Advertisement

Transport Properties for Driven Granular Gases

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

Abstract

Transport properties of granular gases driven by a stochastic bath with friction are determined. This type of thermostats attempt to mimic the effect of the interstitial fluid surrounding the solid particles. As a first step, the steady homogeneous state is analyzed. As with undriven granular gases, it can be seen that the kinetic equation admits a scaled solution where dependence on granular temperature is encoded not only through the scaled velocity \(\mathbf {c}=\mathbf {v}/\upsilon _{\text {th}}\) (\(\upsilon _{\text {th}}\) being the thermal velocity) but also through the (scaled) driven parameters. The Boltzmann kinetic equation is solved then by means of the Chapman–Enskog method around the homogeneous driven state. Momentum and heat fluxes are determined to first-order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. The relevant transport coefficients are identified and compared against computer simulations. Similarly to undriven systems, the theory compares quite well with simulations for conditions of practical interest. Finally, thermal diffusion segregation for driven granular mixtures is also analyzed.

References

  1. 1.
    Yang, X., Huan, C., Candela, D., Mair, R.W., Walsworth, R.L.: Measurements of grain motion in a dense, three-dimensional granular fluid. Phys. Rev. Lett. 88, 044301 (2002)ADSCrossRefGoogle Scholar
  2. 2.
    Huan, C., Yang, X., Candela, D., Mair, R.W., Walsworth, R.L.: NMR experiments on a three-dimensional vibrofluidized granular medium. Phys. Rev. E 69, 041302 (2004)ADSCrossRefGoogle Scholar
  3. 3.
    Abate, A.R., Durian, D.J.: Approach to jamming in an air-fluidized granular bed. Phys. Rev. E 74, 031308 (2006)ADSCrossRefGoogle Scholar
  4. 4.
    Schröter, M., Goldman, D.I., Swinney, H.L.: Stationary state volume fluctuations in a granular medium. Phys. Rev. E 71, 030301(R) (2005)ADSCrossRefGoogle Scholar
  5. 5.
    Möbius, M.E., Lauderdale, B.E., Nagel, S.R., Jaeger, H.M.: Brazil-nut effect: size separation of granular particles. Nature 414, 270 (2001)ADSCrossRefGoogle Scholar
  6. 6.
    Yan, X., Shi, Q., Hou, M., Lu, K., Chan, C.K.: Effects of air on the segregation of particles in a shaken granular bed. Phys. Rev. Lett. 91, 014302 (2003)ADSCrossRefGoogle Scholar
  7. 7.
    Wylie, J.J., Zhang, Q., Xu, H.Y., Sun, X.X.: Drag-induced particle segregation with vibrating boundaries. Europhys. Lett. 81, 54001 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    Puglisi, A., Loreto, V., Marconi, U.M.B., Petri, A., Vulpiani, A.: Clustering and non-Gaussian behavior in granular matter. Phys. Rev. Lett. 81, 3848–3851 (1998)ADSCrossRefGoogle Scholar
  9. 9.
    Puglisi, A., Loreto, V., Marconi, U.M.B., Vulpiani, A.: Kinetic approach to granular gases. Phys. Rev. E 59, 5582–5595 (1999)ADSCrossRefGoogle Scholar
  10. 10.
    Cafiero, R., Luding, S., Herrmann, H.J.: Two-dimensional granular gas of inelastic spheres with multiplicative driving. Phys. Rev. Lett. 84, 6014–6017 (2000)ADSCrossRefGoogle Scholar
  11. 11.
    Prevost, A., Egolf, D.A., Urbach, J.S.: Forcing and velocity correlations in a vibrated granular monolayer. Phys. Rev. Lett. 89, 084301 (2002)ADSCrossRefGoogle Scholar
  12. 12.
    Puglisi, A., Baldassarri, A., Loreto, V.: Fluctuation-dissipation relations in driven granular gases. Phys. Rev. E 66, 061305 (2002)ADSCrossRefGoogle Scholar
  13. 13.
    Visco, P., Puglisi, A., Barrat, A., Trizac, E., van Wijland, F.: Fluctuations of power injection in randomly driven granular gases. J. Stat. Phys. 125, 533–568 (2006)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Fiege, A., Aspelmeier, T., Zippelius, A.: Long-time tails and cage effect in driven granular fluids. Phys. Rev. Lett. 102, 098001 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    Sarracino, A., Villamaina, D., Gradenigo, G., Puglisi, A.: Irreversible dynamics of a massive intruder in dense granular fluids. Europhys. Lett. 92, 34001 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    Kranz, W.T., Sperl, M., Zippelius, A.: Glass transition for driven granular fluids. Phys. Rev. Lett. 104, 225701 (2010)ADSCrossRefGoogle Scholar
  17. 17.
    Hoover, W.G.: Nonequilibrium molecular dynamics. Ann. Rev. Phys. Chem. 34, 103–127 (1983)ADSCrossRefGoogle Scholar
  18. 18.
    Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Liquids. Academic Press, London (1990)zbMATHGoogle Scholar
  19. 19.
    Gradenigo, G., Sarracino, A., Villamaina, D., Puglisi, A.: Non-equilibrium length in granular fluids: from experiment to fluctuating hydrodynamics. Europhys. Lett. 96, 14004 (2011)ADSCrossRefGoogle Scholar
  20. 20.
    Puglisi, A., Gnoli, A., Gradenigo, G., Sarracino, A., Villamaina, D.: Structure factors in granular experiments with homogeneous fluidization. J. Chem. Phys. 136, 014704 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    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
  22. 22.
    Montanero, J.M., Santos, A.: Computer simulation of uniformly heated granular fluids. Granular Matter 2, 53–64 (2000)CrossRefGoogle Scholar
  23. 23.
    Williams, D.R.M., MacKintosh, F.C.: Driven granular media in one dimension: Correlations and equation of state. Phys. Rev. E 54, R9–R12 (1996)ADSCrossRefGoogle Scholar
  24. 24.
    van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam (1981)zbMATHGoogle Scholar
  25. 25.
    Koch, D.L.: Kinetic theory for a monodisperse gas-solid suspension. Phys. Fluids A 2, 1711–1722 (1990)ADSCrossRefGoogle Scholar
  26. 26.
    Koch, D.L., Hill, R.J.: Inertial effects in suspensions and porous-media flows. Ann. Rev. Fluid Mech. 33, 619–647 (2001)ADSCrossRefGoogle Scholar
  27. 27.
    Garzó, V., Tenneti, S., Subramaniam, S., Hrenya, C.M.: Enskog kinetic theory for monodisperse gas-solid flows. J. Fluid Mech. 712, 129–168 (2012)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Garzó, V., Santos, A.: Kinetic Theory of Gases in Shear Flows. Nonlinear Transport. Kluwer Academic Publishers, Dordrecht (2003)CrossRefGoogle Scholar
  29. 29.
    Dufty, J.W., Santos, A., Brey, J.J., Rodríguez, R.F.: Model for nonequilibrium computer simulation methods. Phys. Rev. A 33, 459–466 (1986)ADSCrossRefGoogle Scholar
  30. 30.
    Gradenigo, G., Sarracino, A., Villamaina, D., Puglisi, A.: Fluctuating hydrodynamics and correlation lengths in a driven granular fluid. J. Stat. Mech. P08017, (2011)Google Scholar
  31. 31.
    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
  32. 32.
    Chapman, S., Cowling, T.G.: The Mathematical Theory of Nonuniform Gases. Cambridge University Press, Cambridge (1970)zbMATHGoogle Scholar
  33. 33.
    McLennan, J.A.: Introduction to Nonequilibrium Statistical Mechanics. Prentice-Hall, New Jersey (1989)Google Scholar
  34. 34.
    García de Soria, M.I., Maynar, P., Trizac, E.: Universal reference state in a driven homogeneous granular gas. Phys. Rev. E 85, 051301 (2012)ADSCrossRefGoogle Scholar
  35. 35.
    Chamorro, M.G., Vega Reyes, F., Garzó, V.: Homogeneous steady states in a granular fluid driven by a stochastic bath with friction. J. Stat. Mech. P07013, (2013)Google Scholar
  36. 36.
    Garzó, V., Chamorro, M.G., Vega Reyes, F.: Transport properties for driven granular fluids in situations close to homogeneous steady states. Phys. Rev. E 87, 032201 (2013)ADSCrossRefGoogle Scholar
  37. 37.
    Torquato, S.: Nearest-neighbor statistics for packings of hard spheres and disks. Phys. Rev. E 51, 3170–3182 (1995)ADSCrossRefGoogle Scholar
  38. 38.
    Hayakawa, H.: Hydrodynamics of driven granular gases. Phys. Rev. E 68, 031304 (2003)ADSCrossRefGoogle Scholar
  39. 39.
    García de Soria, M.I., Maynar, P., Trizac, E.: Linear hydrodynamics for driven granular gases. Phys. Rev. E 87, 022201 (2013)ADSCrossRefGoogle Scholar
  40. 40.
    Garzó, V., Chamorro, M.G., Vega Reyes, F.: Erratum: Transport properties for driven granular fluids in situations close to homogeneous steady states. Phys. Rev. E 87, 059906 (2013)ADSCrossRefGoogle Scholar
  41. 41.
    Garzó, V., Montanero, J.M.: Transport coefficients of a heated granular gas. Physica A 313, 336–356 (2002)ADSCrossRefGoogle Scholar
  42. 42.
    Montanero, J.M., Garzó, V.: Shear viscosity for a heated granular binary mixture at low density. Phys. Rev. E 67, 021308 (2003)ADSCrossRefGoogle Scholar
  43. 43.
    Garzó, V., Montanero, J.M.: Shear viscosity for a moderately dense granular binary mixture. Phys. Rev. E 68, 041302 (2003)ADSCrossRefGoogle Scholar
  44. 44.
    Lees, A.W., Edwards, S.F.: The computer study of transport processes under extreme conditions. J. Phys. C 5, 1921–1929 (1972)ADSCrossRefGoogle Scholar
  45. 45.
    Naitoh, T., Ono, S.: The shear viscosity of a hard-sphere fluid via nonequilibrium molecular dynamics. J. Chem. Phys. 70, 4515–4523 (1979)ADSCrossRefGoogle Scholar
  46. 46.
    Montanero, J.M., Santos, A.: Monte Carlo simulation method for the Enskog equation. Phys. Rev. E 54, 438–444 (1996)ADSCrossRefGoogle Scholar
  47. 47.
    Montanero, J.M., Santos, A.: Simulation of the Enskog equation à la Bird. Phys. Fluids 9, 2057–2060 (1997)ADSCrossRefGoogle Scholar
  48. 48.
    Gómez Ordóñez, J., Brey, J.J., Santos, A.: Shear-rate dependence of the viscosity for dilute gases. Phys. Rev. A 39, 3038–3040 (1989)ADSCrossRefGoogle Scholar
  49. 49.
    Barrat, A., Trizac, E.: Lack of energy equipartition in homogeneous heated binary granular mixtures. Granular Matter 4, 57–63 (2002)CrossRefGoogle Scholar
  50. 50.
    Henrique, C., Batrouni, G., Bideau, D.: Diffusion as a mixing mechanism in granular materials. Phys. Rev. E 63, 011304 (2000)ADSCrossRefGoogle Scholar
  51. 51.
    Dorfman, J.R., van Beijeren, H.: The kinetic theory of gases. In: B.J. Berne (ed.) Statistical Mechanics. Part B: Time-Dependent Processes, pp. 65–179. Plenum, New York (1977)Google Scholar
  52. 52.
    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
  53. 53.
    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
  54. 54.
    Brey, J.J., Ruiz-Montero, M.J., Moreno, F.: Hydrodynamic profiles for an impurity in an open vibrated granular gas. Phys. Rev. E 73, 031301 (2006)ADSCrossRefGoogle Scholar
  55. 55.
    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
  56. 56.
    Santos, A., Dufty, J.W.: Critical behavior of a heavy particle in a granular fluid. Phys. Rev. Lett. 86, 4823–4826 (2001)ADSCrossRefGoogle Scholar
  57. 57.
    Garzó, V.: Brazil-nut effect versus reverse Brazil-nut effect in a moderately granular dense gas. Phys. Rev. E (R) 78, 020301 (2008)ADSCrossRefGoogle Scholar
  58. 58.
    Garzó, V.: Segregation by thermal diffusion in moderately dense granular mixtures. Eur. Phys. J. E 29, 261–274 (2009)CrossRefGoogle Scholar
  59. 59.
    Garzó, V., Vega Reyes, F.: Segregation of an intruder in a heated granular gas. Phys. Rev. E 85, 021308 (2012)ADSCrossRefGoogle Scholar
  60. 60.
    Hong, D.C., Quinn, P.V., Luding, S.: Reverse Brazil nut problem: competition between percolation and condensation. Phys. Rev. Lett. 86, 3423–3426 (2001)ADSCrossRefGoogle Scholar
  61. 61.
    Wildman, R.D., Huntley, J.M., Parker, D.J.: Granular temperature profiles in three-dimensional vibrofluidized granular beds. Phys. Rev. E 63, 061311 (2001)ADSCrossRefGoogle Scholar
  62. 62.
    Jenkins, J.T., Yoon, D.K.: Segregation in binary mixtures under gravity. Phys. Rev. Lett. 88, 194301 (2002)ADSCrossRefGoogle Scholar
  63. 63.
    Breu, A.P.J., Ensner, H.M., Kruelle, C.A., Rehberg, I.: Reversing the Brazil-nut effect: Competition between percolation and condensation. Phys. Rev. Lett. 90, 014302 (2003)ADSCrossRefGoogle Scholar
  64. 64.
    Trujillo, L., Alam, M., Herrmann, H.J.: Segregation in a fluidized binary granular mixture: Competition between buoyancy and geometric forces. Europhys. Lett. 64, 190–196 (2003)ADSCrossRefGoogle Scholar
  65. 65.
    Schautz, T., Brito, R., Kruelle, C.A., Rehberg, I.: A horizontal Brazil-nut effect and its reverse. Phys. Rev. Lett. 95, 028001 (2005)ADSCrossRefGoogle Scholar
  66. 66.
    Alam, M., Trujillo, L., Herrmann, H.J.: Hydrodynamic theory for reverse Brazil nut segregation and the non-monotonic ascension dynamics. J. Stat. Phys. 124, 587–623 (2006)ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    Vega Reyes, F., Garzó, V., Khalil, N.: Hydrodynamic granular segregation induced by boundary heating and shear. Phys. Rev. E 89, 052206 (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