Advertisement

Homogeneous Cooling State

  • Vicente GarzóEmail author
Chapter
Part of the Soft and Biological Matter book series (SOBIMA)

Abstract

This chapter deals with the problem of the so-called homogeneous cooling state (namely, a homogeneous state where granular temperature monotonically decays in time) for mono- and multicomponent granular gases. Unlike ordinary or classical gases, the Maxwell–Boltzmann velocity distribution is not a solution to the Boltzmann kinetic equation and the exact form of this solution is still unknown. For long times, however, the kinetic equation admits a scaling solution whose form can be approximately obtained by considering the leading terms in a Sonine (Laguerre) polynomial expansion. A new and surprising result (compared to its ordinary gas counterpart) is found for granular mixtures: the well-known energy equipartition theorem is broken for freely cooling systems.

References

  1. 1.
    Chapman, S., Cowling, T.G.: The Mathematical Theory of Nonuniform Gases. Cambridge University Press, Cambridge (1970)Google Scholar
  2. 2.
    García de Soria, M.I., Maynar, P., Mischler, S., Mouhot, C., Rey, T., Trizac, E.: Towards and H-theorem for granular gases. J. Stat. Mech. P11009 (2015)Google Scholar
  3. 3.
    Haff, P.K.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430 (1983)ADSCrossRefGoogle Scholar
  4. 4.
    Maaß, C.C., Isert, N., Maret, G., Aegerter, C.M.: Experimental investigation of the freely cooling granular gas. Phys. Rev. Lett. 100, 248001 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    Tatsumi, S., Murayama, Y., Hayakawa, H., Sano, M.: Experimental study on the kinetics of granular gases under microgravity. J. Fluid Mech. 641, 521–539 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    Harth, K., Trittel, T., Wegner, S., Stannarius, R.: Free cooling of a granular gas of rodlike particles in microgravity. Phys. Rev. Lett. 120, 213301 (2018)ADSCrossRefGoogle Scholar
  7. 7.
    Brilliantov, N.V., Formella, A., Pöschel, T.: Increasing temperature of cooling granular gases. Nature Comm. 9, 797 (2018)ADSCrossRefGoogle Scholar
  8. 8.
    Ferziger, J.H., Kaper, G.H.: Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972)Google Scholar
  9. 9.
    Brilliantov, N., Pöschel, T.: Kinetic Theory of Granular Gases. Oxford University Press, Oxford (2004)CrossRefGoogle Scholar
  10. 10.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)zbMATHGoogle Scholar
  11. 11.
    van Noije, T.P.C., Ernst, M.H.: Velocity distributions in homogeneous granular fluids: the free and heated case. Granular Matter 1, 57–64 (1998)CrossRefGoogle Scholar
  12. 12.
    Brilliantov, N.V., Pöschel, T.: Breakdown of the Sonine expansion for the velocity distribution of granular gases. Europhys. Lett. 74, 424–430 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Brilliantov, N.V., Pöschel, T.: Erratum: breakdown of the Sonine expansion for the velocity distribution of granular gases. Europhys. Lett. 75, 188 (2006)ADSCrossRefGoogle Scholar
  14. 14.
    Coppex, F., Droz, M., Piasecki, J., Trizac, E.: On the first Sonine correction for granular gases. Physica A 329, 114–126 (2003)Google Scholar
  15. 15.
    Montanero, J.M., Santos, A.: Computer simulation of uniformly heated granular fluids. Granular Matter 2, 53–64 (2000)CrossRefGoogle Scholar
  16. 16.
    Santos, A., Montanero, J.M.: The second and third Sonine coefficients of a freely cooling granular gas revisited. Granular Matter 11, 157–168 (2009)CrossRefGoogle Scholar
  17. 17.
    Goldshtein, A., Shapiro, M.: Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J. Fluid Mech. 282, 75–114 (1995)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Brey, J.J., Ruiz-Montero, M.J., Cubero, D.: Homogeneous cooling state of a low-density granular flow. Phys. Rev. E 54, 3664–3671 (1996)ADSCrossRefGoogle Scholar
  19. 19.
    Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows. Clarendon, Oxford (1994)Google Scholar
  20. 20.
    Huthmann, M., Orza, J.A.G., Brito, R.: Dynamics of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastic particles. Granular Matter 2, 189–199 (2000)CrossRefGoogle Scholar
  21. 21.
    Esipov, S.E., Pöschel, T.: The granular phase diagram. J. Stat. Phys. 86, 1385–1395 (1997)ADSCrossRefGoogle Scholar
  22. 22.
    Krook, M., Wu, T.T.: Formation of Maxwellian tails. Phys. Rev. Lett. 36, 1107–1109 (1976)ADSCrossRefGoogle Scholar
  23. 23.
    Brey, J.J., Cubero, D., Ruiz-Montero, M.J.: High energy tail in the velocity distribution of a granular gas. Phys. Rev. E 59, 1256–1258 (1999)ADSCrossRefGoogle Scholar
  24. 24.
    Jenkins, J.T., Mancini, F.: Balance laws and constitutive relations for plane flows of a dense, binary mixture of smooth, nearly elastic, circular disks. J. Appl. Mech. 54, 27–34 (1987)ADSCrossRefGoogle Scholar
  25. 25.
    Garzó, V., Dufty, J.W.: Homogeneous cooling state for a granular mixture. Phys. Rev. E 60, 5706–5713 (1999)ADSCrossRefGoogle Scholar
  26. 26.
    Martin, P.A., Piasecki, J.: Thermalization of a particle by dissipative collisions. Europhys. Lett. 46, 613–616 (1999)ADSCrossRefGoogle Scholar
  27. 27.
    Boublik, T.: Hard-sphere equation of state. J. Chem. Phys. 53, 471 (1970)ADSCrossRefGoogle Scholar
  28. 28.
    Grundke, E.W., Henderson, D.: Distribution functions of multi-component fluid mixtures of hard spheres. Mol. Phys. 24, 269–281 (1972)ADSCrossRefGoogle Scholar
  29. 29.
    Lee, L.L., Levesque, D.: Perturbation theory for mixtures of simple liquids. Mol. Phys. 26, 1351–1370 (1973)ADSCrossRefGoogle Scholar
  30. 30.
    Montanero, J.M., Garzó, V.: Monte Carlo simulation of the homogeneous cooling state for a granular mixture. Granular Matter 4, 17–24 (2002)CrossRefGoogle Scholar
  31. 31.
    Dahl, S.R., Hrenya, C.M., Garzó, V., Dufty, J.W.: Kinetic temperatures for a granular mixture. Phys. Rev. E 66, 041301 (2002)ADSCrossRefGoogle Scholar
  32. 32.
    Pagnani, R., Marconi, U.M.B., Puglisi, A.: Driven low density granular mixtures. Phys. Rev. E 66, 051304 (2002)ADSCrossRefGoogle Scholar
  33. 33.
    Barrat, A., Trizac, E.: Lack of energy equipartition in homogeneous heated binary granular mixtures. Granular Matter 4, 57–63 (2002)CrossRefGoogle Scholar
  34. 34.
    Barrat, A., Trizac, E.: Molecular dynamics simulations of vibrated granular gases. Phys. Rev. E 66, 051303 (2002)ADSCrossRefGoogle Scholar
  35. 35.
    Clelland, R., Hrenya, C.M.: Simulations of a binary-sized mixture of inelastic grains in rapid shear flow. Phys. Rev. E 65, 031301 (2002)ADSCrossRefGoogle Scholar
  36. 36.
    Montanero, J.M., Garzó, V.: Energy nonequipartition in a sheared granular mixture. Mol. Simul. 29, 357–362 (2003)CrossRefGoogle Scholar
  37. 37.
    Krouskop, P., Talbot, J.: Mass and size effects in three-dimensional vibrofluidized granular mixtures. Phys. Rev. E 68, 021304 (2003)ADSCrossRefGoogle Scholar
  38. 38.
    Wang, H., Jin, G., Ma, Y.: Simulation study on kinetic temperatures of vibrated binary granular mixtures. Phys. Rev. E 68, 031301 (2003)ADSCrossRefGoogle Scholar
  39. 39.
    Brey, J.J., Ruiz-Montero, M.J., Moreno, F.: Energy partition and segregation for an intruder in a vibrated granular system under gravity. Phys. Rev. Lett. 95, 098001 (2005)ADSCrossRefGoogle Scholar
  40. 40.
    Schröter, M., Ulrich, S., Kreft, J., Swift, J.B., Swinney, H.L.: Mechanisms in the size segregation of a binary granular mixture. Phys. Rev. E 74, 011307 (2006)ADSCrossRefGoogle Scholar
  41. 41.
    Wildman, R.D., Parker, D.J.: Coexistence of two granular temperatures in binary vibrofluidized beds. Phys. Rev. Lett. 88, 064301 (2002)ADSCrossRefGoogle Scholar
  42. 42.
    Feitosa, K., Menon, N.: Breakdown of energy equipartition in a 2D binary vibrated granular gas. Phys. Rev. Lett. 88, 198301 (2002)ADSCrossRefGoogle Scholar
  43. 43.
    Dorfman, J.R., van Beijeren, H.: The kinetic theory of gases. In: Berne, B.J. (ed.) Statistical Mechanics. Part B: Time-Dependent Processes, pp. 65–179. Plenum, New York (1977)Google Scholar
  44. 44.
    Lutsko, J.F., Brey, J.J., Dufty, J.W.: Diffusion in a granular fluid. II. Simulation. Phys. Rev. E 65, 051304 (2002)ADSCrossRefGoogle Scholar
  45. 45.
    Santos, A., Dufty, J.W.: Critical behavior of a heavy particle in a granular fluid. Phys. Rev. Lett. 86, 4823–4826 (2001)ADSCrossRefGoogle Scholar
  46. 46.
    Brey, J.J., Dufty, J.W., Santos, A.: Kinetic models for granular flow. J. Stat. Phys. 97, 281–322 (1999)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Stanley, H.: Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, Oxford (1971)Google Scholar
  48. 48.
    Santos, A., Dufty, J.W.: Nonequilibrium phase transition for a heavy particle in a granular fluid. Phys. Rev. E 64, 051305 (2001)ADSCrossRefGoogle Scholar
  49. 49.
    Jenkins, J.T., Savage, S.B.: A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202 (1983)ADSCrossRefGoogle Scholar
  50. 50.
    Lun, C.K.K., Savage, S.B., Jeffrey, D.J., Chepurniy, N.: Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256 (1984)ADSCrossRefGoogle Scholar
  51. 51.
    Jenkins, J.T., Richman, M.W.: Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28, 3485–3493 (1985)ADSCrossRefGoogle Scholar
  52. 52.
    Lun, C.K.K.: Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. J. Fluid Mech. 233, 539–559 (1991)ADSCrossRefGoogle Scholar
  53. 53.
    Huthmann, M., Zippelius, A.: Dynamics of inelastically colliding rough spheres: Relaxation of translational and rotational energy. Phys. Rev. E 56, R6275–R6278 (1997)ADSCrossRefGoogle Scholar
  54. 54.
    Luding, S., Huthmann, M., McNamara, S.: Zippelius: homogeneous cooling of rough, dissipative particles: theory and simulations. Phys. Rev. E 58, 3416–3425 (1998)ADSCrossRefGoogle Scholar
  55. 55.
    McNamara, S., Luding, S.: Energy nonequipartition in systems of inelastic rough spheres. Phys. Rev. E 58, 2247–2250 (1998)ADSCrossRefGoogle Scholar
  56. 56.
    Goldhirsch, I., Noskowicz, S.H., Bar-Lev, O.: Nearly smooth granular gases. Phys. Rev. Lett. 95, 068002 (2005)ADSCrossRefGoogle Scholar
  57. 57.
    Brilliantov, N.V., Pöschel, T., Kranz, W.T., Zippelius, A.: Translations and rotations are correlated in granular gases. Phys. Rev. Lett. 98, 128001 (2007)ADSCrossRefGoogle Scholar
  58. 58.
    Vega Reyes, F., Kremer, G., Santos, A.: Role of roughness on the hydrodynamic homogeneous base state of inelastic hard spheres. Phys. Rev. E (R) 89, 020202 (2014)CrossRefGoogle Scholar
  59. 59.
    Zippelius, A.: Granular gases. Physica A 369, 143–158 (2006)Google Scholar
  60. 60.
    Buck, B., Macaulay, V.A.: Maximum Entropy in Action. Wiley, New York (1991)Google Scholar
  61. 61.
    Santos, A., Kremer, G., dos Santos, M.: Sonine approximation for collisional moments of granular gases of inelastic rough spheres. Phys. Fluids 23, 030604 (2011)ADSCrossRefGoogle Scholar
  62. 62.
    Santos, A., Kremer, G., Garzó, V.: Energy production rates in fluid mixtures of inelastic rough hard spheres. Prog. Theor. Phys. Suppl. 184, 31–48 (2010)ADSCrossRefGoogle Scholar
  63. 63.
    Kremer, G., Santos, A., Garzó, V.: Transport coefficients of a granular gas of inelastic rough hard spheres. Phys. Rev. E 90, 022205 (2014)ADSCrossRefGoogle Scholar
  64. 64.
    Herbst, O., Huthmann, M., Zippelius, A.: Dynamics of inelastically colliding spheres with Coulomb friction: relaxation of translational and rotational energy. Granular Matter 2, 211–219 (2000)CrossRefGoogle Scholar
  65. 65.
    Vega Reyes, F., Santos, A.: Steady state in a gas of inelastic rough spheres heated by a uniform stochastic force. Phys. Fluids 27, 113301 (2015)ADSCrossRefGoogle Scholar
  66. 66.
    Viot, P., Talbot, J.: Thermalization of an anisotropic granular particle. Phys. Rev. E 69, 051106 (2004)ADSCrossRefGoogle Scholar
  67. 67.
    Piasecki, J., Talbot, J., Viot, P.: Angular velocity distribution of a granular planar rotator in a thermalized bath. Phys. Rev. E 75, 051307 (2007)ADSCrossRefGoogle Scholar
  68. 68.
    Santos, A.: Interplay between polydispersity, inelasticity, and roughness in the freely cooling regime of hard-disk granular gases. Phys. Rev. E 98, 012904 (2018)ADSCrossRefGoogle Scholar
  69. 69.
    Vega Reyes, F., Lasanta, A., Santos, A., Garzó, V.: Energy nonequipartition in gas mixtures of inelastic rough hard spheres: the tracer limit. Phys. Rev. E 96, 052901 (2017)ADSCrossRefGoogle Scholar
  70. 70.
    Khalil, N., Garzó, V.: Homogeneous states in driven granular mixtures: Enskog kinetic theory versus molecular dynamics simulations. J. Chem. Phys. 140, 164901 (2014)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx)Universidad de ExtremaduraBadajozSpain

Personalised recommendations