Journal of Scientific Computing

, Volume 8, Issue 1, pp 1–40 | Cite as

A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions

  • I. Goldhirsch
  • M. -L. Tan
  • G. Zanetti
Article

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 separationL0l/(1-e2)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 thenL0 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.

Key words

Granular flow dynamics molecular dynamics numerical simulation 

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References

  1. Araki, S. (1988). The dynamics of particle discs II: Effects of spin degrees of freedom,Icarus 76, 182–198.Google Scholar
  2. 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.Google Scholar
  3. 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.Google Scholar
  4. Bolescu, R. (1975).Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York.Google Scholar
  5. 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.Google Scholar
  6. Brown, R. L., and Richards, J. C. (1970).Principles of Powder Mechanics, Pergamon, London.Google Scholar
  7. Campbell, C. S. (1989). The stress tensor for simple shear flows of a granular material,J. Fluid Mech. 203, 449–473.Google Scholar
  8. Campbell, C. S. (1990). Rapid granular flows,Ann. Rev. Fluid Mech. 22, 57–92.Google Scholar
  9. Campbell, C. S., and Brennan, C. E. (1985). Computer simulations of granular shear flows,J. Fluid Mech. 151, 167–188.Google Scholar
  10. Campbell, C. S., and Gong, A. (1986). The stress tensor in a two-dimensional granular shear flow,J. Fluid Mech. 164, 107–125.Google Scholar
  11. Chapman, S., and Cowling, T. G. (1970).The Mathematical Theory of Nonuniform Gases. Cambridge U. P., 3rd ed. edition.Google Scholar
  12. Evesque, P. (1991). Analysis of the statistics of sandpile avanlanches using soil mechanics: Results and concepts,Phys. Rev. A 43, 2720–2740.Google Scholar
  13. 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.Google Scholar
  14. Goldhirsch, I., and Tan, M.-L. (1992). Unpublished results.Google Scholar
  15. Goldhirsch, I., and Zanetti, G. (1992). Clustering instability in dissipative gases, preprint.Google Scholar
  16. Goldreich, P., and Tremaine, S. (1978). The velocity dispersion in Saturn's rings,Icarus 34, 227–239.Google Scholar
  17. Goodman, M. A., and Cowin, S. C. (1971). Two problems in gravity flows of granular materials,J. Fluid Mech. 45, 321–339.Google Scholar
  18. Hanes, D. M., and Inman, D. L. (1985). Observations of rapidly flowing granular fluid materials,J. Fluid Mech. 150, 357–380.Google Scholar
  19. Hopkins, M. A., and Louge, M. Y. (1991). Inelastic microstructure in rapid granular flows of smooth disks,Phys. Fluids A3(1), 47–57.Google Scholar
  20. 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.Google Scholar
  21. 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.Google Scholar
  22. 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.Google Scholar
  23. 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.Google Scholar
  24. 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.Google Scholar
  25. 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.Google Scholar
  26. Liboff, R. L. (1990).Kinetic Theory: Classical, Quantum and Relativistic Descriptions, Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  27. Lun, C. K. K. (1991). Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres,J. Fluid Mech. 223, 539–559.Google Scholar
  28. 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.Google Scholar
  29. 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.Google Scholar
  30. 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.Google Scholar
  31. Passman, S. L., Nunziato, J. W., and Thomas, J. R. (1980). Gravitational flows of granular materials with incompressible grains,J. Rheol. 24, 395–420.Google Scholar
  32. Rapaport, D. C. (1980). The event scheduling problem in molecular dynamic simulation,J. Comput. Phys. 34, 184–201.Google Scholar
  33. Rapaport, D. C. (1988). Large-scale molecular dynamics simulation using vector and parallel computers,Comput. Phys. Rep. 9, 1–53, and references therein.Google Scholar
  34. 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.Google Scholar
  35. Savage, S. B. (1979). Gravity flow of cohesionless granular materials in chutes and channels,J. Fluid Mech. 92, 53–96.Google Scholar
  36. Savage, S. B. (1992). Instability of an unbounded uniform granular shear flow,J. Fluid Mech. 241, 109–123.Google Scholar
  37. Sokolowski, V. (1965).Statics of Granular Media, Pergamon, Oxford.Google Scholar
  38. Unemera, A., Ogawa, S., and Oshima, N. (1980). On the equations of fully fluidized granular materials,ZAMP 31, 482–493.Google Scholar
  39. van Beijeren, H., and Ernest, M. H. (1979). Kinetic theory of hard spheres,J. Stat. Phys. 21, 125.Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • I. Goldhirsch
    • 1
    • 2
  • M. -L. Tan
    • 2
  • G. Zanetti
    • 2
  1. 1.Department of Fluid Mechanics and Heat Transfer, Faculty of EngineeringTel-Aviv UniversityRamat AvivIsrael
  2. 2.Program in Applied and Computational MathematicsPrinceton UniversityPrinceton
  3. 3.CRS4CagliariItaly

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