Skip to main content Accessibility help
×
Home

Dynamics of dense sheared granular flows. Part 1. Structure and diffusion

  • V. KUMARAN (a1)

Abstract

Shear flows of inelastic spheres in three dimensions in the volume fraction range 0.4–0.64 are analysed using event-driven simulations. 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 the line joining the centres). Here, we have considered both et = +1 and et = en (rough particles) and et = −1 (smooth particles), and the normal coefficient of restitution en was varied in the range 0.6–0.98. Care was taken to avoid inelastic collapse and ensure there are no particle overlaps during the simulation. First, we studied the ordering in the system by examining the icosahedral order parameter Q6 in three dimensions and the planar order parameter q6 in the plane perpendicular to the gradient direction. It was found that for shear flows of sufficiently large size, the system continues to be in the random state, with Q6 and q6 close to 0, even for volume fractions between φ = 0.5 and φ = 0.6; in contrast, for a system of elastic particles in the absence of shear, the system orders (crystallizes) at φ = 0.49. This indicates that the shear flow prevents ordering in a system of sufficiently large size. In a shear flow of inelastic particles, the strain rate and the temperature are related through the energy balance equation, and all time scales can be non-dimensionalized by the inverse of the strain rate. Therefore, the dynamics of the system are determined only by the volume fraction and the coefficients of restitution. The variation of the collision frequency with volume fraction and coefficient of restitution was examined. It was found, by plotting the inverse of the collision frequency as a function of volume fraction, that the collision frequency at constant strain rate diverges at a volume fraction φad (volume fraction for arrested dynamics) which is lower than the random close-packing volume fraction 0.64 in the absence of shear. The volume fraction φad decreases as the coefficient of restitution is decreased from en = 1; φad has a minimum of about 0.585 for coefficient of restitution en in the range 0.6–0.8 for rough particles and is slightly larger for smooth particles. It is found that the dissipation rate and all components of the stress diverge proportional to the collision frequency in the close-packing limit. The qualitative behaviour of the increase in the stress and dissipation rate are well captured by results derived from kinetic theory, but the quantitative agreement is lacking even if the collision frequency obtained from simulations is used to calculate the pair correlation function used in the theory.

Copyright

Corresponding author

Email address for correspondence: kumaran@chemeng.iisc.ernet.in

References

Hide All
Alam, M. & Hrenya, C. M. 2001 Inelastic collapse in simple shear flow of a granular medium. Phys. Rev. E 63, 061308.
Alder, B. & Wainwright, 1970 Decay of the velocity autocorrelation function. Phys. Rev. A 1, 18.
Allen, M. P. & Tildesley, D. J. 1992 Computer Simulation of Liquids. Clarendon.
Bernu, B. & Mazighi, R. 1990 One-dimensional bounce of inelastically colliding marbles on a wall. J. Phys. A 23 57455754.
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.
Brady, J. F. & Morris, J. B. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103139.
Campbell, C. S. 1997 Self-diffusion in granular shear flows. J. Fluid Mech. 348, 85101.
Campbell, C. S. 2002 Granular shear flows at the elastic limit. J. Fluid Mech. 465, 261291.
Campbell, C. S. 2005 Stress-controlled elastic granular shear flows. J. Fluid Mech. 539, 273297.
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-uniform Gases. Cambridge University Press.
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.
Dorfman, J. R. & Cohen, E. G. 1972 Velocity-correlation functions in two and three dimensions: low density. Phys. Rev. A 6, 776.
Dufty, J. 1984 Diffusion in shear flow. Phys. Rev. A 30, 14651476.
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.
Ertas, D. & Halsey, T. C. 2002 Granular gravitational collapse and chute flow. Europhys. Lett. 60, 931937.
Foss, D. R. & Brady, J. F. 1999 Self-diffusion in sheared suspensions by dynamic simulation. J. Fluid Mech. 401, 243274.
Foss, D. R. & Brady, J. F. 2000 Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulation. J. Fluid Mech. 407, 167200.
Garzo, V. & Dufty, J. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895.
Goldhirsch, I. & Zanetti, G. 1993 Clustering instability in dissipative gases. Phys. Rev. Lett. 70, 16191622.
Goldman, D., Shattuck, M. D., Bizon, C., McCormick, W. D., Swift, J. B. & Swinney, H. L. 1998 Absence of inelastic collapse in a realistic three ball model. Phys. Rev. E 57, 48314833.
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.
Hopkins, M. A. & Louge, M. Y. 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 4757.
Jenkins, J. T. 2006 Dense shearing flows of inelastic disks. Phys. Fluids 18, 103307.
Jenkins, J. T. 2007 Dense inclined flows of inelastic spheres. Gran. Matter 10, 4752.
Jenkins, J. T. & Richman, M. W. 1985 Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355377.
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 186202.
Khain, E. 2007 Hydrodynamics of fluid-solid coexistence in dense shear granular flow. Phys. Rev. E 75, 051310.
Khain, E. & Meerson, B. 2006 Shear-induced crystallization of a dense rapid granular flow: hydrodynamics beyond the melting point. Phys. Rev. E 73, 061301.
Kumar, V. S. & Kumaran, V. 2005. Voronoi cell volume distribution and configurational entropy of hard spheres. J. Chem. Phys. 123, 114501114513.
Kumar, V. S. & Kumaran, V. 2006 Bond-orientational analysis of hard-disk and hard-sphere structures. J. Chem. Phys. 124, 204508.
Kumaran, V. 1998 Temperature of a granular material fluidised by external vibrations. Phys. Rev. E 57, 56605664.
Kumaran, V. 2004 Constitutive relations and linear stability of a sheared granular flow. J. Fluid Mech. 506, 143.
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.
Kumaran, V. 2006 b Kinetic theory for the density plateau in the granular flow down an inclined plane. Europhys. Lett. 73, 17.
Kumaran, V. 2006 c Velocity autocorrelations and the viscosity renormalisation in sheared granular flows. Phys. Rev. Lett. 96, 258002258005.
Kumaran, V. 2008 Dense granular flow down an inclined plane: from kinetic theory to granular dynamics. J. Fluid Mech. 599, 120168.
Kumaran, V. 2009 a Dynamics of a dilute sheared inelastic fluid. Part 1. Hydrodynamic modes and the velocity correlation functions. Phys. Rev. E 79, 011301.
Kumaran, V. 2009 b Dynamics of a dilute sheared inelastic fluid. Part 2. The effect of correlations. Phys. Rev. E 79, 011302.
Lois, G., Lematre, A. & Carlson, J. M. 2005 Numerical tests of constitutive laws for dense granular flows. Phys. Rev. E 72, 051303.
Luding, S. & McNamara, S. 1998 How to handle the inelastic collapse of a dissipative hard-sphere gas with the TC model. Gran. Matter 1, 113128.
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223256.
Lutsko, J. F. 2001 Velocity correlations and the structure of nonequilibrium hard-core fluids. Phys. Rev. Lett. 86, 33443347.
McNamara, S. & Young, W. R. 1992 Inelastic collapse and clumping in a one-dimensional granular medium. Phys. Fluids A 4, 496.
McNamara, S. & Young, W. R. 1994 Inelastic collapse in two dimensions. Phys. Rev. E 50, R28.
Mitarai, N. & Nakanishi, H. 2005 Bagnold scaling, density plateau, and kinetic theory analysis of dense granular flow. Phys. Rev. Lett. 94, 128001.
Mitarai, N. & Nakanashi, H. 2007 Velocity correlations in the dense granular shear flows: effects on energy dissipation and normal stress. Phys. Rev. E 031305.
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.
Pusey, P. N. & van Megan, W. 1989 Dynamic light scattering by non-ergodic media. Physica A 157, 705741.
Reddy, K. A. & Kumaran, V. 2007 The applicability of constitutive relations from kinetic theory for dense granular flows. Phys. Rev. E 76, 061305.
Reis, P. M., Ingale, R. A. & Shattuck, M. D. 2007 Caging dynamics in a granular fluid. Phys. Rev. Lett. 98, 188301.
Savage, S. B. & Jeffrey, D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255272.
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 4174.
Sela, N., Goldhirsch, I. & Noskowicz, S. H. 1996 Kinetic theoretical study of a simply sheared two-dimensional granular gas to Burnett order. Phys. Fluids 8, 2337.
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.
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.
Torquato, S. 1995 Nearest neighbour statistics for packings of hard disks and spheres. Phys. Rev. E 51, 31703182.
Vollmayr-Lee, K. & Zippelius, A. 2005 Heterogeneities in the glassy state. Phys. Rev. E 72, 041507.
Weeks, E. R., Crocker, J. C., Levitt, A. C., Schofield, A. & Weitz, D. A. 2000 Three-dimensional direct imaging of structural relaxation near the colloidal glass transition. Science 287, 627631.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Dynamics of dense sheared granular flows. Part 1. Structure and diffusion

  • V. KUMARAN (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed