Abstract
Results of a numerical study of the dynamics of a collection of disks colliding inelastically in a periodic two-dimensional enclosure are presented. The properties of this system, which is perhaps the simplest model for rapidly flowing granular materials, are markedly different from those known for atomic or moleclar gases, in which collisions are of elastic nature. The most prominent feature characterizing granular systems, even in the idealized situation in which no external forcing exists and the initial condition is statistically homogeneous, is their inherent instability to inhomogeneous fluctuations. Granular “gases” are thus generically nonuniform, a fact that suggests extreme caution in pursuing direct analogies with molecular gases. We find that once an inhomogeneous state sets in, the velocity distribution functions differ from the classical Maxwell-Boltzmann distribution. Other characteristics of the system are different from their counterparts in molecular systems as well. For a given value of the coefficient of restitution,e, a granular system forms clusters of typical separationL 0≈l/(1-e 2)1/2, wherel is the mean free path in the corresponding homogeneous system. Most of the fluctuating kinetic energy then resides in the relatively dilute regions that surround the clusters. Systems whose linear dimensions are less thenL 0 do not give rise to clusters; still they are inhomogeneous, the scale of the corresponding inhomogeneity being the longest wavelength allowed by the system's size. The present article is devoted to a detailed numerical study of the above-mentioned clustering phenomenon in two dimensions and in the absence of external forcing. A theoretical framework explaining this phenomenon is outlined. Some general implications as well as practical ramifications are discussed.
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References
Araki, S. (1988). The dynamics of particle discs II: Effects of spin degrees of freedom,Icarus 76, 182–198.
Babic, M. (1991). Particle clustering: An instability of rapid granular flows, inProc. U.S.-Japan Seminar on the Micromechanics of Granular Materials, Potsdam, New York.
Bagnold, R. A. (1954). Experiments on a gravity free dispersion of large solid particles in a newtonian fluid under shear,Proc. R. Soc. London A 225, 49–63.
Bolescu, R. (1975).Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York.
Boyle, E. J., and Massoudi, M. A. (1990). A theory for granular materials exhibiting normal stress effects based on Enskog's dense gas theory,Int. J. Eng. Sci. 28(12), 1261–1275, and references therein.
Brown, R. L., and Richards, J. C. (1970).Principles of Powder Mechanics, Pergamon, London.
Campbell, C. S. (1989). The stress tensor for simple shear flows of a granular material,J. Fluid Mech. 203, 449–473.
Campbell, C. S. (1990). Rapid granular flows,Ann. Rev. Fluid Mech. 22, 57–92.
Campbell, C. S., and Brennan, C. E. (1985). Computer simulations of granular shear flows,J. Fluid Mech. 151, 167–188.
Campbell, C. S., and Gong, A. (1986). The stress tensor in a two-dimensional granular shear flow,J. Fluid Mech. 164, 107–125.
Chapman, S., and Cowling, T. G. (1970).The Mathematical Theory of Nonuniform Gases. Cambridge U. P., 3rd ed. edition.
Evesque, P. (1991). Analysis of the statistics of sandpile avanlanches using soil mechanics: Results and concepts,Phys. Rev. A 43, 2720–2740.
Goldhirsch, I. (1991). Clustering instability in granular gases, in Peters, W. C., Plasynski, S. I., and Roco, M. C. (eds.),Proceedings of the Joint DOE/NSF Workshop on Flow of Particulates and Fluids, pp. 211–235, W.P.I., Worcester, Massachusetts.
Goldhirsch, I., and Tan, M.-L. (1992). Unpublished results.
Goldhirsch, I., and Zanetti, G. (1992). Clustering instability in dissipative gases, preprint.
Goldreich, P., and Tremaine, S. (1978). The velocity dispersion in Saturn's rings,Icarus 34, 227–239.
Goodman, M. A., and Cowin, S. C. (1971). Two problems in gravity flows of granular materials,J. Fluid Mech. 45, 321–339.
Hanes, D. M., and Inman, D. L. (1985). Observations of rapidly flowing granular fluid materials,J. Fluid Mech. 150, 357–380.
Hopkins, M. A., and Louge, M. Y. (1991). Inelastic microstructure in rapid granular flows of smooth disks,Phys. Fluids A3(1), 47–57.
Jeffrey, D. J., Lun, C. K. K., Savage, S. B., and Chepurnyi, N. (1984). Kinetic theories of granular flow: Inelastic particles in a Couette flow and slightly inelastic particles in a general flow field,J. Fluid Mech. 140, 223–256, and references therein.
Jenkins, J. T., and Askari, E. (1991). Rapid granular shear flows driven by identical, bumpy, frictionless boundaries, in Peters, W. C., Plasynski, S. I., and Roco, M. C. (eds.),Proceedings of the Joint DOE/NSF Workshop on Flow of Particulates and Fluids, W.P.I., Worcester, Massachusetts.
Jenkins, J. T., and Richman, M. W. (1985). Grad's 13-moment system for a dense gas of inelastic spheres,Arch. Rat. Mech. Anal. 87, 355–377.
Jenkins, J. T., and Savage, S. B. (1983). A theory for rapid flow of identical, smooth, nearly elastic, spherical particles,J. Fluid Mech. 130, 187–202, and references therein.
Johnson, P. C., and Jackson, R. (1987). Frictional-collisional constitutive relations for granular materials with applications to plane shearing,J. Fluid Mech. 176, 67–93, and references therein.
Kim, H., Walton, O. R., and Rosato, A. D. (1991). Microstructure and stress differences in shearing flows, in Adeli, H., and Sierakowski, R. L. (eds.),Mechanics Computing in 1990's and Beyond, Vol. 2, pp. 1249–1253, ASCE, New York.
Liboff, R. L. (1990).Kinetic Theory: Classical, Quantum and Relativistic Descriptions, Prentice Hall, Englewood Cliffs, New Jersey.
Lun, C. K. K. (1991). Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres,J. Fluid Mech. 223, 539–559.
Lun, C. K. K., and Savage, S. B. (1987). A simple kinetic theory for granular flow of rough inelastic spherical particles,J. Appl. Mech. 154, 47–53, and references therein.
Nott, P., Johnson, P. C., and Jackson, R. (1990). Frictional-collisional equations of motion of particulate flows and their applications to chutes,J. Fluid Mech. 210, 501–535.
Ogawa, S. (1978). Multitemperature theory of granular materials, inPprc. U.S.-Japan Seminar on Continuum Mechanics and Statistical Approaches to the Mechanics of Granular Matter, Vol. 31, pp. 208–217, Gukujutsu Bunken Fukyukai, Tokyo.
Passman, S. L., Nunziato, J. W., and Thomas, J. R. (1980). Gravitational flows of granular materials with incompressible grains,J. Rheol. 24, 395–420.
Rapaport, D. C. (1980). The event scheduling problem in molecular dynamic simulation,J. Comput. Phys. 34, 184–201.
Rapaport, D. C. (1988). Large-scale molecular dynamics simulation using vector and parallel computers,Comput. Phys. Rep. 9, 1–53, and references therein.
Richman, M. W. (1989). The source of the second moment in dilute granular flows of highly inelastic spheres,J. Rheol. 33(8), 1293–1306, and references therein.
Savage, S. B. (1979). Gravity flow of cohesionless granular materials in chutes and channels,J. Fluid Mech. 92, 53–96.
Savage, S. B. (1992). Instability of an unbounded uniform granular shear flow,J. Fluid Mech. 241, 109–123.
Sokolowski, V. (1965).Statics of Granular Media, Pergamon, Oxford.
Unemera, A., Ogawa, S., and Oshima, N. (1980). On the equations of fully fluidized granular materials,ZAMP 31, 482–493.
van Beijeren, H., and Ernest, M. H. (1979). Kinetic theory of hard spheres,J. Stat. Phys. 21, 125.
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Goldhirsch, I., Tan, M.L. & Zanetti, G. A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. J Sci Comput 8, 1–40 (1993). https://doi.org/10.1007/BF01060830
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DOI: https://doi.org/10.1007/BF01060830