Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-13T01:26:27.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield

Published online by Cambridge University Press:  20 April 2006

C. K. K. Lun ,
S. B. Savage ,
D. J. Jeffrey  and
N. Chepurniy
C. K. K. Lun
Affiliation:
Faculty of Engineering, McGill University, 817 Sherbrooke St West, Montreal, Canada
S. B. Savage
Affiliation:
Faculty of Engineering, McGill University, 817 Sherbrooke St West, Montreal, Canada
D. J. Jeffrey
Affiliation:
Faculty of Engineering, McGill University, 817 Sherbrooke St West, Montreal, Canada
N. Chepurniy
Affiliation:
Faculty of Engineering, McGill University, 817 Sherbrooke St West, Montreal, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles is studied using statistical methods analogous to those used in the kinetic theory of gases. Two theories are developed: one for the Couette flow of particles having arbitrary coefficients of restitution (inelastic particles) and a second for the general flow of particles with coefficients of restitution near 1 (slightly inelastic particles). The study of inelastic particles in Couette flow follows the method of Savage & Jeffrey (1981) and uses an ad hoc distribution function to describe the collisions between particles. The results of this first analysis are compared with other theories of granular flow, with the Chapman-Enskog dense-gas theory, and with experiments. The theory agrees moderately well with experimental data and it is found that the asymptotic analysis of Jenkins & Savage (1983), which was developed for slightly inelastic particles, surprisingly gives results similar to the first theory even for highly inelastic particles. Therefore the ‘nearly elastic’ approximation is pursued as a second theory using an approach that is closer to the established methods of Chapman-Enskog gas theory. The new approach which determines the collisional distribution functions by a rational approximation scheme, is applicable to general flowfields, not just simple shear. It incorporates kinetic as well as collisional contributions to the constitutive equations for stress and energy flux and is thus appropriate for dilute as well as dense concentrations of solids. When the collisional contributions are dominant, it predicts stresses similar to the first analysis for the simple shear case.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 140 , March 1984 , pp. 223 - 256
Copyright
© 1984 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
Brush, S. G. 1972 Kinetic Theory, vol. 3. Pergamon.
Campbell, C. S. 1982 Shear flow of granular materials. Ph.D. dissertation, Calif. Inst. Tech.
Campbell, C. S. & Brennen, C. E. 1982 Computer simulation of shear flows of granular material. In Proc. 2nd U.S.-Japan Seminar on New Models and Constitutive Relations in the Mechanics of Granular Material. Elsevier.
Carnahan, N. F. & Starling, K. E. 1969 Equations of state for non-attracting rigid spheres. J. Chem. Phys. 51, 635–636.Google Scholar
Chapman, S. 1916 On the kinetic theory of a gas. Phil. Trans. R. Soc. Lond. A 217, 115–197.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press.
Condiff, D. W., Lu, W. K. & Dahler, J. S. 1965 Transport properties of polyatomic fluids, a dilute gas of perfectly rough spheres. J. Chem. Phys. 42, 3445–3475.Google Scholar
Dahler, J. S. & Theodosopulu, M. 1975 The kinetic theory of dense polyatomic fluids. Adv. Chem. Phys. 31, 155–229.Google Scholar
Davis, H. T. 1973 Kinetic theory of dense fluids and liquids revisited. Adv. Chem. Phys. 24, 257–343.Google Scholar
Ferziger, J. H. & Kaper, H. G. 1972 Mathematical Theory of Transport Processes in Gases. Elsevier.
Goldsmith, W. 1960 Impact: The Theory and Physical Behavior of Colliding Solids. Arnold.
Guggenheim, E. A. 1960 Elements of the Kinetic Theory of Gases. Pergamon.
Hirschfelder, J. O., Curtis, C. R. & Bird, R. B. 1954 The Molecular Theory of Gases and Liquids. Wiley.
Jeans, J. 1967 An Introduction to the Kinetic Theory of Gases. Cambridge University Press.
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 187–202.Google Scholar
Ogawa, S., Umemura, A. & Oshima, N. 1980 On the equations of fully fluidized granular materials. Z. angew. Math. Phys. 31, 483–493.Google Scholar
Present, R. D. 1958 Kinetic Theory of Gases. McGraw-Hill.
Reif, F. 1965 Fundamentals of Statistical and Thermal Physics. McGraw-Hill.
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. 1980 Experiments on dry cohesionless materials in an annular shear cell at high strain rates. Presented at EUROMECH 133 - Statics and Dynamics of Granular Materials, Oxford University.
Shen, H. 1981 Constitutive relationships for fluid-solid mixtures. Ph.D. dissertation, Clarkson Coll. Tech., Potsdam.