Granular flow down an inclined plane: Bagnold scaling and rheology

Leonardo E. Silbert; Deniz Ertaş; Gary S. Grest; Thomas C. Halsey; Dov Levine; Steven J. Plimpton

doi:10.1103/PhysRevE.64.051302

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Granular flow down an inclined plane: Bagnold scaling and rheology

Leonardo E. Silbert¹, Deniz Ertaş², Gary S. Grest¹, Thomas C. Halsey², Dov Levine³, and Steven J. Plimpton¹

¹Sandia National Laboratories, Albuquerque, New Mexico 87185
²Corporate Strategic Research, ExxonMobil Research and Engineering, Annandale, New Jersey 08801
³Department of Physics, Technion, Haifa 32000, Israel

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Phys. Rev. E 64, 051302 – Published 25 October, 2001

DOI: https://doi.org/10.1103/PhysRevE.64.051302

Abstract

We have performed a systematic, large-scale simulation study of granular media in two and three dimensions, investigating the rheology of cohesionless granular particles in inclined plane geometries, i.e., chute flows. We find that over a wide range of parameter space of interaction coefficients and inclination angles, a steady-state flow regime exists in which the energy input from gravity balances that dissipated from friction and inelastic collisions. In this regime, the bulk packing fraction (away from the top free surface and the bottom plate boundary) remains constant as a function of depth z, of the pile. The velocity profile in the direction of flow $v_{x} (z)$ scales with height of the pile H, according to $v_{x} (z) \propto H^{α}$ , with $α = 1.52 \pm 0.05$ . However, the behavior of the normal stresses indicates that existing simple theories of granular flow do not capture all of the features evidenced in the simulations.

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References (49)

R. A. Bagnold, Proc. R. Soc. London, Ser. A 255, 49 (1954).
D. Ertaş, G. S. Grest, T. C. Halsey, D. Levine, and L. E. Silbert, Europhy. Lett. (unpublished).
L. E. Silbert, G. S. Grest, S. J. Plimpton, and D. Levine (unpublished).
S.-S. Hsiau and H.-W. Jang, J. Chin. Inst. Chem. Eng. 22, 93 (1999).
D. W. Howell, R. P. Behringer, and C. T. Veje, Chaos 9, 559 (1999).
A. Barois-Cazenave, P. Marchal, V. Falk, and L. Choplin, Powder Technol. 103, 58 (1999).
M. Medved, D. Dawson, H. M. Jaeger, and S. R. Nagel, Chaos 9, 691 (1999).
T. Takahashi, Annu. Rev. Fluid Mech. 13, 57 (1981).
C. S. Campbell, P. W. Cleary, and M. Hopkins, J. Geophys. Res. 100, 8267 (1995).
J. T. Jenkins and E. Askari, Chaos 9, 654 (1999).
P.-A. Lemieux and D. J. Durian, Phys. Rev. Lett. 85, 4273 (2000).
O. Pouliquen and N. Renaut, J. Phys. II 6, 923 (1996).
O. Hungr and N. R. Morgenstern, Geotechnique 3, 405 (1984).
T. G. Drake, J. Geophys. Res. 95, 8681 (1990).
E. Azanza, F. Chevoir, and P. Moucheront, J. Fluid Mech. 400, 199 (1999).
D. M. Hanes and O. R. Walton, Powder Technol. 109, 133 (2000).
S. B. Savage, J. Fluid Mech. 377, 1 (1998).
H. Ahn, C. E. Brennen, and R. H. Sabersky, J. Appl. Mech. 59, 109 (1992).
P. Mills, D. Loggia, and M. Tixier, Europhys. Lett. 45, 733 (1999).
D. V. Kharkhar, J. J. McCarthy, and J. M. Ottino, Powder Technol. 86, 219 (1996).
D. Hirshfeld and D. C. Rapaport, Phys. Rev. E 56, 2012 (1997).
O. R. Walton and R. L. Braun, J. Rheol. 30, 949 (1986).
O. R. Walton, Mech. Mater. 16, 239 (1993).
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X. M. Zheng and J. M. Hill, Comp. Mech. 22, 160 (1998).
S. Dippel and D. E. Wolf, Comput. Phys. Commun. 121, 284 (1999).
P. A. Cundall and O. D. L. Strack, Geotechnique 29, 47 (1979).
O. Pouliquen, Phys. Fluids 11, 542 (1999).
L. Vanel, D. Howell, D. Clark, R. P. Behringer, and E. Clement, Phys. Rev. E 60, R5040 (1999).
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B. C. Vemuri, L. C. L. Vu-Quoc, X. Zhang, and O. Walton, Mech. Mater. 31, 235 (1999).
J. J. Moreau, Eur. J. Mech. A/Solids 13, 93 (1994).
We have also studied the role of dynamic friction in 2D simulations. This model has $u_{t}$ set to zero always, and only includes the tangential velocity damping term $τ_{t} v_{t} .$ Using this model, we never observed a steady state flow for our chute flow simulations.
T. C. Halsey and D. Ertaş, Phys. Rev. Lett. 83, 5007 (1999).
R. D. Mindlin and H. Deresiewicz, L. Appl. Mech. 20, 327 (1953).
K. L. Johnson, Contact Mechanics (Cambridge University, Cambridge, 1999).
J. Schafer, S. Dippel, and D. E. Wolf, J. Phys. I 6, 5 (1996).
M. P. Allen and D. J. Tilldesley, Computer Simulations of Liquids (Oxford University, Oxford, 1999).
S. J. Plimpton, J. Comput. Phys. 117, 1 (1995).
Although we do not see slip in this study, we have observed slip in our 2D simulations using a smoother bottom wall constructed from spheres of the same diameter as those in the bulk.
T. S. Komatsu, S. Inagaki, N. Nakagawa, and S. Nasuno, Phys. Rev. Lett. 86, 1757 (2001).
W. Losert, L. Bocquet, T. C. Lubensky, and J. P. Gollub, Phys. Rev. Lett. 85, 1428 (2000).
R. Ojha, N. Menon, and D. J. Durian, Phys. Rev. E 62, 4442 (2000).
R. M. Nedderman, Statics and Kinematics of Granular Materials (Cambridge University, Cambridge, 1992).
We also find that the mean-square velocity fluctuations or granular temperature scale similarly with ${\dot{γ}}^{2}$ .
Although the overall fit is good, deviations from the simple scaling law occur even at depths where the density has reached its bulk value. Thus, the rheology of the system does not appear to be entirely local, with surface effects penetrating farther into the pile than is suggested by the density profile.
H. A. Makse, D. L. Johnson, and L. M. Schwartz, Phys. Rev. Lett. 84, 4160 (2000).
P. A. Thompson and G. S. Grest, Phys. Rev. Lett. 67, 1751 (1991).

[c1] R. A. Bagnold, Proc. R. Soc. London, Ser. A 255, 49 (1954).

[c2] D. Ertaş, G. S. Grest, T. C. Halsey, D. Levine, and L. E. Silbert, Europhy. Lett. (unpublished).

[c3] L. E. Silbert, G. S. Grest, S. J. Plimpton, and D. Levine (unpublished).

[c4] S.-S. Hsiau and H.-W. Jang, J. Chin. Inst. Chem. Eng. 22, 93 (1999).

[c5] D. W. Howell, R. P. Behringer, and C. T. Veje, Chaos 9, 559 (1999).

[c6] A. Barois-Cazenave, P. Marchal, V. Falk, and L. Choplin, Powder Technol. 103, 58 (1999).

[c7] M. Medved, D. Dawson, H. M. Jaeger, and S. R. Nagel, Chaos 9, 691 (1999).

[c8] T. Takahashi, Annu. Rev. Fluid Mech. 13, 57 (1981).

[c9] C. S. Campbell, P. W. Cleary, and M. Hopkins, J. Geophys. Res. 100, 8267 (1995).

[c10] J. T. Jenkins and E. Askari, Chaos 9, 654 (1999).

[c11] P.-A. Lemieux and D. J. Durian, Phys. Rev. Lett. 85, 4273 (2000).

[c12] O. Pouliquen and N. Renaut, J. Phys. II 6, 923 (1996).

[c13] O. Hungr and N. R. Morgenstern, Geotechnique 3, 405 (1984).

[c14] T. G. Drake, J. Geophys. Res. 95, 8681 (1990).

[c15] E. Azanza, F. Chevoir, and P. Moucheront, J. Fluid Mech. 400, 199 (1999).

[c16] D. M. Hanes and O. R. Walton, Powder Technol. 109, 133 (2000).

[c17] S. B. Savage, J. Fluid Mech. 377, 1 (1998).

[c18] H. Ahn, C. E. Brennen, and R. H. Sabersky, J. Appl. Mech. 59, 109 (1992).

[c19] P. Mills, D. Loggia, and M. Tixier, Europhys. Lett. 45, 733 (1999).

[c20] D. V. Kharkhar, J. J. McCarthy, and J. M. Ottino, Powder Technol. 86, 219 (1996).

[c21] D. Hirshfeld and D. C. Rapaport, Phys. Rev. E 56, 2012 (1997).

[c22] O. R. Walton and R. L. Braun, J. Rheol. 30, 949 (1986).

[c23] O. R. Walton, Mech. Mater. 16, 239 (1993).

[c24] T. Pöschel, J. Phys. II 3, 27 (1993).

[c25] X. M. Zheng and J. M. Hill, Powder Technol. 86, 219 (1996).

[c26] X. M. Zheng and J. M. Hill, Comp. Mech. 22, 160 (1998).

[c27] S. Dippel and D. E. Wolf, Comput. Phys. Commun. 121, 284 (1999).

[c28] P. A. Cundall and O. D. L. Strack, Geotechnique 29, 47 (1979).

[c29] O. Pouliquen, Phys. Fluids 11, 542 (1999).

[c30] L. Vanel, D. Howell, D. Clark, R. P. Behringer, and E. Clement, Phys. Rev. E 60, R5040 (1999).

[c31] L. Vu-Quoc and X. Zhang, Mech. Mater. 31, 235 (1999).

[c32] B. C. Vemuri, L. C. L. Vu-Quoc, X. Zhang, and O. Walton, Mech. Mater. 31, 235 (1999).

[c33] J. J. Moreau, Eur. J. Mech. A/Solids 13, 93 (1994).

[c34] We have also studied the role of dynamic friction in 2D simulations. This model has $u_{t}$ set to zero always, and only includes the tangential velocity damping term $τ_{t} v_{t} .$ Using this model, we never observed a steady state flow for our chute flow simulations.

[c35] T. C. Halsey and D. Ertaş, Phys. Rev. Lett. 83, 5007 (1999).

[c36] R. D. Mindlin and H. Deresiewicz, L. Appl. Mech. 20, 327 (1953).

[c37] K. L. Johnson, Contact Mechanics (Cambridge University, Cambridge, 1999).

[c38] J. Schafer, S. Dippel, and D. E. Wolf, J. Phys. I 6, 5 (1996).

[c39] M. P. Allen and D. J. Tilldesley, Computer Simulations of Liquids (Oxford University, Oxford, 1999).

[c40] S. J. Plimpton, J. Comput. Phys. 117, 1 (1995).

[c41] Although we do not see slip in this study, we have observed slip in our 2D simulations using a smoother bottom wall constructed from spheres of the same diameter as those in the bulk.

[c42] T. S. Komatsu, S. Inagaki, N. Nakagawa, and S. Nasuno, Phys. Rev. Lett. 86, 1757 (2001).

[c43] W. Losert, L. Bocquet, T. C. Lubensky, and J. P. Gollub, Phys. Rev. Lett. 85, 1428 (2000).

[c44] R. Ojha, N. Menon, and D. J. Durian, Phys. Rev. E 62, 4442 (2000).

[c45] R. M. Nedderman, Statics and Kinematics of Granular Materials (Cambridge University, Cambridge, 1992).

[c46] We also find that the mean-square velocity fluctuations or granular temperature scale similarly with ${\dot{γ}}^{2}$ .

[c47] Although the overall fit is good, deviations from the simple scaling law occur even at depths where the density has reached its bulk value. Thus, the rheology of the system does not appear to be entirely local, with surface effects penetrating farther into the pile than is suggested by the density profile.

[c48] H. A. Makse, D. L. Johnson, and L. M. Schwartz, Phys. Rev. Lett. 84, 4160 (2000).

[c49] P. A. Thompson and G. S. Grest, Phys. Rev. Lett. 67, 1751 (1991).

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Granular flow down an inclined plane: Bagnold scaling and rheology

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