Hostname: page-component-cd4964975-ppllx Total loading time: 0 Render date: 2023-03-31T12:52:43.631Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Dynamics of dense sheared granular flows. Part II. The relative velocity distributions

Published online by Cambridge University Press:  27 July 2009

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
Email address for correspondence:


The distribution of relative velocities between colliding particles in shear flows of inelastic spheres is analysed in the volume fraction range 0.4–0.64. Particle interactions are considered to be due to instantaneous binary collisions, and the collision model has a normal coefficient of restitution en (negative of the ratio of the post- and pre-collisional relative velocities of the particles along the line joining the centres) and a tangential coefficient of restitution et (negative of the ratio of post- and pre-collisional velocities perpendicular to line joining the centres).

The distribution of pre-collisional normal relative velocities (along the line joining the centres of the particles) is found to be an exponential distribution for particles with low normal coefficient of restitution in the range 0.6–0.7. This is in contrast to the Gaussian distribution for the normal relative velocity in an elastic fluid in the absence of shear. A composite distribution function, which consists of an exponential and a Gaussian component, is proposed to span the range of inelasticities considered here. In the case of rough particles, the relative velocity tangential to the surfaces at contact is also evaluated, and it is found to be close to a Gaussian distribution even for highly inelastic particles.

Empirical relations are formulated for the relative velocity distribution. These are used to calculate the collisional contributions to the pressure, shear stress and the energy dissipation rate in a shear flow. The results of the calculation were found to be in quantitative agreement with simulation results, even for low coefficients of restitution for which the predictions obtained using the Enskog approximation are in error by an order of magnitude. The results are also applied to the flow down an inclined plane, to predict the angle of repose and the variation of the volume fraction with angle of inclination. These results are also found to be in quantitative agreement with previous simulations.

Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Baran, O., Ertas, D., Halsey, T. C., Grest, G. S. & Lechman, J. B. 2006 Velocity correlations in dense gravity-driven granular chute flow. Phys. Rev. E 74, 051302.CrossRefGoogle ScholarPubMed
Bocquet, L., Errami, J. & Lubensky, T. C. 2002 Hydrodynamic model for a dynamical jammed-to-flowing transition in gravity driven granular media. Phys. Rev. Lett. 89, 184301184304.CrossRefGoogle ScholarPubMed
Campbell, C. S. 1997 Self-diffusion in granular shear flows. J. Fluid Mech. 348, 85101.CrossRefGoogle Scholar
Campbell, C. S. 2002 Granular shear flows at the elastic limit. J. Fluid Mech. 465, 261291.CrossRefGoogle Scholar
Campbell, C. S. 2005 Stress-controlled elastic granular shear flows. J. Fluid Mech. 539, 273297.CrossRefGoogle Scholar
Campbell, C. S. 2006 Granular material flows – An overview. Powder Tech. 162, 208229.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Cole, D. M. & Peters, J. F. 2007 A physically based approach to granular media mechanics: grain-scale experiments, initial results and implications to numerical modelling. Granular Matter 9, 309321.CrossRefGoogle Scholar
Cole, D. M. & Peters, J. F. 2008 Grain-scale mechanics of geologic materials and lunar simulants under normal loading. Granular Matter 10, 171185.CrossRefGoogle Scholar
Delannay, R., Louge, M., Richard, P., Taberlet, N. & Valance, A. 2007 Towards a theoretical picture of dense granular flows down inclines. Nature Mater. 6, 99108.CrossRefGoogle Scholar
Dorfman, J. R. & Cohen, E. G. 1972 Velocity-correlation functions in two and three dimensions: low density. Phys. Rev. A 6, 776.CrossRefGoogle Scholar
Dufty, J. 1984 Diffusion in shear flow. Phys. Rev. A, 30, 14651476.CrossRefGoogle Scholar
Ernst, M. H., Cichocki, B., Dorfman, J. R., Sharma, J. & van Beijeren, H. 1978 Kinetic theory of nonlinear viscous flow in two and three dimensions. J. Stat. Phys. 18, 237270.CrossRefGoogle Scholar
Ertas, D. & Halsey, T. C. 2002 Granular gravitational collapse and chute flow. Europhys. Lett. 60, 931937.CrossRefGoogle Scholar
Foerster, S. F., Louge, M. Y., Chang, H. & Allia, K. 1994 Measurements of the collision properties of small spheres. Phys. Fluids 6, 11081115.CrossRefGoogle Scholar
Foss, D. R. & Brady, J. F. 2000 Structure, diffusion and rheology of Brownian suspensions by Stokesian Dynamics simulation. J. Fluid Mech. 407, 167200.CrossRefGoogle Scholar
Garzo, V. & Dufty, J. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895.CrossRefGoogle ScholarPubMed
GDR MiDi 2004 On dense granular flows. Euro. Phys. J. E 14, 341365.CrossRefGoogle ScholarPubMed
Goldschmidt, M. J. V., Beetstra, R. & Kuipers, J. A. M. 2002 Hydrodynamic modelling of dense gas-fluidised beds: comparison of the kinetic theory of granular flow with three-dimensional hard-sphere discrete particle simulations. Chem. Engng Sci. 57, 20592075.CrossRefGoogle Scholar
Jenkins, J. T. 2006 Dense shearing flows of inelastic disks. Phys. Fluids 18, 103307.CrossRefGoogle Scholar
Jenkins, J. T. 2007 Dense inclined flows of inelastic spheres. Granular Matter 10, 4752.CrossRefGoogle Scholar
Khain, E. 2007 Hydrodynamics of fluid–solid coexistence in dense shear granular flow. Phys. Rev. E 75, 051310.CrossRefGoogle ScholarPubMed
Khain, E. & Meerson, B. 2006 Shear-induced crystallization of a dense rapid granular flow: hydrodynamics beyond the melting point. Phys. Rev. E 73, 061301.CrossRefGoogle ScholarPubMed
Kirkpatrick, T. R. & Nieuwoudt, J. 1986 Stability analysis of a dense hard-sphere fluid subjected to large shear-induced ordering. Phys. Rev. Lett. 56, 885888.CrossRefGoogle Scholar
Kumar, V. S. & Kumaran, V. 2006 Bond-orientational analysis of hard-disk and hard-sphere structures. J. Chem. Phys. 124, 204508.CrossRefGoogle Scholar
Kumaran, V. 1998 Temperature of a granular material fluidised by external vibrations. Phys. Rev. E 57, 56605664.CrossRefGoogle Scholar
Kumaran, V. 2004 Constitutive relations and linear stability of a sheared granular flow. J. Fluid Mech. 506, 143.CrossRefGoogle Scholar
Kumaran, V. 2006 a The constitutive relations for the granular flow of rough particles, and its application to the flow down an inclined plane. J. Fluid Mech. 561, 142.CrossRefGoogle Scholar
Kumaran, V. 2006 b Kinetic theory for the density plateau in the granular flow down an inclined plane. Europhys. Lett. 73, 17.CrossRefGoogle Scholar
Kumaran, V. 2006 c Velocity autocorrelations and the viscosity renormalisation in sheared granular flows. Phys. Rev. Lett. 96, 258002258005.CrossRefGoogle Scholar
Kumaran, V. 2008 Dense granular flows down an inclined plane – from kinetic theory to granular dynamic. J. Fluid Mech. 599, 121168.CrossRefGoogle Scholar
Kumaran, V. 2009 a Dynamics of a dilute sheared inelastic fluid. I. Hydrodynamic modes and the velocity correlation functions. Phys. Rev. E 79, 011301.CrossRefGoogle ScholarPubMed
Kumaran, V. 2009 b Dynamics of a dilute sheared inelastic fluid. II. The effect of correlations. Phys. Rev. E 79, 011302.CrossRefGoogle Scholar
Lahiri, R. & Ramaswamy, S. 1994 Shear-Induced melting and re-entrance: A model. Phys. Rev. Lett. 73, 10431046.CrossRefGoogle Scholar
Liu, A. J. & Nagel, S. R. 1998 Jamming is not just cool anymore. Nature 396, 2122.CrossRefGoogle Scholar
Lois, G., Lemaitre, A. & Carlson, J. M. 2005 Numerical tests of constitutive laws for dense granular flows. Phys. Rev. E 72, 051303.CrossRefGoogle ScholarPubMed
Luding, S. 2001 Global equation of state of two-dimensional hard sphere systems. Phys. Rev. E 63, 042201.CrossRefGoogle Scholar
Lutsko, J. F. 1996 Molecular chaos, pair correlations, and shear-induced ordering of hard spheres. Phys. Rev. Lett. 77, 22252228.CrossRefGoogle ScholarPubMed
Lutsko, J. F. 2001 Velocity correlations and the structure of nonequilibrium hard-core fluids. Phys. Rev. Lett. 86, 33443347.CrossRefGoogle ScholarPubMed
Lutsko, J. F & Dufty, J. W. 1986 Possible instability for shear-induced order-disorder transition. Phys. Rev. Lett. 57, 27752778.CrossRefGoogle Scholar
Mitarai, N. & Nakanishi, H. 2005 Bagnold scaling, density plateau, and kinetic theory analysis of dense granular flow. Phys. Rev. Lett. 94, 128001.CrossRefGoogle ScholarPubMed
Mitarai, N. & Nakanashi, H. 2007 Velocity correlations in the dense granular shear flows: effects on energy dissipation and normal stress. Phys. Rev. E 031305.Google Scholar
Orpe, A. V., Kumaran, V., Reddy, K. A. & Kudrolli, A. 2008 Fast decay of the velocity autocorrelation function in dense shear flow of inelastic hard spheres. Europhys. Lett. 84, 64003.CrossRefGoogle Scholar
Pouliquen, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11, 542548.CrossRefGoogle Scholar
Reddy, K. A. & Kumaran, V. 2007 The applicability of constitutive relations from kinetic theory for dense granular flows. Phys. Rev. E 76, 061305.CrossRefGoogle Scholar
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 4174.CrossRefGoogle Scholar
Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C., Levine. D. & Plimpton, S. J. 2001 Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 51302.CrossRefGoogle Scholar
Silbert, L. E., Grest, G. S., Brewster, R. E. & Levine, A. J. 2007 Rheology and contact lifetimes in dense granular flows. Phys. Rev. Lett. 99, 068002.CrossRefGoogle ScholarPubMed
Stevens, M. J. & Robbins, M. O. 1993 Simulations of shear-induced melting and ordering. Phys. Rev. E 48, 37783792.CrossRefGoogle ScholarPubMed
Theodosopulu, M. & Dahler, J. S. 1974 a Kinetic theory of polyatomic liquids. I. The generalized moment method. J. Chem. Phys. 9, 35673582.CrossRefGoogle Scholar
Theodosopulu, M. & Dahler, J. S. 1974 b Kinetic theory of polyatomic liquids. I. The rough sphere, rigid ellipsoid and square-well ellipsoid models. J. Chem. Phys. 9, 40484057.CrossRefGoogle Scholar
Torquato, S. 1995 Nearest neighbour statistics for packings of hard disks and spheres. Phys. Rev. E 51, 31703182.CrossRefGoogle Scholar