Abstract
Unlike the basic units of molecular systems, the elementary constituents of granular matter (the grains) experience dissipative interactions. This fact is the root cause of many of the difficulties encountered in the study of granular materials and among its major consequences are the existence of unique states and instabilities (e.g. collapse and clustering) as well as the multistable/metastable nature of most granular states. Another central consequence is the inherent lack of scale separation. The latter is responsible e.g. for the prominent normal stress differences (anisotropic pressures), long-range correlations, scale-dependent stress (and other) fields and, in general, the rheological nature of these materials. Constitutive relations and boundary conditions for dilute and near-elastic rapid granular flows have been derived from the pertinent Boltzmann equation with extensions to moderate densities obtained by employing the Enskog-Boltzmann equation or (systematically, via) response theory. Unlike the rapid flows, the dense, static and quasistatic regimes have not been treated in a systematic fashion heretofore. Preliminary results on elasticity in the static regime are presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Knowlton, T. M., J. W. Carson, G. E. Klitzing, and W.-C. Yang. April 1994. Particle technology: The importance of storage, transfer and collection. Chemical Engineering Progress 44, 44–54.
Jenike, A. W. 1964. Storage and flow of solids. Bulletin of the University of Utah Engineering Experiment Station 123.
Johnson, P. C., P. Nott, and R. Jackson. 1990. Frictional-collisional equations of motion of particulate flows and their application to chutes. Journal of Fluid Mechanics 210, 501–535.
Campbell, C. S. 1990. Rapid granular flows. Annual Reviews of Fluid Mechanics 22, 57–92, and references therein.
Ogawa, S., A. Unemura, and N. Oshima. 1980. On the equations of motion of fully fluidized granular materials. Zeitschrift für Mechanik und Physik 31, 482–493, and references therein.
For pioneering works in this field, see [5] and references therein, and Lun, C. K. K. 1991. Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. Journal of Fluid Mechanics 223, 539–559, and references therein.
Goldshtein, A., and M. Shapiro. 1995. Mechanics of collisional motion of granular materials, Part I: General hydrodynamic equations. Journal of Fluid Mechanics 282, 75–114.
Sela, N., and I. Goldhirsch. 1995. Hydrodynamics of a one-dimensional granular medium. Physics of Fluids 7(3), 507–525.
Goldhirsch, I., N. Sela, and S. H. Noskowicz. 1996. Kinetic theoretical study of a simply sheared granular gas—to Burnett order. Physics of Fluids 8(9), 2337–2353.
Goldhirsch, I., and N. Sela. 1996. Origin of normal stress differences in rapid granular flows. Physical Review E 54(4), 4458–4461 (1996).
Sela, N., and I. Goldhirsch. 1998. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. Journal of Fluid Mechanics 361, 41–74, and references therein.
Brey, J. J., F. Moreno, and J. W. Dufty. 1996. Model kinetic equations for low density granular flow. Physical Review E 54(1), 445–456.
Brey, J. J., J. W. Dufty, C. S. Kim, and A. Santos. 1998. Hydrodynamics for granular flow at low density. Physical Review E 58, 4638–4653.
Chapman, S., and T. G. Cowling. 1970. The Mathematical Theory of Nonuniform Gases. Cambridge: Cambridge University Press.
Tan, M.-L., and I. Goldhirsch. 1998. Rapid granular flows as mesoscopic systems. Physical Review Letters 81(14), 3022–3025.
Jaeger, H. M., S. R. Nagel, and R. P. Behringer. 1996. Granular solids, liquids and gases. Reviews of Modern Physics 68(4), 1259–1273, and references therein.
Haff, P. K. 1983. Grain flow as a fluid mechanical phenomenon. Journal of Fluid Mechanics 134, 401–430.
Goldhirsch, I., and G. Zanetti. 1993. Clustering instability in dissipative gases. Physical Review Letters 70, 1619–1622. See also Goldhirsch, I., M.-L. Tan, and G. Zanetti. 1993. A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. Journal of Scientific Computation 8(1), 1–40.
Burnett, D. 1935. The distribution of molecular velocities and the mean motion in a nonuniform gas. Proceedings of the London Mathematical Society 40, 382–435.
See, for example, Bobylev, A. V. 1984. Exact solutions of the nonlinear Boltzmann equation and the theory of Maxwell gas relaxation. Theoretical and Mathematical Physics 60(2) 280–310. See also Gorban, A. N., and I. V. Karlin. 1992. Structure and approximations of the Chapman-Enskog expansion for linearized Grad equations. Transport Theory and Statistical Physics 21, 101–117.
Rosenau, P. 1989. Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. Physical Review A 40, 7193–7196.
Slemrod, M. 1999. Constitutive relations for monoatomic gases based on a generalized rational approximation to the sum of the Chapman-Enskog expansions. Archive for Rational Mechanics and Analysis 146, 73–90. See also Jin, S., and M. Slemrod. Regularization of the Burnett equations for fast granular flows via relaxation. 2000. Preprint.
Ernst, M. H., and J. R. Dorfman. 1976. Nonanalytic dispersion relations for classical fluids II: The general fluid. Journal of Statistical Physics 12(4), 311–357.
Lutsko, J. F. 1996. Molecular chaos, pair correlations and shear induced ordering of hard spheres. Physical Review Letters 77, 2225–2228.
Luding, S., M. Miiller, and S. McNamara. 1998. The validity of molecular chaos in granular flows. World Congress on Particle Technology, Brighton. See also Soto, R., unpublished.
Tan, M.-L., and I. Goldhirsch. 1997. Intercluster interactions in rapid granular shear flows. Physics of Fluids 9(4), 856–869, and references therein.
Glasser, B. J., and I. Goldhirsch. 1999. Scale dependence, correlations and fluctuations of stresses in rapid granular flows. Preprint.
Miller, B., C. O’Hern, and R. P. Behringer. 1996. Stress fluctuations for continuously sheared granular materials. Physical Review Letters 77, 3110–3113.
Babic, M. 1997. Average balance equations for granular materials. International Journal of Engineering Science 35, 523–548.
Murdoch, A. I. 1998. On effecting averages and changes of scale via weighting functions. Archives of Mechanics 50, 531–539, and references therein.
Hopkins, M. A., and M. Y. Louge. 1991. Inelastic microstructure in rapid granular flows of smooth disks. Physics of Fluids A 3(1), 47–57.
McNamara, S., and W. R. Young. 1992. Inelastic collapse and clumping in a one-dimensional granular medium. 1992. Physics of Fluids A 4, 496–504. Recent work is cited in Kadanoff, L. P. 1999. Built upon sand: Theoretical ideas inspired by granular flows. Reviews of Modern Physics 71(1), 435–444.
Goldhirsch, I., and T. P. C. van Noije. 2000. Green-Kubo relations for granular fluids. Physical Review E 61(3), 3241–3244.
For previous work on boundary conditions for granular gases, see Jenkins, J. T., and E. Askari. 1991. Boundary conditions for rapid granular flows. Journal of Fluid Mechanics 223, 497–508, and references therein.
Goldhirsch, I. 1999. Scales and kinetics of granular flows. Chaos 9(3), 659–672.
Ronis, D. 1979. Statistical mechanics of systems nonlinearly displaced from equilibrium I. Physica 99A, 403–434.
See, for example, Liao, C.-L., T.-P. Chang, D.-H. Young, and C. S. Chang. 1997. Stress-strain relationship for granular materials based on the hypothesis of best fit. International Journal of Solids and Structures 34, 4087–4100.
Goldenberg, C., and I. Goldhirsch. 2000. Elasticity of microscopically inhomogeneous systems. Preprint.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this paper
Cite this paper
Goldhirsch, I. (2001). Kinetic and Continuum Descriptions of Granular Flows. In: Aref, H., Phillips, J.W. (eds) Mechanics for a New Mellennium. Springer, Dordrecht. https://doi.org/10.1007/0-306-46956-1_22
Download citation
DOI: https://doi.org/10.1007/0-306-46956-1_22
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-7156-4
Online ISBN: 978-0-306-46956-5
eBook Packages: Springer Book Archive