Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-29T12:52:53.360Z Has data issue: false hasContentIssue false
  • English
  • Français

The stress tensor in a granular flow at high shear rates

Published online by Cambridge University Press:  20 April 2006

S. B. Savage  and
D. J. Jeffrey
S. B. Savage
Affiliation:
Department of Civil Engineering and Applied Mechanics, McGill University, Canada
D. J. Jeffrey
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The stress tensor in a granular shear flow is calculated by supposing that binary collisions between the particles comprising the granular mass are responsible for most of the momentum transport. We assume that the particles are smooth, hard, elastic spheres and express the stress as an integral containing probability distribution functions for the velocities of the particles and for their spatial arrangement. By assuming that the single-particle velocity distribution function is Maxwellian and that the spatial pair distribution function is given by a formula due to Carnahan & Starling, we reduce this integral to one depending upon a single non-dimensional parameter R: the ratio of the characteristic mean shear velocity to the root mean square of the precollisional particle-velocity perturbation. The integral is evaluated asymptotically for R [Gt ] 1 and R [Lt ] 1 and numerically for intermediate values. Good agreement is found between the stresses measured in experiments on dry granular materials and the theoretical predictions when R is given the value 1·7. This case is probably the one for which the present analysis is most appropriate. For moderate and large values of R, the theory predicts both shear and normal stresses that are proportional to the square of the particle diameter and the square of the shear rate, and depend strongly on the solids volume fraction. A provisional comparison is made between the stresses predicted in the limit R → ∞ and the experimental results of Bagnold for shear flow of neutrally buoyant wax spheres suspended in water. The predicted stresses are of the correct order of magnitude and yield the proper variation of stress with concentration. When R [Lt ] 1, the shear stress is linear in the shear rate, and the analysis can be applied to shear flow in a fluidized bed, although such an application is not developed further here.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 110 , September 1981 , pp. 255 - 272
Copyright
© 1981 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

Alder, B. J. & Hoover, W. G. 1968 Numerical statistical mechanics. In Physics of Simple Liquids (ed. H. N. V. Temperley, J. S. Rowlinson & G. S. Rushbrooke). North Holland.
Bagnold, R. A. 1954 Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. Roy. Soc. A 225, 49–63.Google Scholar
Batchelor, G. K. 1976 Developments in microhydrodynamics. In Theoretical and Applied Mechanics (ed. W. T. Koiter). Proc. IUTAM Congress, Delft, North Holland.
Baxter, R. J. 1968 Percus — Yevick equation for hard spheres with surface adhesion. J. Chem. Phys. 49, 2770–2774.Google Scholar
Baxter, R. J. 1971 Distribution functions. Physical Chemistry, An Advanced Treatise. Vol. VIII A. Liquid State (ed. D. Henderson). Academic.
Born, M. & Green, H. S. 1947 A general kinetic theory of liquids. III. Dynamical properties. Proc. Roy. Soc. A 190, 455–474.Google Scholar
Carlos, C. R. & Richardson, J. F. 1968 Solids movement in liquid fluidized beds. I. Particle velocity distribution. Chem. Engng Sci. 23, 813–824.Google Scholar
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. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press.
Cheng, D. C.-H. & Richmond, R. A. 1978 Some observations on the rheological behaviour of dense suspensions. Rheol. Acta 17, 446–453.Google Scholar
Croxton, C. A. 1974 Liquid State Physics. Cambridge University Press.
Faber, T. E. 1972 An Introduction to the Theory of Liquid Metals. Cambridge University Press.
Gadala-Maria, F. 1979 The rheology of concentrated suspensions. Ph.D. dissertation, Stanford University.
Goldsmith, H. L. & Mason, S. G. 1967 Microrheology of dispersions. Rheology: Theory and Applications, vol. 4 (ed. F. R. Eirich). Academic.
Green, H. S. 1969 The Molecular Theory of Fluids. Dover.
Hansen, J. P. & McDonald, I. R. 1976 Theory of Simple Liquids. Academic.
Henderson, D. 1971 Physical Chemistry, An Advanced Treatise, vol. 8A, Liquid State. Academic.
Jeffrey, D. J. & Acrivos, A. 1976 The rheological properties of suspensions of rigid particles. Am. Inst. Chem. Engng J. 22, 417–432.Google Scholar
Jenkins, J. T. & Cowin, S. C. 1979 Theories for flowing granular materials. Mechanics Applied to the Transport of Bulk Materials. A.S.M.E. AMD-31 (ed. S. C. Cowin).
Kanatani, K. 1979 A micropolar continuum theory for the flow of granular materials. Int. J. Engng Sci. 17, 419–432.Google Scholar
Kirkwood, J. G., Buff, F. P. & Green, M. S. 1949 The statistical mechanical theory of transport processes. III. The coefficients of shear and bulk viscosity of liquids. J. Chem. Phys. 17, 988–994.Google Scholar
Lebowitz, J. L. 1964 Exact solution of generalized Percus — Yevick equation for a mixture of hard spheres. Phys. Rev. A 133, 895–899.Google Scholar
Mansoori, G. A., Carnahan, N. F., Starling, K. E. & Leland, T. W. 1971 Equilibrium thermodynamic properties of the mixture of hard spheres. J. Chem. Phys. 54, 1523–1525.Google Scholar
Marble, F. E. 1964 Mechanism of particle collision in the one-dimenstional dynamics of gasparticle mixtures. Phys. Fluids 7, 1270–1282.Google Scholar
Mctigue, D. F. 1978 A model for stresses in shear flow of granular material. Proc. U.S.–Japan Seminar on Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, pp. 266–271. Tokyo: Gakujutsu Bunken Fukyu-kai.
Mctigue, D. F. 1979 A nonlinear continuum model for flowing granular materials. Ph.D. dissertation, Stanford University.
Ree, F. H. 1971 Computer calculations for model systems. Physical Chemistry, An Advanced Treatise, vol. 8 A, Liquid State (ed. D. Henderson). Academic.
Reed, T. H. & Gubbins, K. E. 1973 Applied Statistical Mechanics. McGraw-Hill.
Savage, S. B. 1979 Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 53–96.Google Scholar
Savage, S. B. & Sayed, M. 1980 Experiments on dry cohensionless materials in an annular shear cell at high strain rates. Presented at EUROMECH 133 –Statics and Dynamics of Granular Materials. Oxford University.
Sohofield, A. N. & Wroth, C. P. 1968 Critical State Soil Mechanics. McGraw-Hill.
Soo, S. L. 1967 Fluid Dynamics of Multiphase Systems. Blaisdell.
Temperley, H. N. V., Rowlinson, J. S. & Rushbrooke, G. S. 1968 Physics of Simple Liquids. North Holland.
Thiele, E. J. 1963 Equation of state for hard spheres. J. Chem. Phys. 39, 474–479.Google Scholar
Wertheim, M. S. 1963 Exact solution of the Percus — Yevick integral equation for hard spheres. Phys. Rev. Lett. 10, 321–323.Google Scholar
Wood, W. W. 1968 Monte Carlo studies of simple liquid models. Physics of Simple Liquids (ed. H. N. V. Temperley, J. S. Rowlinson & G. S. Rushbrooke). North Holland.
Ziman, J. M. 1979 Models of Disorder. Cambridge University Press.