Form of Velocity Distribution Function for Rapid Granular Flows
Publication: Journal of Engineering Mechanics
Volume 116, Issue 6
Abstract
A simple delta function form of the velocity distribution function is proposed for smooth, uniform, slightly inelastic spheres in rapidly sheared granular flow. The collisional stresses are computed for simple‐shear flow and compared to existing theories. Good agreement is observed with more elaborate kinetic theories, numerical simulations, and shear cell data. The kinetic formulation is placed in the physically transparent form proposed by Bagnold. This formulation is used to show the sources of the discrepancy in results between existing data and the theory of Shen and Ackermann, which is based on Bagnold's formulation. It is shown that the discrepancies can be substantially alleviated by incorporating more detail into Shen and Ackermann's phenomenological picture of collisions, and by improving their estimate of the mean‐free path. Thus, it is shown that using the proper form for averaging has a much more significant effect on the magnitude of the predicted stresses than the form of the velocity distribution function. The delta function form can be employed to simplify calculations while retaining an acceptable level of accuracy.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Ahmadi, G., and Ma, D. (1986). “A kinetic model for granular flows including interstitial fluid effects.” Int. Symp. on Slurry Flows, M. C. Roco and W. Wiedenroth, eds., FED, American Society of Mechanical Engineers, 38.
2.
Babić, M., and Shen, H. H. (1989). “A simple mean free path theory for stresses in a rapid granular flow.” J. Engrg. Mech. Div., ASCE, 115(3), 1262–1282.
3.
Bagnold, R. A. (1954). “Experiments on a gravity‐free dispersion of large solid spheres in a Newtonian fluid under shear.” Proc. Royal Soc. London, A225, 49–63.
4.
Carnahan, N. F., and Starling, K. E. (1969). “Equations of state for non‐attracting rigid spheres.” J. Chem. Phys., 51, 635–636.
5.
Chapman, S., and Cowling, T. G. (1970). The mathematical theory of non‐uniform gases. 3rd Ed., Cambridge Univ. Press, Cambridge.
6.
Hopkins, M. A. (1985). “Collisional stresses in a rapidly deforming granular flow,” thesis presented to Clarkson University, at Potsdam, N.Y., in partial fulfillment of the requirements for the degree of Master of Science.
7.
Jenkins, J. T., and Richman, M. W. (1985a). “Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks.” Phys. Fluids, 28(12), 3485–3494.
8.
Jenkins, J. T., and Richman, M. W. (1985b). “Grad's 13‐moment for a dense gas of inelastic spheres.” Arch. for Rational Mech. and Anal., Springer‐Verlag, 87(4), 355–377.
9.
Jenkins, J. T., and Richman, M. W. (1988). “Plane simple shear flow of smooth, inelastic, circular disks: The anisotropy of the second moment in the dilute and dense limits.” J. Fluid Mech., 192, 313–328.
10.
Jenkins, J. T., and Savage, S. B. (1983). “A theory for the rapid flow of identical, smooth, nearly elastic particles.” J. Fluid Mech., 130, 187–202.
11.
Lun, C. K. K., et al. (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.
12.
Pasquarell, G. C., et al. (1988). “Collisional stress in granular flows: Bagnold revisited.” J. Engrg. Mech. Div., ASCE, 114(1), 49–64.
13.
Savage, S. B., and Jeffrey, D. J. (1981). “The stress tensor in a granular flow at high shear rates.” J. Fluid Mech., 110, 255–272.
14.
Savage, S. B., and Sayed, M. (1984). “Stresses developed by dry cohesionless granular materials sheared in an annular shear cell.” J. Fluid Mech., 142, 391–430.
15.
Shen, H. H., and Ackermann, N. L. (1982). “Constitutive relationships for fluidsolid mixtures.” J. Engrg. Mech. Div., 108(5), 748–763.
16.
Walton, O. K., and Braun, R. L. (1987). “Stress calculations for assemblies of inelastic spheres in uniform shear.” Acta Mech., 63, 73–86.
Information & Authors
Information
Published In
Copyright
Copyright © 1990 ASCE.
History
Published online: Jun 1, 1990
Published in print: Jun 1990
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.