Skip to main content Accessibility help
×
Home

Rapid granular flows down inclined planar chutes. Part 1. Steady flows, multiple solutions and existence domains

  • MARK J. WOODHOUSE (a1), ANDREW J. HOGG (a1) and ALISTAIR A. SELLAR (a1)

Abstract

The highly agitated flow of grains down an inclined chute is modelled using a kinetic theory for inelastic collisions. Solutions corresponding to steady, fully developed flows are obtained by solving numerically a nonlinear system of ordinary differential equations using a highly accurate pseudospectral method based on mapped Chebyshev polynomials. The solutions are characterized by introducing macroscopic, depth-integrated variables representing the mass flux of flowing material per unit width, its centre-of-mass and the mass supported within the flowing layer, and the influence of the controlling parameters on these solutions is investigated. It is shown that, in certain regions of parameter space, multiple steady solutions can be found for a specified mass flux of material. An asymptotic analysis of the governing equations, appropriate to highly agitated flows, is also developed and these results aid in the demarcation of domains in parameter space where steady solutions can be obtained.

Copyright

Corresponding author

Present address: School of Mathematics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address for correspondence: mark.woodhouse@manchester.ac.uk

References

Hide All
Abramowitz, M. & Stegun, I. A. (Ed.) 1965 Handbook of Mathematical Functions, 9th edn. Dover.
Ahn, H., Brennen, C. E. & Sabersky, R. H. 1991 Measurements of velocity, velocity fluctuation, density, and stresses in chute flows of granular materials. J. Appl. Mech. 58, 792803.
Ahn, H., Brennen, C. E. & Sabersky, R. H. 1992 Analysis of the fully developed chute flow of granular materials. J. Appl. Mech. 59, 109119.
Anderson, K. G. & Jackson, R. 1992 A comparison of the solutions of some proposed equations of motion of granular materials for fully developed flow down inclined planes. J. Fluid Mech. 241, 145168.
Azanza, E., Chevoir, F. & Moucheront, P. 1999 Experimental study of collisional granular flows down an inclined plane. J. Fluid Mech. 400, 199227.
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.
Brey, J. J., Ruiz-Montero, M. J. & Moreno, F. 2001 Hydrodynamics of an open vibrated granular system. Phys. Rev. E 63, 061305/1–10.
Campbell, C. S. 1990 Rapid granular flows. Annu. Rev. Fluid Mech. 22, 5792.
Campbell, C. S. & Brennen, C. E. 1985 Chute flows of granular material: some computer simulations. J. Appl. Mech. 52, 172178.
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory Of Non-Uniform Gases, 3rd edn. Cambridge University Press.
Drake, T. G. 1991 Granular flow: physical experiments and their implications for microstructural theories. J. Fluid Mech. 225, 121152.
Forterre, Y. & Pouliquen, O. 2001 Longitudinal vortices in granular flows. Phys. Rev. Lett. 86, (26), 58865889.
Forterre, Y. & Pouliquen, O. 2002 Stability analysis of rapid granular chute flows: formation of longitudinal vortices. J. Fluid Mech. 467, 361387.
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 124.
Garzó, V. & Dufty, J. W. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, (5), 58955911.
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267293.
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401430.
Hanes, D. M. & Walton, O. R. 2000 Simulations and physical measurements of glass spheres flowing down a bumpy incline. Powder Technol. 109, 133144.
Huan, C., Yang, X., Candela, D., Mair, R. W. & Walsworth, R. L. 2004 NMR experiments on a three-dimensional vibrofluidized granular medium. Phys. Rev. E 69, 041302/1–13.
Jaeger, H. M., Nagel, S. R. & Behringer, R. P. 1996 Granular solids, liquids, and gases. Rev. Mod. Phys. 68, (4), 12591273.
Jenkins, J. T. 2006 Dense shearing flows of inelastic disks. Phys. Fluids 18, 103307/1–9.
Jenkins, J. T. 2007 Dense inclined flows of inelastic spheres. Granul. Matter 10, 4752.
Jenkins, J. T. & Hanes, D. M. 1993 The balance of momentum and energy at an interface between colliding and freely flying grains in a rapid granular flow. Phys. Fluids A 5, (3), 781783.
Jenkins, J. T. & Richman, M. W. 1985 Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, (4), 355377.
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187202.
Jenkins, J. T. & Zhang, C. 2002 Kinetic theory for identical, frictional, nearly elastic spheres. Phys. Fluids 14, 12281235.
Johnson, P. C. & Jackson, R. 1987 Frictional–collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 6793.
Johnson, P. C., Nott, P. & Jackson, R. 1990 Frictional–collisional equations of motion for particulate flows and their applications to chutes. J. Fluid Mech. 210, 501535.
Kumaran, V. 1998 a Kinetic theory for a vibro-fluidized bed. J. Fluid Mech. 364, 163185.
Kumaran, V. 1998 b Temperature of a granular material “fluidized” by external vibrations. Phys. Rev. E 57 (5), 56605664.
Kumaran, V. 2008 Dense granular flow down an inclined plane: from kinetic theory to granular dynamics. J. Fluid Mech. 599, 121168.
Lun, C. K. K. & Savage, S. B. 1986 The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials. Acta Mech. 63, 1544.
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 flowfield. J. Fluid Mech. 140, 223256.
McNamara, S. & Young, W. R. 1992 Inelastic collapse and clumping in a one-dimensional granular medium. Phys. Fluids A 4, (3), 496504.
McNamara, S. & Young, W. R. 1994 Inelastic collapse in two dimensions. Phys. Rev. E 50, (1), 2831.
Mitarai, N. & Nakanishi, H. 2004 Linear stability analysis of rapid granular flow down a slope and density wave formation. J. Fluid Mech. 507, 309334.
Mitarai, N. & Nakanishi, H. 2005 Bagnold scaling, density plateau, and kinetic theory analysis of dense granular flow. Phys. Rev. Lett. 94, 128001/1–4.
Nott, P. & Jackson, R. 1992 Frictional–collisional equations of motion for granular materials and their application to flow in aerated chutes. J. Fluid Mech. 241, 125144.
Ramírez, R. & Soto, R. 2003 Temperature inversion in granular fluids under gravity. Physica A 322, 7380.
Rheinboldt, W. C. 1986 Numerical Analysis of Parametrized Nonlinear Equations, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 7. John Wiley & Sons.
Richman, M. W. 1988 Boundary conditions based upon a modified Maxwellian velocity distribution for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227240.
Richman, M. W. & Marciniec, R. P. 1990 Gravity-driven granular flows of smooth, inelastic spheres down bumpy inclines. J. Appl. Mech. 57, 10361043.
Savage, S. B. 1979 Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 5396.
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 4174.
Sellar, A. A. 2003 Free-surface rapid granular flows. PhD thesis, School of Mathematics, University of Bristol.
Silbert, L. E., Ertaş, 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, 051302/1–14.
Silbert, L. E., Grest, G. S., Brewster, R. & Levine, A. J. 2007 Rheology and contact lifetimes in dense granular flows. Phys. Rev. Lett. 99, 068002/1–4.
Silbert, L. E., Grest, G. S., Plimpton, S. J. & Levine, D. 2002 Boundary effects and self-organization in dense granular flows. Phys. Fluids 14, (8), 26372646.
Soto, R., Mareschal, M. & Risso, D. 1999 Departure from Fourier's law for fluidized granular media. Phys. Rev. Lett. 83, (24), 50035006.
Walton, O. R. 1993 Numerical simulation of inclined chute flows of monodisperse, inelastic, frictional spheres. Mech. Mater. 16, 239247.
Woodhouse, M. J. & Hogg, A. J. 2010 Rapid granular flows down inclined planar chutes. Part 2. Linear stability analysis of steady flow solutions. J. Fluid Mech. Forthcoming.
Zheng, X. M. & Hill, J. M. 1996 Molecular dynamics modelling of granular chute flow: density and velocity profiles. Powder Technol. 86, 219227.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Rapid granular flows down inclined planar chutes. Part 1. Steady flows, multiple solutions and existence domains

  • MARK J. WOODHOUSE (a1), ANDREW J. HOGG (a1) and ALISTAIR A. SELLAR (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