Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-19T15:39:31.171Z Has data issue: false hasContentIssue false
  • English
  • Français

Instability of unbounded uniform granular shear flow

Published online by Cambridge University Press:  26 April 2006

S. B. Savage
S. B. Savage
Affiliation:
Department of Civil Engineering and Applied Mechanics. McGill University, Montreal H3A 2K6, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper presents a linear stability analysis of cohesionless granular materials during rapid shear flow. The analysis is based on the governing equations developed in the kinetic theory of Lun et al. (1984) for granular flows of smooth, nearly elastic, uniform spherical particles. The primary flow is taken to be a uniform, simple shear flow and the effects of small perturbations in velocity components, granular temperature and solids fraction are considered. The inelasticity of the particles is characterized by a constant coefficient of restitution which is assumed to be close to unity. Some permissible solutions are sinusoidal plane waves in which the wavenumber vector is continuously turned by the mean shear flow and its magnitude varied as time proceeds. The initial growth (or decay) rates for these perturbations are sought. The resulting linearized equations for the flow perturbations turn out to be exceedingly long and complex; they are determined by the use of computer algebra. It is found that, in general, long wavelengths are the most unstable and that short wavelengths are dampened by ‘viscous action’. ‘Instability’ increases with decreasing coefficient of restitution. Numerical results for initial growth rates were obtained for several values of mean solids fraction and particle coefficient of restitution. Flows tend to be more stable at both high and very low concentrations than at moderate concentrations. These results appear to be consistent in the main with recent computer simulations of granular flows of disk-like particles by Hopkins & Louge (1991).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 109 - 123
Copyright
© 1992 Cambridge University Press

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.)

References

Bagnold R. A. 1954 Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid under shear Proc. R. Soc. Lond. A 225, 49–63.Google Scholar
Barnes H. A. 1989 Shear-thickening (‘Dilatancy’) in suspensions of nonaggregating solid particles dispersed in Newtonian liquids. J. Rheol. 33, 329–366.Google Scholar
Batchelor, G. K. & Janse van Rensburg R. W. 1986 Structure formation in bidisperse sedimentation. J. Fluid Mech. 166, 379–407.Google Scholar
Betchov, R. & Criminale W. O. 1967 Stability of Parallel Flows. Academic.
Brady, J. F. & Bossis G. 1988 Stokesian dynamics. Ann. Rev. Fluid Mech. 20, 111–157.Google Scholar
Campbell C. S. 1989 The stress tensor for simple shear flows of a granular material. J. Fluid Mech. 203, 449–473.Google Scholar
Campbell C. S. 1990 Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57–92.Google Scholar
Campbell, C. S. & Brennen C. E. 1985 Computer simulation of granular shear flows. J. Fluid Mech. 151, 167–188.Google Scholar
Campbell, C. S. & Gong. A. 1986 The stress tensor in a two-dimensional granular shear flow. J. Fluid Mech. 164, 107–125.Google Scholar
Carnahan, N. F. & Starling K. E. 1969 Equations of state for non-attracting rigid spheres. J. Chem. Phys. 42, 635–636.Google Scholar
Green, D. & Homsy G. M. 1987 Instabilities in self-fluidized beds - I: Theory. Intl J. Multiphase Flow 13, 443–45.Google Scholar
Gutt, G. M. & Haff P. K. 1988 Boundary conditions on continuum theories of granular flow. Brown Bag Preprint Series BB-70, Division of Physics, Mathematics, and Astronomy, Caltech, 24 pp.
Haff P. K. 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430.Google Scholar
Haff P. K. 1986 A physical picture of kinetic granular flow. J. Rheol. 30, 931–948.Google Scholar
Hanes D. M., Jenkins, J. T. & Richman M. W. 1988 The thickness of steady plane shear flows of circular disks driven by identical boundaries. J. Appl. Mech. 110, 969–974.Google Scholar
Hearn A. C. 1987 REDUCE Users Manual, Version 3.3. Rand Publication CP78, Rev. 7/87, The Rand Corporation. Santa Monica.
Hopkins, M. A. & Louge. M. Y. 1991 Inelastic microstructure in rapid granular flows of smooth disks Phys. Fluids A 3, 47–57.Google Scholar
Hopkins, M. A. & Shen. H. 1988 A Monte-Carlo simulation of a rapid simple shear flow of granular materials. In Micromechanics of Granular Materials (ed. M. Satake & J. T. Jenkins), pp. 349–358. Elsevier.
Hui K., Haff P. K., Ungar, J. E. & Jackson R. 1984 Boundary conditions for high shear grain flows. J. Fluid Mech. 145, 223–233.Google Scholar
Jackson R. 1985 Hydrodynamic stability of fluid particle systems. In Fluidization (ed. J. F. Davidson, R. Clift & D. Harrison). Academic.
Jenkins J. T. 1987a Balance laws and constitutive relations for rapid flows of granular materials. In Constitutive Models of Deformations (ed. J. Chandra & R. Srivastav), p. 109. Philadelphia: SIAM.
Jenkins J. T. 1987b Rapid flows of granular materials. Ing Non-classical Continuum Mechanics: Abstract Techniques and Applications (ed. R. J. Kops & A. A. Lacey), pp. 213–224. Cambridge University Press.
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.Google Scholar
Jenkins, J. T. & Richman M. W. 1986 Boundary conditions for plane flows of smooth nearly elastic, circular disks. J. Fluid Mech. 171, 53–69.Google 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.Google Scholar
Johnson, P. C. & Jackson R. 1987 Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 67–93.Google Scholar
KytoUmaa, H. K. & Brennen C. E. 1990 Kinematic instability of three-component flows. Intl J. Multiphase Flow (to be submitted).Google Scholar
Lees, A. W. & Edwards S. F. 1972 The computer study of transport processes under extreme conditions J. Phys. C 5, 1921–9.Google Scholar
Liu J. T. C. 1990 Coherent structures in transitional and turbulent free shear flows. Ann. Rev. Fluid Mech. 21, 285–315.Google Scholar
Lun C. K. K., Savage S. B., Jeffrey, D. J. & Chepurnity N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256.Google Scholar
Moffatt H. K. 1967 The interaction of turbulence with strong wind shear. In Atmospheric Turbulence and Radio Wave Propagation (ed. A. M. Yaglom & V. I. Tatarsky), pp. 135–56. Moscow: Nauka.
Phillips O. M. 1969 Shear-flow turbulence. Ann. Rev. Fluid Mech. 1, 245–64.Google Scholar
Rayna G. 1987 REDUCE. Software for Algebraic Computation. Springer.
Rich A., Rich, J. & Stoutemyer D. 1990 DERIVE, Version 2, User Manual. Soft Warehouse, Honolulu, Hawaii.
Richman M. W. 1988 Boundary conditions based upon a modified Maxwellian velocity distribution for flows of identical, smooth nearly elastic, spheres. Acta Mech. 75, 227–240.Google Scholar
Richman M. W. 1989 The source of second moment in dilute granular flows of highly inelastic spheres. J. Rheol. 33, 1293–1306.Google Scholar
Richman, M. W. & Chou C. S. 1988 Boundary effects on granular shear flows of smooth disks. Z. angew. Math. Phys. 39, 885–901.Google Scholar
Savage S. B. 1983 Granular flows down rough inclines - review and extension. Mechanics of Granular Materials: New Models and Constitutive Relations (ed. J. T. Jenkins & M. Satake), pp. 261–282. Elsevier.
Savage S. B. 1984 The mechanics of rapid granular flows. Adv. Appl. Mech. 24, 289–366.Google Scholar
Savage S. B. 1989 Flow of granular materials. In Theoretical and Applied Mechanics (ed. P. Germain, M. Piau & D. Caillerie), pp. 241–266. Elsevier.
Savage S. B. 1991 Numerical simulations of Couette flow of granular materials; spatio-temporal coherence and 1/f noise. In Physics of Granular Media (ed. J. Dodds & D. Bideau). New York: Nova Scientific (in press).
Savage, S. B. & Jeffrey D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255–272.Google Scholar
Savage, S. B. & Sayed M. 1984 Stresses developed by dry cohesionless granular materials sheared in an annular shear cell. J. Fluid Mech. 142, 391–430.Google Scholar
Shen, H. & Ackermann N. L. 1982 Constitutive relationships for fluid–solid mixtures. J. Engng Mech. Div. ASCE 108, 748–763.Google Scholar
Shen, H. & Ackermann N. L. 1984 Constitutive equations for a simple shear flow of disk shaped granular material. Intl J. Engng Sci. 22, 829–843.Google Scholar
Shlesinger, M. F. & West B. 1988 1/f versus 1fa noise. Random Fluctuations and Pattern Growth (ed. H. E. Stanley & N. Ostrowsky), pp. 320–324. Dordrecht: Kluwer Academic.
Voss R. F. 1988 Fractals in nature: From characterization to simulation. In The Science of Fractal Images (ed. H.-O. Peitgen & D. Saupe), pp. 21–70. Springer.
Walton, O. R. & Bkaun R. L. 1986a Stress calculations for assemblies of inelastic spheres in uniform shear. Acta Mech. 63, 73–86.Google Scholar
Walton, O. R. & Braun R. L. 1986b Viscosity and temperature calculations for assemblies of inelastic frictional disks. J. Rheol. 30, 949–980.Google Scholar
Walton O. R., Braun R. L., Mallon, R. G. & Cervelli D. M. 1987 Particle-dynamics calculations of gravity flow of inelastic, frictional spheres. In Micromechanics of Granular Materials (ed. M. Satake & J. T. Jenkins), pp. 153–162. Elsevier.
Walton O. R., Kim, H. & Rosato A. 1991 Micro-structure and stress differences in shearing flows. In Mechanics Computing in 1990′s and Beyond (ed. H. Adeli & R. L. Sierakowski), vol. 2, pp. 1249–1253. ASCE.
Weiland R. H., Fessas, Y. P. & Ramarao B. V. 1984 On instabilities arising during sedimentation of two-component mixtures of solids. J. Fluid Mech. 142, 383–389.Google Scholar
Wolfram S. 1991 MATHEMATICA, A system for doing mathematics by computer (2nd edn). Addison Wesley.