Chaos
Feedback | Help | Exit
Chaos
   
 
 
 
Previous Article
Hydraulic theory for a debris flow supported on a collisional shear layer
We consider a heap of grains driven by gravity down an incline. We assume that the heap is supported at its base on a relatively thin carpet of intensely sheared, highly agitated grains that interact ...
Next Article
Cluster-growth in freely cooling granular media
When dissipative particles are left alone, their fluctuation energy decays due to collisional interactions, clusters build up and grow with time until the system size is reached. When the effective di...

Scales and kinetics of granular flows

Chaos 9, 659 (1999); doi:10.1063/1.166440

Issue Date: September 1999

You are not logged in to this journal. Log in

I. Goldhirsch
Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel
When a granular material experiences strong forcing, as may be the case, e.g., for coal or gravel flowing down a chute or snow (or rocks) avalanching down a mountain slope, the individual grains interact by nearly instantaneous collisions, much like in the classical model of a gas. The dissipative nature of the particle collisions renders this analogy incomplete and is the source of a number of phenomena which are peculiar to "granular gases," such as clustering and collapse. In addition, the inelasticity of the collisions is the reason that granular gases, unlike atomic ones, lack temporal and spatial scale separation, a fact manifested by macroscopic mean free paths, scale dependent stresses, "macroscopic measurability" of "microscopic fluctuations" and observability of the effects of the Burnett and super-Burnett "corrections." The latter features may also exist in atomic fluids but they are observable there only under extreme conditions. Clustering, collapse and a kinetic theory for rapid flows of dilute granular systems, including a derivation of boundary conditions, are described alongside the mesoscopic properties of these systems with emphasis on the effects, theoretical conclusions and restrictions imposed by the lack of scale separation. ©1999 American Institute of Physics.
History: Received 31 December 1998; accepted 25 June 1999
Permalink: http://link.aip.org/link/?CHAOEH/9/659/1
BUY THIS ARTICLE   (US$28)
Download PDF (175 kB) View Cart

KEYWORDS and PACS

Keywords
PACS
  • 45.70.Mg
    Classical mechanics of discrete systems Granular systems Granular flow: mixing, segregation and stratification
  • 47.55.Kf
    Fluid dynamics Nonhomogeneous flows Multiphase and particle-laden flows
  • 05.20.Dd
    Statistical physics, thermodynamics, and nonlinear dynamical systems Classical statistical mechanics Kinetic theory
  • YEAR: 1999

PUBLICATION DATA

ISSN:
1054-1500 (print)   1089-7682 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (56)
View in Separate Window/Tab
For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. A. Coulomb, Memoires de Mathematiques et de Physique Presentes a l'Academie Royale des Sciences par Divers Savants et lus dans les Assemblees, L'Imprimerie Royale, Paris, 1776.
  2. G. Hagen, Berlin Monatsberichte Akad. Wis. 35, 442 (1852).
  3. O. Reynolds, Philos. Mag. 20, 469–481 (1885).
  4. cf., e.g., Physics of Dry Granular Media, edited by H. J. Herrmann, J.-P. Hovi, and S. Luding, NATO ASI Series E: Applied Sciences, Vol. 350, 1998;
  5. also some recent reviews (a) R. P. Behringer, Nonlinear Sci. Today 3, 1 (1993);
  6. (b) J. M. Jaeger and S. R. Nagel, Rev. Mod. Phys. 68, 1259–1273 (1996).
  7. P. B. Umbanhowar, F. Melo, and H. L. Swinney, Nature (London) 382, 793 (1996).
  8. An old reference proposing an explanation for the normal stress differences is M. W. Richman, J. Rheol. 33, 1293 (1989), and references therein. See also other references on this topic in the review paper by Campbell, Ref. 10 below.
  9. I. Goldhirsch, N. Sela, and S. H. Noskowicz, Phys. Fluids 8, 2337 (1996).
  10. I. Goldhirsch and N. Sela, Phys. Rev. E 54, 4458 (1996).
  11. N. Sela and I. Goldhirsch, J. Fluid Mech. 361, 41 (1998), and references therein.
  12. Review on rapid granular flows: C. S. Campbell, Annu. Rev. Fluid Mech. 22, 57 (1990).
  13. M. Hopkins and H. Shen, J. Fluid Mech. 244, 477–491 (1992).
  14. D. Burnett, Proc. London Math. Soc. 40, 382 (1935).
  15. M. K. Kogan, Rarefied Gas Dynamics (Plenum, New York, 1969).
  16. S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge Univ. Press, Cambridge, 1970).
  17. For example, (a) L. C. Woods, J. Fluid Mech. 93, 585 (1979);
    18. (b) K. Fiscko and D. Chapman, AIAA J. 118, 374 (1989).
  18. M. A. Hopkins and M. Y. Louge, Phys. Fluids A 3, 47 (1991).
  19. (a) I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993);
    20. (b) I. Goldhirsch, M-L. Tan, and G. Zanetti, J. Sci. Comput. 8, 1 (1993);
    (c) M-L. Tan and I. Goldhirsch, Phys. Fluids 9, 856 (1997).
  20. (a) I. Goldhirsch and M-L. Tan, Phys. Fluids 8, 1752–1763 (1996);
    21. (b) M-L. Tan and I. Goldhirsch, Phys. Fluids 9, 856–869 (1997).
  21. S. McNamara and W. R. Young, Phys. Fluids A 4, 496 (1992).
  22. S. McNamara and W. R. Young, Phys. Fluids A 5, 34 (1993).
  23. S. McNamara and W. R. Young, Phys. Rev. E 50, R28–R31 (1994).
  24. P. K. Haff, J. Fluid Mech. 134, 401–948 (1983).
  25. M-L. Tan and I. Goldhirsch, Phys. Rev. Lett. 81, 3022–3025 (1998).
  26. cf. e.g., (a) A. V. Bobylev, Theor. Math. Phys. 60, 280–310 (1984);
    27. (b) A. N. Gorban and I. V. Karlin, Transp. Theory Stat. Phys. 21, 101–117 (1992).
  27. P. Rosenau, Phys. Rev. A 40, 7193–7196 (1989);
    28. see also M. Slemrod, Arch. Ration. Mech. Anal. 146, 73 (1999).
  28. M. H. Ernst and J. R. Dorfman, J. Stat. Phys. 12, 311 (1976).
  29. I. Oppenheim, in Correlation Functions and Quasiparticle Interactions in Condensed Matter, edited by J. Woods Halley (Plenum, 1978), and references therein;
    30. D. Ronis, Physica A 99, 403 (1979).
  30. (a) H. Mori, Phys. Rev. 111, 694 (1958);
    31. (b) H. Mori, 112, 1829 (1958).
  31. I. Goldhirsch and T. P. C. van Noije, "Green-Kubo relations for granular fluids," preprint, 1998.
  32. S. Harris, Introduction to the Theory of the Boltzmann Equation (Holt, Reinhart, and Winston, New York, 1971).
  33. C. Cercignani, Theory and Application of the Boltzmann Equation (Scottish Academic, Edinburgh and London, 1975).
  34. T. P. C. van Noije, M. H. Ernst, R. Brito, and J. A. G. Orza, Phys. Rev. Lett. 79, 411 (1997).
  35. J. F. Lutsko, Phys. Rev. Lett. 77, 2225 (1996), and references therein.
  36. A. I. Murdoch, Arch. Mech. 50, 531 (1998), and references therein.
  37. M. Babic, Int. J. Eng. Sci. 35, 523 (1997).
  38. B. J. Glasser and I. Goldhirsch, preprint, 1998.
  39. S. B. Savage, in Disorder, Diffusion and Structure Formation in Granular Flows, edited by D. Bideau and A. Hansen [Random Materials and Processes Ser., edited by H. E. Stanley and E. Guyon (North Holland, Amsterdam, 1993)].
  40. C. O'Hern, B. Miller, and R. P. Behringer, Phys. Rev. E 54, 4458 (1996).
  41. J. J. Brey, F. Moreno, and J. W. Dufty, Phys. Rev. E 54, 1270 (1996).
  42. J. J. Brey, J. W. Dufty, C. S. Kim, and A. Santos, Phys. Rev. E 58, 4638 (1998).
  43. S. H. Noskowicz, O. Bar-Lev, and I. Goldhirsch (unpublished).
  44. (a) I. Kuscer and M. M. R. Williams, Phys. Fluids 10, 1922 (1967);
    45. (b) F. Schurrer and G. Kugerl, Phys. Fluids A 2, 609 (1990).
  45. (a) S. E. Esipov and T. Poschel, J. Stat. Phys. 86, 1385 (1997);
    46. (b) J. J. Brey, D. Cubero, and M. J. Ruiz-Montero, Phys. Rev. E 59, 1256 (1999).
  46. (a) J. T. Jenkins and S. B. Savage, J. Fluid Mech. 130, 187 (1983);
    47. (b) C. K. K. Lun, S. B. Savage, D. J. Jeffrey, and N. Chepurniy, 140, 223–256 (1984);
    (c) J. T. Jenkins and M. W. Richman, Phys. Fluids 28, 3485 (1985);
    (d) C. K. K. Lun and S. B. Savage, J. Appl. Mech. 154, 47 (1987);
    (e) J. T. Jenkins and M. W. Richman, J. Fluid Mech. 192, 313–328 (1988).
  47. J. T. Jenkins and M. W. Richman, Arch. Ration. Mech. Anal. 87, 355–377 (1985).
  48. C. K. K. Lun, J. Fluid Mech. 223, 539 (1991).
  49. E. J. Boyle and M. Massoudi, Int. J. Eng. Sci. 28, 1261–1275 (1990).
  50. (a) O. R. Walton and R. L. Braun, Acta Mech. 63, 73–86 (1986).
    51. (b) O. R. Walton and R. L. Braun, J. Rheol. 30, 949–980 (1986).
  51. H. Grad, Commun. Pure Appl. Math. 2, 331–407 (1949).
  52. N. Sela and I. Goldhirsch, "Boundary Conditions for Granular and Molecular Gases: a Systematic Approach" (unpublished).
  53. Some previous work on boundary conditions for granular gases: (a) J. T. Jenkins and M. W. Richman, J. Fluid Mech. 171, 53–69 (1986);
    54. (b) M. W. Richman, Acta Mech. 75, 227–240 (1988);
    (c) J. T. Jenkins, J. Appl. Mech. 59, 120–127 (1992);
    (c) J. T. Jenkins and E. Askari, J. Fluid Mech. 223, 497–508 (1991).
  54. C. L. Pekeris, Proc. Natl. Acad. Sci. USA 41, 661–664 (1955).
  55. N. Sela and I. Goldhirsch, Phys. Fluids 7, 507–525 (1995).
  56. Advances in Thermodynamics, edited by S. Salomon and S. Sienintycz (Francis and Taylor, New York, 1991), and numerous references therein.
  57. L. P. Kadanoff, Rev. Mod. Phys. 71, 435–444 (1999).
  58. J. J. Brey, M. J. Ruiz-Montero, and D. Cubero (unpublished).

CITING ARTICLES
For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.


Copyright © American Institute of Physics