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Dense granular flow down an inclined plane: from kinetic theory to granular dynamics

Published online by Cambridge University Press:  06 March 2008

V. KUMARAN
V. KUMARAN*
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
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Abstract

The hydrodynamics of the dense granular flow of rough inelastic particles down an inclined plane is analysed using constitutive relations derived from kinetic theory. The basic equations are the momentum and energy conservation equations, and the granular energy conservation equation contains a term which represents the dissipation of energy due to inelastic collisions. A fundamental length scale in the flow is the ‘conduction length’ δ=(d/(1-en)1/2), which is the length over which the rate of conduction of energy is comparable to the rate of dissipation. Here, d is the particle diameter and en is the normal coefficient of restitution. For a thick granular layer with height h ≫ δ, the flow in the bulk is analysed using an asymptotic analysis in the small parameter δ/h. In the leading approximation, the rate of conduction of energy is small compared to the rates of production and dissipation, and there is a balance between the rate of production due to mean shear and the rate of dissipation due to inelastic collisions. A direct consequence of this is that the volume fraction in the bulk is a constant in the leading approximation. The first correction due to the conduction of energy is determined using asymptotic analysis, and is found to be O(δ/h)2 smaller than the leading-order volume fraction. The numerical value of this correction is found to be negligible for systems of practical interest, resulting in a lack of variation of volume fraction with height in the bulk.

The flow in the ‘conduction boundary layers’ of thickness comparable to the conduction length at the bottom and top is analysed. Asymptotic analysis is used to simplify the governing equations to a second-order differential equation in the scaled cross-stream coordinate, and the resulting equation has the form of a diffusion equation. However, depending on the parameters in the constitutive model, it is found that the diffusion coefficient could be positive or negative. Domains in the parameter space where the diffusion coefficients are positive and negative are identified, and analytical solutions for the boundary layer equations, subject to appropriate boundary conditions, are obtained when the diffusion coefficient is positive. There is no boundary layer solution that matches the solution in the bulk for parameter regions where the diffusion coefficient is negative, indicating that a steady solution does not exist. An analytical result is derived showing that a boundary layer solution exists (diffusion coefficient is positive) if, and only if, the numerical values of the viscometric coefficients are such that volume fraction in the bulk decreases as the angle of inclination increases. If the numerical values of the viscometric coefficients are such that the volume fraction in the bulk increases as the angle of inclination increases, a boundary layer solution does not exist.

The results are extended to dense flows in thin layers using asymptotic analysis. Use is made of the fact that the pair distribution function is numerically large for dense flows, and the inverse of the pair distribution function is used as a small parameter. This approximation results in a nonlinear second-order differential equation for the pair distribution function, which is solved subject to boundary conditions. For a dissipative base, it is found that a flowing solution exists only when the height is larger than a critical value, whereas the temperature decreases to zero and the flow stops when the height becomes smaller than this critical value. This is because the dissipation at the base becomes a larger fraction of the total dissipation as the height is decreased, and there is a minimum height below which the rate of production due to shear is not sufficient to compensate for the rate of dissipation at the base. The scaling of the minimum height with dissipation in the base, the bulk volume fraction and the parameters in the constitutive relations are determined. From this, the variation of the minimum height on the angle of inclination is obtained, and this is found to be in qualitative agreement with previous experiments and simulations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 599 , 25 March 2008 , pp. 121 - 168
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Anderson, J. D. 2005 Ludwig Prandtl's boundary layer. Physics Today 58, 42–48.CrossRefGoogle Scholar
Baran, O., Ertas, D., Halsey, T. C., Grest, G. S. & Lechman, J. B. 2006 Velocity correlations in dense gravity-driven granular chute flows. Phys. Rev. E 74, 051302–051311.Google Scholar
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, 184301–184304.CrossRefGoogle ScholarPubMed
Bonamy, D., Daviaud, F., Laurent, L., Bonetti, M. & Bouchaud, J. P. 2002 Multiscale clustering in granular surface flows. Phys. Rev. Lett. 89, 034301–034304.CrossRefGoogle ScholarPubMed
Brey, J. J. Ruiz-Montero, M. J. & Moreno, F. 2001 Hydrodynamics of an open vibrated granular system. Phys. Rev. E 63, 061305–061314.Google ScholarPubMed
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Delannay, R., Louge, M., Richard, P., Taberlet, N. & Valance, A. 2007 Towards a theoretical picture of dense granular flows down inclines. Nature Materials 6, 99–108.CrossRefGoogle ScholarPubMed
Ernst, M. H., Cichocki, B., Dorfman, J. R., Sharma, J. & vanBeijeren, H. Beijeren, H. 1978 Kinetic theory of nonlinear viscous flow in two and three dimensions. J. Statist. Phys. 18, 237–270.CrossRefGoogle Scholar
Ertas, D. & Halsey, T. C. 2002 Europhys. Lett. 60, 931–934.CrossRefGoogle Scholar
GDR MiDi 2004 On dense granular flows. Eur. Phys. J. E 14, 341–365.Google Scholar
Jenkins, J. T. 2006 Dense shearing flows of inelastic disks. Phys. Fluids 18, 103307–103315.CrossRefGoogle Scholar
Jenkins, J. T. & Askari, E. 1999 Hydraulic theory for a debris flow supported on a collisional shear layer. Chaos 9, 654–658.CrossRefGoogle ScholarPubMed
Jenkins, J. T. & Richman, M. W. 1985 Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355–377.CrossRefGoogle Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 186–202.CrossRefGoogle Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441, 727–730.CrossRefGoogle ScholarPubMed
Kumaran, V. 1998 Temperature of a granular material fluidised by external vibrations. Phys. Rev. E 57, 5660–5664.Google Scholar
Kumaran, V. 2004 Constitutive relations and linear stability of a sheared granular flow. J. Fluid Mech. 506, 1–43.CrossRefGoogle Scholar
Kumaran, V. 2006a The constitutive relations for the granular flow of rough particles, and its application to the flow down an inclined plane. J. Fluid Mech. 561, 1–42.CrossRefGoogle Scholar
Kumaran, V. 2006b Kinetic theory for the density plateau in the granular flow down an inclined plane. Europhys. Lett. 73, 1–7.CrossRefGoogle Scholar
Kumaran, V. 2006c Velocity autocorrelations and the viscosity renormalisation in sheared granular flows. Phys. Rev. Lett. 96, 258002–258005.CrossRefGoogle Scholar
Lois, G., Carlson, J. & Lemaitre, A. 2005 Numerical tests of constitutive laws for dense granular flows. Phys. Rev. E 72, 051303.Google ScholarPubMed
Lois, G., Carlson, J. & Lemaitre, A. 2006 Emergence of multi-contact interactions in contact dynamics simulations. Eur. Phys. Lett. 76, 318–324.CrossRefGoogle Scholar
Louge, M.-Y. 2003 Model for dense granular flows down bumpy surfaces Phys. Rev. E 67, 061303–061313.Google Scholar
Louge, M.-Y. & Keast, S. C. 2001 On dense granular flows down flat frictional inclines. Phys. Fluids 13, 1213–1233.CrossRefGoogle Scholar
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, 223–256.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
Pouliquen, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11, 542–548.CrossRefGoogle Scholar
Pouliquen, O. 2004 Velocity correlations in dense granular flows. Phys. Rev. Lett. 93, 248001–248004.CrossRefGoogle ScholarPubMed
Pouliquen, O. & Chevoir, F. 2002 Dense flows of dry granular material. C. R. Phys. 3, 163–175.CrossRefGoogle Scholar
Reddy, K. A. & Kumaran, V. 2007 The applicability of constitutive relations from kinetic theory for dense granular flows. Phys. Rev. E 76, 061305–061313.Google ScholarPubMed
Savage, S. B. & Jeffrey, D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255–272.CrossRefGoogle Scholar
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41–74.CrossRefGoogle Scholar
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.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.Google ScholarPubMed
Silbert, L. E., Grest, G. S., Plimpton, S. J. & Levine, D. 2002 Boundary effects and self-organization in dense granular flows. Phys. Fluids 14, 2637–2646.CrossRefGoogle Scholar
Silbert, L. E., Landry, J. W. & Grest, G. S. 2003 Boundary effects and self-organization in dense granular flows. Phys. Fluids 15, 1–10.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
Soto, R., Mareschal, M. & Risso, D. 1999 Departure from Fourier's law for fluidized granular media. Phys. Rev. Lett. 83, 5003–5006.CrossRefGoogle Scholar
Torquato, S. 1995 Nearest neighbour statistics for packings of hard disks and spheres. Phys. Rev. E 51, 3170–3182.Google ScholarPubMed