15. 15.
    Koval, G., Roux, J.-N.: Annular shear of cohesionless granular materials: from the inertial to quasistatic regime. Phys. Rev. E 79, 021306 (2009) Google Scholar
  16. 16.
    Silbert, L.E., Ertaz, D., Grest, G.S.: Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 051302 (2001)ADSCrossRefGoogle Scholar
  17. 17.
    Da Cruz, F.: Ph.D. thesis, Ecole Nationale de Ponts et Chaussées, France (2004)Google Scholar
  18. 18.
    Xu, N., O’Hern, C.S., Kondic, L.: Stabilization of nonlinear velocity profiles in athermal systems undergoing planar shear flow. Phys. Rev. E 72, 041504 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Radjai, F., Roux, S.: Turbulentlike fluctuations in quasistatic flow of granular media. Phys. Rev. Lett. 89, 064302 (2003)ADSCrossRefGoogle Scholar
  20. 20.
    Baran, O., Ertaz, D., Halsey, T.C., Grest, G.S., Lechman, J.B.: Velocity correlations in dense gravity-driven granular chute flow. Phys. Rev. E 74, 051302 (2006)ADSCrossRefGoogle Scholar
  21. 21.
    Staron, L.: Correlated motion in the bulk of dense granular flows. Phys. Rev. E 77, 051304 (2008)ADSCrossRefGoogle Scholar
  22. 22.
    Pouliquen, O.: Velocity correlations in dense granular flows. Phys. Rev. Lett. 93, 248001 (2004)ADSCrossRefGoogle Scholar
  23. 23.
    Staron, L., Lagrée, P.Y., Josserand, C., Lhuillier, D.: Flow and jamming of a two-dimensional granular bed: toward a nonlocal rheology? Phys. Fluids 22, 113303 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    Bonamy, D., Diviaud, F., Laurent, L.: Multiscale clustering in granular surface flows. Phys. Rev. Lett. 89, 034301 (2002)ADSCrossRefGoogle Scholar
  25. 25.
    Richefeu, V., Combe, G., Viggiani, G.: An Experimental Assessment of Displacement Fluctuations in a 2D Granular Material Subjected to Shear. Géotechnique Letters, vol. 2 (2012)Google Scholar
  26. 26.
    Pöschel, T., Schwager, T.: Computational Granular Dynamics (Models and Algorithms). Springer, Berlin (2005)Google Scholar
  27. 27.
    Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. Géotechnique 29(1), 47–65 (1979)CrossRefGoogle Scholar
  28. 28.
    Savage, S.B., Sayed, M.: Stresses developed by dry cohesionless granular materials sheared in an annular shear cell. J. Fluid. Mech. 142, 391–430 (1984)ADSCrossRefGoogle Scholar
  29. 29.
    Lun, C.K.K., Bent, A.A.: Numerical simulation of inelastic frictional spheres in simple shear flow. J. Fluid Mech. 258, 335–353 (1994)ADSCrossRefGoogle Scholar
  30. 30.
    Reis, P.M., Ingale, R.A., Shattuck, M.D.: Caging dynamics in a granular fluid. Phys. Rev. Lett. 98, 188301 (2007)ADSCrossRefGoogle Scholar
  31. 31.
    Solomon, T.H., Weeks, E.R., Swinney, H.L.: Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. Phys. Rev. Lett. 71, 3975–3978 (1993)ADSCrossRefGoogle Scholar
  32. 32.
    Hurley, M., Harrowell, P.: Kinetic structure of a two-dimensional liquid. Phys. Rev. E 52, p1694 (1995)ADSCrossRefGoogle Scholar
  33. 33.
    Marty, G., Dauchot, O.: Subdiffusion and cage effect in a sheared granular material. Phys. Rev. Lett. 94, 015701 (2005)ADSCrossRefGoogle Scholar
  34. 34.
    Alder, B.J., Wainwright, T.E.: Velocity autocorrelations for hard spheres. Phys. Rev. Lett. 18, 988 (1967)ADSCrossRefGoogle Scholar
  35. 35.
    Ernst, M.H., Hauge, E.H., Van Leeuven, J.M.J.: Asymptotic time behavior of correlation functions and mode-mode coupling theory. Phys. Lett. A 34, 419 (1971)ADSCrossRefGoogle Scholar
  36. 36.
    Shlesinger, M.F., Zaslavsky, G.M., Klafter, J.: Strange kinetics. Nature 363, 31 (1993)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR-6303CNRS-Université de BourgogneDijon CedexFrance

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