Non-Newtonian Steady States for Granular Gases

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


This chapter addresses the study of non-Newtonian transport properties of several steady laminar flows in granular gases. As a first step, it analyzes the well-known simple or uniform shear flow where a granular gas under constant shear rate and uniform temperature and density supports a steady state. In this state, collisional cooling compensates locally for viscous heating, hence the viscosity function and the two viscometric functions are nonlinear functions of the coefficient of restitution. Following this, a special class of steady Couette flows is presented. As occurs with the uniform shear flow state, in all flows of this class (referred to as the LTu class) viscous heating is exactly balanced by inelastic cooling leading to a uniform heat flux. While the rheological functions of the LTu flows are identical to those obtained in the uniform shear flow state problem, generalized thermal conductivity coefficients can be identified. Determination of the non-Newtonian transport coefficients is done by following analytical and computational routes. Comparison between theoretical predictions and simulation results shows in general good agreement, even for conditions of strong inelasticity and large velocity and temperature gradients.


  1. 1.
    Garzó, V., Santos, A.: Kinetic Theory of Gases in Shear Flows: Nonlinear Transport. Kluwer Academic Publishers, Dordrecht (2003)CrossRefGoogle Scholar
  2. 2.
    Campbell, C.S., Gong, A.: The stress tensor in a two-dimensional granular shear flow. J. Fluid Mech. 164, 107–125 (1986)ADSCrossRefGoogle Scholar
  3. 3.
    Walton, O.R., Braun, R.L.: Viscosity and temperature calculations for shearing assemblies of inelastic, frictional disks. J. Rheol. 30, 949–980 (1986)ADSCrossRefGoogle Scholar
  4. 4.
    Campbell, C.S.: The stress tensor for simple shear flows of a granular material. J. Fluid Mech. 203, 449–473 (1989)ADSCrossRefGoogle Scholar
  5. 5.
    Hopkins, M.A., Shen, H.H.: A Monte Carlo solution for rapidly shearing granular flows based on the kinetic theory of dense gases. J. Fluid Mech. 244, 477–491 (1992)ADSCrossRefGoogle Scholar
  6. 6.
    Alam, M., Luding, S.: How good is the equipartition assumption for the transport properties of a granular mixture? Granular Matter 4, 139–142 (2002)CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Alam, M., Luding, S.: Rheology of bidisperse granular mixtures via event-driven simulations. J. Fluid Mech. 476, 69–103 (2003)ADSCrossRefGoogle Scholar
  9. 9.
    Dahl, S.R., Clelland, R., Hrenya, C.M.: The effects of continuous size distributions on the rapid flow of inelastic particles. Phys. Fluids 14, 1972–1984 (2002)ADSCrossRefGoogle Scholar
  10. 10.
    Polashenski, W., Zamankhan, P., Mäkiharju, S., Zamankhan, P.: Fine structures of granular flows. Phys. Rev. E 66, 021303 (2002)ADSCrossRefGoogle Scholar
  11. 11.
    Dahl, S.R., Hrenya, C.M.: Size-segregation in collisional granular flows with continuous size distributions. Phys. Fluids 16, 1–13 (2004)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Iddir, H., Arastoopour, H., Hrenya, C.M.: Analysis of binary and ternary granular mixtures behavior using the kinetic theory approach. Powder Technol. 151, 117–125 (2005)CrossRefGoogle Scholar
  13. 13.
    Santos, A., Garzó, V., Dufty, J.W.: Inherent rheology of a granular fluid in uniform shear flow. Phys. Rev. E 69, 061303 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    Campbell, C.S.: Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57–92 (1990)ADSCrossRefGoogle Scholar
  15. 15.
    Goldhirsch, I.: Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293 (2003)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kumaran, V.: Kinetic theory for sheared granular flows. C. R. Phys. 16, 51–61 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    Reif, R.: Fundamentals of Statistical and Thermal Physics. McGraw-Hill, New York (1965)Google Scholar
  18. 18.
    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
  19. 19.
    Kawasaki, K., Gunton, J.D.: Theory of nonlinear transport processes: nonlinear shear viscosity and normal stress effects. Phys. Rev. A 8, 2048–2064 (1973)ADSCrossRefGoogle Scholar
  20. 20.
    Hoover, W.G.: Nonequilibrium molecular dynamics. Annu. Rev. Phys. Chem. 34, 103–127 (1983)ADSCrossRefGoogle Scholar
  21. 21.
    Evans, D.J., Morriss, G.P.: Non-Newtonian molecular dynamics. Comput. Phys. Rep. 1, 299–343 (1984)ADSCrossRefGoogle Scholar
  22. 22.
    Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Liquids. Academic Press, London (1990)zbMATHGoogle Scholar
  23. 23.
    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
  24. 24.
    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
  25. 25.
    Jenkins, J.T., 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, 313–328 (1988)ADSCrossRefGoogle Scholar
  26. 26.
    Lun, C.K.K., Bent, A.A.: Numerical simulation of inelastic frictional spheres in simple shear flow. J. Fluid Mech. 258, 335–353 (1994)Google Scholar
  27. 27.
    Goldhirsch, I., Tan, M.L.: The single-particle distribution function for rapid granular shear flows of smooth inelastic disks. Phys. Fluids 8, 1752–1763 (1996)ADSCrossRefGoogle Scholar
  28. 28.
    Sela, N., Goldhirsch, I., Noskowicz, S.H.: Kinetic theoretical study of a simply sheared two-dimensional granular gas to Burnett order. Phys. Fluids 8, 2337–2353 (1996)ADSCrossRefGoogle Scholar
  29. 29.
    Goldhirsch, I., Sela, N.: Origin of normal stress differences in rapid granular flows. Phys. Rev. E 54, 4458–4461 (1996)ADSCrossRefGoogle Scholar
  30. 30.
    Brey, J.J., Ruiz-Montero, M.J., Moreno, F.: Steady uniform shear flow in a low density granular gas. Phys. Rev. E 55, 2846–2856 (1997)ADSCrossRefGoogle Scholar
  31. 31.
    Chou, C.S., Richman, M.W.: Constitutive theory for homogeneous granular shear flows of highly inelastic hard spheres. Physica A 259, 430–448 (1998)Google Scholar
  32. 32.
    Montanero, J.M., Garzó, V., Santos, A., Brey, J.J.: Kinetic theory of simple granular shear flows of smooth hard spheres. J. Fluid Mech. 389, 391–411 (1999)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Tij, M., Tahiri, E., Montanero, J.M., Garzó, V., Santos, A., Dufty, J.W.: Nonlinear Couette flow in a low density granular gas. J. Stat. Phys. 103, 1035–1068 (2001)CrossRefGoogle Scholar
  34. 34.
    Vega Reyes, F., Santos, A., Garzó, V.: Steady base states for non-Newtonian granular hydrodynamics. J. Fluid Mech. 719, 431–464 (2013)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Vega Reyes, F., Urbach, J.S.: Steady base states for Navier–Stokes granular hydrodynamics with boundary heating and shear. J. Fluid Mech. 636, 279–293 (2009)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Vega Reyes, F., Santos, A., Garzó, V.: Non-Newtonian granular hydrodynamics. What do the inelastic simple shear flow and the elastic Fourier flow have in common? Phys. Rev. Lett. 104, 028001 (2010)ADSCrossRefGoogle Scholar
  37. 37.
    Vega Reyes, F., Garzó, V., Santos, A.: Class of dilute granular Couette flows with uniform heat flux. Phys. Rev. E 83, 021302 (2011)ADSCrossRefGoogle Scholar
  38. 38.
    Lees, A.W., Edwards, S.F.: The computer study of transport processes under extreme conditions. J. Phys. C 5, 1921–1929 (1972)ADSCrossRefGoogle Scholar
  39. 39.
    Bird, R.B.: Non-Newtonian behavior of polymeric liquids. Physica A 118, 3–16 (1983)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    McLennan, J.A.: Introduction to Nonequilibrium Statistical Mechanics. Prentice-Hall, New Jersey (1989)Google Scholar
  41. 41.
    Kumaran, V.: Asymptotic solution of the Boltzmann equation for the shear flow of smooth inelastic disks. Physica A 275, 483–504 (2000)Google Scholar
  42. 42.
    Kumaran, V.: Hydrodynamic modes of a sheared granular flow from the Boltzmann and Navier–Stokes equations. Phys. Fluids 13, 2258–2268 (2001)ADSCrossRefGoogle Scholar
  43. 43.
    Buck, B., Macaulay, V.A.: Maximum Entropy in Action. Wiley, New York (1991)Google Scholar
  44. 44.
    Saha, S., Alam, M.: Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad’s moment method. J. Fluid Mech. 757, 251–296 (2014)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Garzó, V.: Tracer diffusion in granular shear flows. Phys. Rev. E 66, 021308 (2002)ADSCrossRefGoogle Scholar
  46. 46.
    Astillero, A., Santos, A.: Uniform shear flow in dissipative gases: computer simulations of inelastic hard spheres and frictional elastic hard spheres. Phys. Rev. E 72, 031309 (2005)ADSCrossRefGoogle Scholar
  47. 47.
    Gupta, V.K., Torrilhon, M.: Automated Boltzmann collision integrals for moment equations. In: Mareschal, M., Santos, A. (eds.) 28th International Symposium on Rarefied Gas Dynamics. AIP Conference Proceedings, vol. 1501, pp. 67–74 (2012)Google Scholar
  48. 48.
    Chamorro, M.G., Vega Reyes, F., Garzó, V.: Non-Newtonian hydrodynamics for a dilute granular suspension under uniform shear flow. Phys. Rev. E 92, 052205 (2015)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Brey, J.J., Dufty, J.W., Santos, A.: Kinetic models for granular flow. J. Stat. Phys. 97, 281–322 (1999)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Montanero, J.M., Garzó, V.: Rheological properties in a low-density granular mixture. Phys. A 310, 17–38 (2002)CrossRefGoogle Scholar
  51. 51.
    Montanero, J.M., Santos, A., Garzó, V.: DSMC evaluation of the Navier–Stokes shear viscosity of a granular fluid. In: Capitelli, M. (ed.) 24th International Symposium on Rarefied Gas Dynamics. AIP Conference Proceedings, vol. 762, pp. 797–802 (2005)Google Scholar
  52. 52.
    Astillero, A., Santos, A.: Aging to non-Newtonian hydrodynamics in a granular gas. Europhys. Lett. 78, 24002 (2007)ADSCrossRefGoogle Scholar
  53. 53.
    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
  54. 54.
    Garzó, V.: Transport coefficients for an inelastic gas around uniform shear flow: linear stability analysis. Phys. Rev. E 73, 021304 (2006)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    Richman, M.W.: The source of second moment in dilute granular flows of highly inelastic spheres. J. Rheol. 33, 1293–1305 (1989)ADSCrossRefGoogle Scholar
  56. 56.
    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
  57. 57.
    Jenkins, J.T., Richman, M.W.: Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Ration. Mech. Anal. 87, 355–377 (1985)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Saitoh, K., Hayakawa, H.: Rheology of a granular gas under a plane shear. Phys. Rev. E 75, 021302 (2007)ADSCrossRefGoogle Scholar
  59. 59.
    Jenkins, J.T., Mancini, F.: Kinetic theory for binary mixtures of smooth, nearly elastic spheres. Phys. Fluids A 1, 2050–2057 (1989)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    Arnarson, B., Willits, J.T.: Thermal diffusion in binary mixtures of smooth, nearly elastic spheres with and without gravity. Phys. Fluids 10, 1324–1328 (1998)ADSCrossRefGoogle Scholar
  61. 61.
    Willits, J.T., Arnarson, B.: Kinetic theory of a binary mixture of nearly elastic disks. Phys. Fluids 11, 3116–3122 (1999)ADSCrossRefGoogle Scholar
  62. 62.
    Zamankhan, Z.: Kinetic theory for multicomponent dense mixtures of slightly inelastic spherical particles. Phys. Rev. E 52, 4877–4891 (1995)ADSCrossRefGoogle Scholar
  63. 63.
    Garzó, V., Montanero, J.M.: Effect of energy nonequipartition on the transport properties in a granular mixture. Granular Matter 5, 165–168 (2003)CrossRefGoogle Scholar
  64. 64.
    Lutsko, J.F.: Rheology of dense polydisperse granular fluids under shear. Phys. Rev. E 70, 061101 (2004)ADSCrossRefGoogle Scholar
  65. 65.
    Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Marín, C., Garzó, V., Santos, A.: Transport properties in a binary mixture under shear flow. Phys. Rev. E 52, 3812–3819 (1995)ADSCrossRefGoogle Scholar
  67. 67.
    Montanero, J.M., Garzó, V.: Energy nonequipartition in a sheared granular mixture. Mol. Simul. 29, 357–362 (2003)CrossRefGoogle Scholar
  68. 68.
    Ciccotti, G., Tenenbaum, A.: Canonical ensemble and nonequilibrium states by molecular dynamics. J. Stat. Phys. 23, 767–772 (1980)ADSCrossRefGoogle Scholar
  69. 69.
    Tenenbaum, A., Ciccotti, G., Gallico, R.: Stationary nonequilibrium states by molecular dynamics. Fourier’s law. Phys. Rev. A 25, 2778–2787 (1982)ADSCrossRefGoogle Scholar
  70. 70.
    Mareschal, M., Kestemont, E., Baras, F., Clementi, E., Nicolis, G.: Nonequilibrium states by molecular dynamics: transport coefficients in constrained fluids. Phys. Rev. A 35, 3883–3893 (1987)ADSCrossRefGoogle Scholar
  71. 71.
    Clause, P.J., Mareschal, M.: Heat transfer in a gas between parallel plates: moment method and molecular dynamics. Phys. Rev. A 38, 4241–4252 (1988)ADSCrossRefGoogle Scholar
  72. 72.
    Santos, A., Astillero, A.: System of elastic hard spheres which mimics the transport properties of a granular gas. Phys. Rev. E 72, 031308 (2005)ADSCrossRefGoogle Scholar
  73. 73.
    Vega Reyes, F., Garzó, V., Santos, A.: Granular mixtures modeled as elastic hard spheres subject to a drag force. Phys. Rev. E 75, 061306 (2007)ADSCrossRefGoogle Scholar
  74. 74.
    Brey, J.J., Cubero, D.: Steady state of a fluidized granular medium between two walls at the same temperature. Phys. Rev. E 57, 2019–2029 (1998)ADSCrossRefGoogle Scholar
  75. 75.
    Brey, J.J., Cubero, D., Ruiz-Montero, M.J., Moreno, F.: Fourier state of a fluidized granular gas. Europhys. Lett. 53, 432–437 (2001)ADSCrossRefGoogle Scholar
  76. 76.
    Brey, J.J., Khalil, N., Ruiz-Montero, M.J.: The Fourier state of a dilute granular gas described by the inelastic Boltzmann equation. J. Stat. Mech. P08019 (2009)Google Scholar
  77. 77.
    Brey, J.J., Khalil, N., Dufty, J.W.: Thermal segregation beyond Navier–Stokes. New J. Phys. 13, 055019 (2011)ADSCrossRefGoogle Scholar
  78. 78.
    Brey, J.J., Khalil, N., Dufty, J.W.: Thermal segregation of intruders in the Fourier state of a granular gas. Phys. Rev. E 85, 021307 (2012)ADSCrossRefGoogle Scholar
  79. 79.
    Vega Reyes, F., Garzó, V., Santos, A.: Impurity in a granular gas under nonlinear Couette flow. J. Stat. Mech. P09003 (2008)Google Scholar
  80. 80.
    Vega Reyes, F., Santos, A., Garzó, V.: Computer simulations of an impurity in a granular gas under planar Couette flow. J. Stat. Mech. P07005 (2011)Google 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