Advertisement

Rapid Granular Flows: Kinetics and Hydrodynamics

  • Isaac Goldhirsch
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

The kinetics of a monodisperse collection of spherical particles whose collisions are characterized by a fixed coefficient of normal restitution is analyzed in one, two and three dimensions, the major goal being the derivation of hydrodynamic equations of motion and boundary conditions. A generalized Chapman-Enskog expansion serves to extract such equations from the Boltzmann equation, appropriately modified to account for inelastic collisions. Among the results obtained is an explanation of the anisotropic pressure (“normal stress differences”) in granular gases as a Burnett effect. Boundary conditions are developed by employing a novel (systematic) method which is relevant to both inelastic particles and molecules. The limitations of the Chapman-Enskog approach applied to granular gases are discussed as are the limitations of the hydrodynamic equations. A biased outlook on future developments is presented.

Keywords

Boltzmann Equation Knudsen Number Hydrodynamic Equation Granular Flow Normal Stress Difference 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ARa]
    Arn Arson B.Ö. and Willits J.T., Thermal Diffusion in Binary Mixtures of Smooth, Nearly Elastic Spheres in the Presence and Absence of Gravity, Phys. Fluids., 10(6) (1998), 1324–1328.CrossRefGoogle Scholar
  2. [BEa]
    Behringer R.P., The Dynamics of Flowing Sand, Nonlinear Science Today, 3(1993), 1–15.CrossRefGoogle Scholar
  3. [BBa]
    Bobylev A.V., Exact Solutions of the Nonlinear Boltzmann Equation and the Theory of Maxwell Gas Relaxation, Theor. Math. Phys., 60(2) (1984), 280–310.MathSciNetCrossRefGoogle Scholar
  4. [BOa]
    Boyle E.J. and Massoudi M., A Theory for Granular Materials Exhibiting Normal Stress Effects Based on Enskog’s Dense Gas Theory, Int. J. Engng. Sei., 28 (1990), 1261–1275.MATHCrossRefGoogle Scholar
  5. [BRa]
    Brey J.J., Moreno F., and Dufty J.W., Model Kinetic Equations for Low Density Granular Flow, Phys. Rev. E, 54(1) (1996), 445–456.CrossRefGoogle Scholar
  6. [BRb]
    Brey J.J., Dufty J.W., Kim C.S., and Santos A., Hydrodynamics for Granular Flow at Low Density,Phys. Rev. E, 58 (1998), 4638–4653.CrossRefGoogle Scholar
  7. [BUa]
    Burnett D., The Distribution of Molecular Velocities and the Mean Motion in a Nonuniform Gas,Proc. Lond. Math. Soc., 40 (1935), 382–435.MathSciNetGoogle Scholar
  8. [CAa]
    Campbell C.S., Rapid Granular Flows, Ann. Revs. Fluid Mech., 22 (1990), 57–92.CrossRefGoogle Scholar
  9. [CEa]
    Cercignani C., Theory and Application of the Boltzmann Equation, Scottish Academic Press (1975).MATHGoogle Scholar
  10. [CHa]
    Chapman S. and Cowling T.G., The Mathematical Theory of Nonuniform Gases, Cambridge University Press (1970).Google Scholar
  11. [COa]
    Coulomb C.A., Essai sur une Application dies Regies de Maximis et Minimis a Quelques Problèmes de Statique, Relatifs à l’Architecture, Mémoires de Mathématiques et de Physique, Présentes à l’Academie Royale des Sciences par Divers Savants et lus dans les Assemblees, 7, L’Imprimerie Royale, Paris (1776), 343–382.Google Scholar
  12. [Fia]
    Fiscko K. and Chapman D., Comparison of Burnett, Super-Burnett and Monte-Carlo Solutions for Hypersonic Shock Structure, Prog. Aerounau. Astronaut., 118 (1989), 374–395.Google Scholar
  13. [GOa]
    Goldhirsch I., Sela N., and Noskowicz S.H., Kinetic Theoretical Study of a Simply Sheared Granular Gas - to Burnett Order, Phys. Fluids., 8 (9) (1996), 2337–2353.MATHCrossRefGoogle Scholar
  14. [GOb]
    Goldhirsch I. and Sela N., Origin of Normal Stress Differences in Rapid Granular Flows, Phys. Rev. E, 54 (4) (1996), 4458–4461.CrossRefGoogle Scholar
  15. [GOc]
    Goldhirsch I. and Zanetti G., Clustering Instability in Dissipa- tive Gases, Phys. Rev. Lett., 70 (1993), 1619–1622.CrossRefGoogle Scholar
  16. [GOd]
    Goldhirsch I., Tan M-L., and Zanetti G., A Molecular Dynamical Study of Granular Fluids I: The Unforced Granular Gas in Two Dimensions, J. Sei. Comp., 8(1) (1993), 1–40.MATHCrossRefGoogle Scholar
  17. [GOe]
    Goldhirsch I. and Tan M-L., The Single Particle Distribution Function for Rapid Granular Shear Flows of Smooth Inelastic Disks, Phys. Fluids, 8(7) (1996), 1752–1763.CrossRefGoogle Scholar
  18. [GOf]
    Goldhirsch I. and VAN Noije T.P.C., Green-Kubo Relations for Granular Fluids, preprint (1998).Google Scholar
  19. [GDa]
    Goldshtein A. and Shapiro M., Mechanics of Collisional Motion of Granular Materials, Part I: General Hydrodynamic Equations, J. Fluid Mech., 282 (1995), 75 114.MathSciNetGoogle Scholar
  20. [GBa]
    Gorban A.N. and Karlin I.V., Structure and approximations of the Chapman-Enskog expansion for the linearized Grad equations, Trans. Theory Stat. Phys., 21 (1992), 101–117.MathSciNetMATHCrossRefGoogle Scholar
  21. [GRa]
    Grad H., On the Kinetic: Theory of Rarefied Gases, Comm. Pure Appl. Math., 2 (1949), 331 407.Google Scholar
  22. [HFa]
    Haff P.K., Grain Flow as a Fluid Mechanical Phenomenon, J. Fluid Mech., 134 (1983), 401 430.Google Scholar
  23. [HAa]
    Hagen G., Über den Druck und die Bewegung des Trockenen Sandes, Monatsberichte der Königlich, Preußischen Akademie der Wissenschaften zu Berlin (Jan 19, 1852), 35–442.Google Scholar
  24. [HRa]
    Harris S., Introduction to the Theory of the Boltzmann equation, Holt, Reinhart and Winston (1971).Google Scholar
  25. [HHa]
    Herrmann H.J., Hovi J.-P., and Luding S. Eds., Physics of Dry Granular Media, NATO ASI Series E: Applied Sciences, Vol. 350 (1998).MATHGoogle Scholar
  26. [HOa]
    Hopkins M. A. and Louge M.Y., Inelastic Microstructure in Rapid Granular Flows of Smooth Disks, Phys. Fluids A, 3(1) (1991), 47–57.CrossRefGoogle Scholar
  27. [HOb]
    Hopkins M.A. and Shen H.H., A Monte-Carlo Solution for Rapidly Shearing Granular Flows Based on the Kinetic Theory of Dense Gases, J. Fluid Mech., 244 (1992), 477–491.MATHCrossRefGoogle Scholar
  28. [JAa]
    Jaeger J.M. and Nagel S.R., Granular Solids, Liquids and Gases, Rev. Mod. Phys., 68 (1996), 1259–1273.CrossRefGoogle Scholar
  29. [JEa]
    Jenkins J.T. and Savage S.B., A Theory for Rapid Granular Flow of Identical, Smooth, Nearly Elastic, Spherical Particles, J. Fluid Mech., 130 (1983), 187–202.MATHCrossRefGoogle Scholar
  30. [JEb]
    Jenkins J.T. and Richman M.W., Grad’s 13-Moment System for a Dense Gas of Inelastic Particles, Phys. Fluids, 28 (1985), 3485–3494.MATHCrossRefGoogle Scholar
  31. [JEc]
    Jenkins J.T. and Richman M.W., Plane Simple Shear of Smooth Inelastic Circular Disks: the Anisotropy of the Second Moment, in the Dilute and Dense Limits, J. Fluid. Mech., 192 (1988), 313–328.MATHCrossRefGoogle Scholar
  32. [JEd]
    Jenkins J.T. and Richman M.W., Grad’s 13-Moment System for A Dense Gas of Inelastic Spheres, Arch. Ration. Mech. Anal., 87 (1985), 355–377.MathSciNetMATHCrossRefGoogle Scholar
  33. [JEe]
    Jenkins J.T. and Richman M.W., Plane Simple Shear of Smooth Inelastic Circular Disks: the Anisotropy of the Second Moment in the Dilute and Dense Limits, J. Fluid. Mech., 192 (1988), 313–328.MATHCrossRefGoogle Scholar
  34. [JEf]
    Jenkins J.T. and Richman M.W., Boundary Conditions for Plane Flows of Smooth, Nearly Elastic, Circular Disks, J. Fluid Mech., 171 (1986), 53–69.MATHCrossRefGoogle Scholar
  35. [JEg]
    Jenkins J.T., Boundary Conditions for Rapid Granular Flow: Flat, Frictional Walls, J. Appl. Mech., 114 (1992), 120–127.CrossRefGoogle Scholar
  36. [JEH]
    Jenkins J.T. and Askari E., Boundary Conditions for Rapid Granular Flows, J. Fluid Mech., 223 (1991), 497–508.CrossRefGoogle Scholar
  37. [JEi]
    Jenkins J.T. and Mancini F., Balance Laws and Constitutive Relations for Plane Flows of a Dense, Binary Mixture of Smooth, Nearly Elastic Disks, J. Appl. Mech., 109 (1987), 27–34.CrossRefGoogle Scholar
  38. [JEj]
    Jenkins J.T. and Mancini F., Kinetic Theory for Smooth, Nearly Elastic Spheres, Phys. Fluids, A 1 (1989), 2050–2057.Google Scholar
  39. [KOa]
    Kogan M.K., Rarefied Gas Dynamics, Plenum Press (1969).Google Scholar
  40. [LUa]
    Lun C.K.K., Savage S.B., Jeffrey D.J. and Chepurniy N., Kinetic Theories of Granular Flow: Inelastic Particles in a Couette Flow and Slightly Inelastic Particles in a General Flow Field, J. Fluid Mech., 140 (1984), 223–256.MATHCrossRefGoogle Scholar
  41. [LUb]
    Lun C.K.K, and Savage S.B., A Simple Kinetic Theory for Granular Flow of Rough Inelastic Spheres, J. Appl. Mech., 154 (1987), 47–53.CrossRefGoogle Scholar
  42. [LUc]
    Lun C.K.K., Kinetic Theory for Granular Flow of Dense, Slightly Inelastic, Slightly Rough Spheres, J. Fluid Mech., 223 (1991), 539–559.CrossRefGoogle Scholar
  43. [MNa]
    Mcnamara S. and Young W.R., Inelastic Collapse in Two Dimensions, Phys. Rev. E, 50(1994), R28-R31.CrossRefGoogle Scholar
  44. [MNb]
    Mcnamara S. and Young W.R., Inelastic Collapse and Clumping in a One-Dimensional Granular Medium, Phys. Fluids, A 4 (1992), 496–504.Google Scholar
  45. [MNc]
    Mcnamara S. and Young W.R., Kinetics of a One-Dimensional Granular Medium in the Quasielastic Limit, Phys. Fluids, A5(L) (1993), 34–45.MathSciNetGoogle Scholar
  46. [MOa]
    Mori H., Time-Correlation Functions in the Statistical Mechanics of Transport Processes, Phys. Rev., 111 (1958), 694–706.MathSciNetMATHCrossRefGoogle Scholar
  47. [MOb]
    Mori H., Statistical Mechanical Theory of Transport in Fluids, Phys. Rev., 112 (1958), 1829–1842.MathSciNetMATHCrossRefGoogle Scholar
  48. [NOa]
    Noskowicz S.H., Bar-LEV, O., and Goldhirsch I, Velocity Distribution Function of an Isotropic and Homogeneous Granular Medium, preprint (1998).Google Scholar
  49. [NOb]
    Noskowicz S.H., BAR-LEV O., and GOLDHIRSCH I., Hydrodynamics of Collections of Frictional Spherical Grains, unpublished (1998).Google Scholar
  50. [OPa]
    Oppenheim I., Nonlinear Response Theory, in Correlation Functions and Quasiparticle Interactions in Condensed Matter, ed. J. Woods Halley, Plenum (1978), 235–258 and references therein.Google Scholar
  51. [PEa]
    Pekeris C.L., Solution of the Boltzmann-Hilbert Integral Equation, Proc. N.A.S., 41 (1955), 661–664.MathSciNetMATHCrossRefGoogle Scholar
  52. [RAa]
    Rapaport D.C., Large-Scale Molecular Dynamics Simulations Using Vector and Parallel Computers,Computer Phys. Repts., 9 (1988), 1–53.CrossRefGoogle Scholar
  53. [REa]
    Reynolds O., On the Dilatancy of Media Composed of Rigid Particles in Contact, Phil. Mag., 8 (1885), 22–53.Google Scholar
  54. [RIa]
    Richman M.W., Boundary Conditions Based Upon a Modified Maxwellian Velocity Distribution for Flows of Identical, Smooth, Nearly Elastic Spheres, Acta Mech., 75 (1988), 227–240.CrossRefGoogle Scholar
  55. [ROa]
    Rosenau P., Extending Hydrodynamics via the Regularization of the Chapman-Enskog Expansion,Phys. Rev. A, 40 (1989), 7193–7196.MathSciNetCrossRefGoogle Scholar
  56. [SAa]
    Savage S.B., Analyses of Slow, High Concentration Flows of Granular Materials, J. Fluid Mech., 377 (1999), 1–26.MathSciNetCrossRefGoogle Scholar
  57. [SEa]
    Sela N. and Goldhirsch I., Hydrodynamic Equations for Rapid Flows of Smooth Inelastic Spheres, to Burnett order, J. Fluid Mech., 361 (1998), 41–74 and references therein.MathSciNetMATHCrossRefGoogle Scholar
  58. [SEb]
    Sela N. and Goldhirsch I., Hydrodynamics of a One-Dimensional Granular Medium, Phys. Fluids, 7(3) (1995), 507–525.MATHCrossRefGoogle Scholar
  59. [SEc]
    Sela N. and Goldhirsch I., Boundary Conditions for Granular and Molecular Gases: a Systematic Approach, unpublished (1998).Google Scholar
  60. [SLa]
    Slemrod M., Constitutive Relations for Monoatomic Gases Based on a Generalized Rational Approximation to the Sum of the Chapman-Enskog Expansions, preprint (1998).Google Scholar
  61. [TAa]
    Tan M-L. and Goldhirsch I., Intercluster Interactions in Rapid Granular Shear Flows, Phys. Fluids, 9(4) (1997), 856–869.CrossRefGoogle Scholar
  62. [TAb]
    Tan M-L. and Goldhirsch I., Rapid Granular Flows as Mesoscopic Systems, Phys. Rev. Lett., 81(14) (1998), 3022–3025.CrossRefGoogle Scholar
  63. [UMa]
    Umbanhowar P.B., Melo F., and Swinney H.L., Localized Excitations in a Vertically Vibrated Layer, Nature, 382 (1996), 793–796.CrossRefGoogle Scholar
  64. [WAa]
    Walton O.R. and Braun R.L., Stress Calculations for Assemblies of Inelastic Spheres in Uniform Shear, Acta Mech., 63 (1986), 73–86.CrossRefGoogle Scholar
  65. [WAb]
    Walton O.R. and Braun R.L., Viscosity and Temperature Calculations for Shearing Assemblies of Inelastic, Frictional Disks, J. Rheol., 30 (1986), 949–980.CrossRefGoogle Scholar
  66. [WOa]
    Woods L.C., Transport Processes in Dilute Gases Over the Whole Range of Knudsen Numbers. Part I: General Theory, J. Fluid. Mech 93 (1979), 585–607.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Isaac Goldhirsch
    • 1
  1. 1.Department of Fluid Mechanics and Heat Transfer, Faculty of EngineeringTel-Aviv UniversityRamat-Aviv, Tel-AvivIsrael

Personalised recommendations