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Modelling and numerical methods for granular gases

  • Lorenzo Pareschi
  • Giuseppe Toscani
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

We discuss certain kinetic models of dilute granular systems of spheres with dissi- pative collisions and variable coefficient of restitution. Under the assumption of weak inelastic- ity the cooling process of the system is studied and some hydrodynamical models are derived. Accurate numerical methods based on a spectral representation in velocity are also presented and the development of fast algorithms is considered.

Keywords

Boltzmann Equation Inelastic Collision Granular Flow Collision Operator Kernel Mode 
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.

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References

  1. 1.
    A. Baldassarri, U. Marini Bettolo Marconi, A. Puglisi. Kinetic models of inelastic gases. Mat. Mod. Meth. Appl. Sci. 12 (2002) 965-983.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    N. Bellomo, M. Esteban, M. Lachowitz. Nonlinear kinetic equations with dissipative collisions. Appl. Math. Letters 8 (1995), 46-52.CrossRefGoogle Scholar
  3. 3.
    N. Ben-Nairn, P. Krapivski. Multiscaling in inelastic collisions. Phys. Rev. £, 61 (2000), R5-R8.Google Scholar
  4. 4.
    D. Benedetto, E. Caglioti, M. Pulvirenti. A kinetic equation for granular media. Mat. Mod. Numer. Anal. 31 (1997), 615-641.MathSciNetMATHGoogle Scholar
  5. 5.
    D. Benedetto, E. Caglioti, J.A. Carrillo, M. Pulvirenti. A non maxwellian distribution for one-dimensional granular media./Statist. Phys. 91 (1998), 979-990.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    D. Benedetto, E. Caglioti, F. Golse, M. Pulvirenti. A hydrodynamic model arising in the context of granular media. Comput. Math. Appl. Computers and Math, with Applications 38(1999), 121-131.MathSciNetMATHCrossRefGoogle Scholar
  7. 7. M. Bisi, G. Spiga, G. Toscani. Hydrodynamics from Grad's equations for weakly inelastic granular flows, (preprint).Google Scholar
  8. 8.
    A.V. Bobylev, J.A-Carrillo, I. Gamba. On some properties of kinetic and hydrodynamics equations for inelastic interactions J. Statist. Phys. 98 (2000), 743-773.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    A.V. Bobylev, C. Cercignani. Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions. J. Statist. Phys. 110 (2003), 333-375.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    A.V. Bobylev, C. Cercignani, G. Toscani. Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials. J. Statist. Phys. 111 (2003), 403-17.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    A.V. Bobylev, I.M. Gamba, V. Panferov. Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, (preprint), (2003).Google Scholar
  12. 12.
    J.J. Brey, J.W. Dufty, A. Santos. Dissipative dynamics for hard spheres, J. Statist. Phys. 87(1997), 1051-1068.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    N.V. Brilliantov, T. Poschel. Granular Gases-The early stage. In: Coherent Structures in Classical Systems, Miguel Rubi, Ed., Lecture Notes in Physics, Vol. 567, Springer-Verlag (2001), p. 408-419.Google Scholar
  14. 15.
    J. A. Carrillo, C. Cercignani, I.M. Gamba. Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E (3) 62 no. 6, part A, (2000), 7700-7707.MathSciNetCrossRefGoogle Scholar
  15. 16.
    C. Cercignani, R. Illner, M. Pulvirenti. The mathematical theory of dilute gases. Springer Series in Applied Mathematical Sciences, Vol. 106, Springer-Verlag, 1994.Google Scholar
  16. 17.
    C. Cercignani. Recent developments in the mechanism of granular materials. Fisica Matematica e ingegneria delle strutture, Pitagora Editrice, Bologna, 1995.Google Scholar
  17. 18.
    J.W. Dufty. Kinetic theory and hydrodynamics for rapid granular flow-A perspective. arXiv: cond-mat/0108444vl (2001).Google Scholar
  18. 19.
    Y. Du, H. Li, L.P. Kadanoff. Breakdown of hydrodynamics in a one-dimensional system of inelastic particles. Phys. Rev. Lett. 74 (1995), 1268-1271.CrossRefGoogle Scholar
  19. 20.
    M.H. Ernst, R. Brito. High energy tails for inelastic Maxwell models. Europhys. Lett 43 (2002), 497-502.Google Scholar
  20. 21.
    M.H. Ernst, R. Brito. Scaling solutions of inelastic Boltzmann equation with over- populated high energy tails. J. Statist. Phys. 109 (2002), 407-432.MathSciNetMATHCrossRefGoogle Scholar
  21. 22.
    S. Esipov, T. Poschel. The granular phase diagram. J. Stat. Phys. 86, (1997) 1385-1395.MATHCrossRefGoogle Scholar
  22. 23.
    F. Filbet, L. Pareschi, G. Toscani. Accurate numerical solution for the collisional motion of (heated) granular flows, (preprint), (2004).Google Scholar
  23. 24.
    I.M. Gamba, V. Panferov, C. Villani. On the inelastic Boltzmann equation with diffusive forcing. Nonlinear problems in mathematical physics and related topics, II, In Honor of Professor O.A. Ladyzhenskaya 179-192, Int. Math. Ser. (N.Y.) 2, Kluwer-Plenum, New York, 2002.Google Scholar
  24. 25. I.M. Gamba, V. Panferov, C. Villani. On the Boltmann Equation for diffusively excited granular media. Comm. Math. Phys. (to appear).Google Scholar
  25. 26.
    D. Goldman, M.D. Shattuck, C. Bizon, W.D. McCormick, J.B. Swift, H.L. Swinney. Absence of inelastic collapse in a realistic three ball model. Phys. Rew. E, 57 (1998), 4831-4833.Google Scholar
  26. 27.
    I. Goldhirsch. Scales and kinetics of granular flows. Chaos9 (1999), 659-672.MATHCrossRefGoogle Scholar
  27. 28.
    P.K. Haff. Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134 (1983), 401-430.MATHCrossRefGoogle Scholar
  28. 29. Hailiang Li, G. Toscani. Long-time asymptotics of kinetic models of granular flows. Arch. Rational Mech. Anal, (to appear).Google Scholar
  29. 30.
    J.T. Jenkins, M. W. Richman. Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rational Mech. Anal. 87 (1985), 355-377.MathSciNetMATHCrossRefGoogle Scholar
  30. 31.
    S. McNamara, W.R. Young. Inelastic collapse and clumpingin a one-dimensional granular medium. Phys. Fluids A 4 (1992), 496-504.CrossRefGoogle Scholar
  31. 32.
    S. McNamara, W.R. Young. Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A 5 (1993), 34-45.MathSciNetCrossRefGoogle Scholar
  32. 33.
    G. Naldi, L. Pareschi, G. Toscani. Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit. RAIRO Model Math. Anal. Numer. 37 (2003), 73-90.MathSciNetMATHCrossRefGoogle Scholar
  33. 34.
    L. Pareschi. On the fast evaluation of kinetic equations for driven granular flows, Proceedings ENUMATH 2001, Springer-Italia (2003).Google Scholar
  34. 35.
    L. Pareschi, B. Perthame. A Fourier spectral method for homogeneous Boltzmann equations, Trans. Theo. and Stat. Phys. 25, 369-383 (1996).MathSciNetMATHCrossRefGoogle Scholar
  35. 36.
    L. Pareschi, G. Russo. Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator, SI AM J. Numer. Anal 37 (2000), 1217-1245.MathSciNetMATHCrossRefGoogle Scholar
  36. 37.
    L. Pareschi, G. Russo. An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM Proceedings, 10 (2001), 35-75.MathSciNetMATHCrossRefGoogle Scholar
  37. 38.
    L. Pareschi, G. Toscani, C. Villani. Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit. Numer. Math. 93 (2003), 527-548.MathSciNetMATHCrossRefGoogle Scholar
  38. 39.
    R. Ramirez, T. Poschel, N. V. Brilliantov, T. Schwager. Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. £60 (1999), 4465-472.Google Scholar
  39. 40.
    G. Toscani. One-dimensional kinetic models of granular flows. RAIRO Model Math. Anal. Numer. 34 (2000), 1277-1292.MathSciNetMATHCrossRefGoogle Scholar
  40. 41. G. Toscani. Kinetic and hydrodynamic models of nearly elastic granular flows. Monatsch. Math, (to appear).Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Lorenzo Pareschi
    • 1
  • Giuseppe Toscani
    • 2
  1. 1.Department of MathematicsUniversity of FerraraFerraraItaly
  2. 2.Department of MathematicsUniversity of PaviaPaviaItaly

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