Skip to main content
SpringerLink
Log in
Book cover

Granular Gaseous Flows pp 141–175Cite as

Navier–Stokes Transport Coefficients for Monocomponent Granular Gases. II. Simulations and Applications

  • Chapter
  • First Online:

  • 534 Accesses

Part of the book series: Soft and Biological Matter ((SOBIMA))

Abstract

The theoretical results derived in Chap. 3 for the Navier–Stokes transport coefficients are compared in this chapter with computer simulations carried out by several research groups. The comparison shows generally good agreement even in conditions of strong collisional dissipation. From this we appreciate the ability of kinetic theory to quantitatively capture the influence of dissipation on the transport properties of granular fluids. Once the reliability of granular kinetic theory has been assessed, several interesting applications of this theory are offered. First, a linear stability analysis of the Navier–Stokes hydrodynamic equations is performed to determine the critical size of the system beyond which the homogeneous cooling state becomes unstable. Theoretical predictions compare again very favorably with computer simulations for conditions of practical interest. As a second application, granular hydrodynamics is employed to obtain the temperature and density profiles of a granular fluid in a steady state with gravity. In agreement with simulations and experiments, the theory predicts that the temperature (density) profile as a function of the height of the system presents a minimum (maximum) whose value is determined by the coefficient of normal restitution of the gas. The chapter finishes with a short summary of the theoretical results derived at the level of the Navier–Stokes transport coefficients for other collisional models.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    See the website http://hrenya.colorado.edu.

  2. 2.

    See the website, https://mfix.netl.doe.gov.

References

  1. Garzó, V., Dufty, J.W.: Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895–5911 (1999)

    Article  ADS  Google Scholar 

  2. Garzó, V., Santos, A., Montanero, J.M.: Modified Sonine approximation for the Navier-Stokes transport coefficients of a granular gas. Physica A 376, 94–107 (2007)

    Google Scholar 

  3. Noskowicz, S.H., Bar-Lev, O., Serero, D., Goldhirsch, I.: Computer-aided kinetic theory and granular gases. Europhys. Lett. 79, 60001 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  4. Brey, J.J., Ruiz-Montero, M.J., Cubero, D.: On the validity of linear hydrodynamics for low-density granular flows described by the Boltzmann equation. Europhys. Lett. 48, 359–364 (1999)

    Article  ADS  Google Scholar 

  5. Brey, J.J., Ruiz-Montero, M.J.: Simulation study of the Green-Kubo relations for dilute granular gases. Phys. Rev. E 70, 051301 (2004)

    Article  ADS  Google Scholar 

  6. Brey, J.J., Ruiz-Montero, M.J., Maynar, P., García de Soria, M.I.: Hydrodynamic modes, Green-Kubo relations, and velocity correlations in dilute granular gases. J. Phys. Condens. Matter 17, S2489–S2502 (2005)

    Article  ADS  Google Scholar 

  7. 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, vol. 762, pp. 797–802. AIP Conference Proceedings (2005)

    Google Scholar 

  8. Montanero, J.M., Santos, A.: Monte Carlo simulation method for the Enskog equation. Phys. Rev. E 54, 438–444 (1996)

    Article  ADS  Google Scholar 

  9. Montanero, J.M., Santos, A.: Simulation of the Enskog equation à la Bird. Phys. Fluids 9, 2057–2060 (1997)

    Article  ADS  Google Scholar 

  10. Montanero, J.M., Santos, A., Garzó, V.: First-order Chapman-Enskog velocity distribution function in a granular gas. Physica A 376, 75–93 (2007)

    Google Scholar 

  11. Goldhirsch, I., Zanetti, G.: Clustering instability in dissipative gases. Phys. Rev. Lett. 70, 1619–1622 (1993)

    Article  ADS  Google Scholar 

  12. McNamara, S.: Hydrodynamic modes of a uniform granular medium. Phys. Fluids A 5, 3056–3069 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  13. Brey, J.J., Dufty, J.W., Kim, C.S., Santos, A.: Hydrodynamics for granular flows at low density. Phys. Rev. E 58, 4638–4653 (1998)

    Article  ADS  Google Scholar 

  14. Goldhirsch, I., Tan, M.L., Zanetti, G.: A molecular dynamical study of granular fluids I: the unforced granular gas in two dimensions. J. Sci. Comput. 8, 1–40 (1993)

    Article  Google Scholar 

  15. McNamara, S., Young, W.R.: Inelastic collapse in two dimensions. Phys. Rev. E 50, R28–R31 (1994)

    Article  ADS  Google Scholar 

  16. McNamara, S., Young, W.R.: Dynamics of a freely evolving, two-dimensional granular medium. Phys. Rev. E 53, 5089–5100 (1996)

    Article  ADS  Google Scholar 

  17. Deltour, P., Barrat, J.L.: Quantitative study of a freely cooling granular medium. J. Phys. I (France) 7, 137–151 (1997)

    Article  Google Scholar 

  18. Brito, R., Ernst, M.H.: Extension of Haff’s cooling law in granular flows. Europhys. Lett. 43, 497–502 (1998)

    Article  ADS  Google Scholar 

  19. Luding, S.: Herrmann: cluster-growth in freely cooling granular media. Chaos 9, 673–680 (1999)

    Article  ADS  Google Scholar 

  20. Ben-Naim, E., Chen, S.Y., Doolen, G.D., Redner, S.: Shocklike dynamics of inelastic gases. Phys. Rev. Lett. 83, 4069–4072 (1999)

    Article  ADS  Google Scholar 

  21. Fullmer, W.D., Hrenya, C.M.: The clustering instability in rapid granular and gas-solid flows. Annu. Rev. Fluid Mech. 49, 485–510 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  22. Résibois, P., de Leener, M.: Classical Kinetic Theory of Fluids. Wiley, New York (1977)

    MATH  Google Scholar 

  23. Garzó, V.: Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, 021106 (2005)

    Article  ADS  Google Scholar 

  24. van Noije, T.P.C., Ernst, M.H.: Cahn-Hilliard theory for unstable granular fluids. Phys. Rev. E 61, 1765–1782 (2000)

    Article  ADS  Google Scholar 

  25. Brey, J.J., Ruiz-Montero, M.J., Moreno, F.: Instability and spatial correlations in a dilute granular gas. Phys. Fluids 10, 2976–2982 (1998)

    Article  ADS  Google Scholar 

  26. Brey, J.J., Ruiz-Montero, M.J., Cubero, D.: Origin of density clustering in a freely evolving granular gas. Phys. Rev. E 60, 3150–3157 (1999)

    Article  ADS  Google Scholar 

  27. Soto, R., Mareschal, M., Malek Mansour, M.: Nonlinear analysis of the shearing instability in granular gases. Phys. Rev. E 62, 3836–3842 (2000)

    Article  ADS  Google Scholar 

  28. Mitrano, P.P., Dhal, S.R., Cromer, D.J., Pacella, M.S., Hrenya, C.M.: Instabilities in the homogeneous cooling of a granular gas: a quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, 093303 (2011)

    Article  ADS  Google Scholar 

  29. Kudrolli, A., Wolpert, M., Gollub, J.P.: Cluster formation due to collisions in granular material. Phys. Rev. Lett. 78, 1383–1386 (1997)

    Article  ADS  Google Scholar 

  30. Olafsen, J.S., Urbach, J.S.: Clustering, order, and collapse in a driven granular monolayer. Phys. Rev. Lett. 81, 4369–4372 (1998)

    Article  ADS  Google Scholar 

  31. Falcon, E., Wunenburger, R., Èvesque, P., Fauve, S., Chabot, C., Garrabos, Y., Beysens, D.: Cluster formation in a granular medium fluidized by vibrations in low gravity. Phys. Rev. Lett. 83, 440–443 (1999)

    Article  ADS  Google Scholar 

  32. Nie, X., Ben-Naim, E., Chen, S.: Dynamics of freely cooling granular gases. Phys. Rev. Lett. 89, 204301 (2002)

    Article  ADS  Google Scholar 

  33. Mitrano, P.P., Garzó, V., Hilger, A.M., Ewasko, C.J., Hrenya, C.M.: Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases. Phys. Rev. E 85, 041303 (2012)

    Article  ADS  Google Scholar 

  34. Mitrano, P.P., Zenk, J.R., Benyahia, S., Galvin, J.E., Dhal, S.R., Hrenya, C.M.: Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime. J. Fluid Mech. 738, R2 (2014)

    Article  ADS  Google Scholar 

  35. Mitrano, P.P., Dhal, S.R., Hilger, A.M., Ewasko, C.J., Hrenya, C.M.: Dual role of friction in granular flows: attenuation versus enhancement of instabilities. J. Fluid Mech. 729, 484–495 (2013)

    Article  ADS  Google Scholar 

  36. Clement, E., Rajchenbach, J.: Fluidization of a bidimensional powder. Europhys. Lett. 16, 133–138 (1991)

    Article  ADS  Google Scholar 

  37. 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)

    Article  ADS  Google Scholar 

  38. 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)

    Article  ADS  Google Scholar 

  39. Gallas, J.A.C., Herrmann, H.J., Sokolowski, S.: Molecular dynamics simulation of powder fluidizaton in two dimensions. Physica A 189, 437–446 (1992)

    Google Scholar 

  40. Lee, J.: Scaling behavior of granular particles in a vibrating box. Physica A 219, 305–326 (1995)

    Google Scholar 

  41. Helal, K., Biben, T., Hansen, J.P.: Local fluctuations in a fluidized granular medium. Physica A 240, 361–373 (1997)

    Google Scholar 

  42. McNamara, S., Barrat, J.L.: Energy flux into a fluidized granular medium at a vibrating wall. Phys. Rev. E 55, 7767–7770 (1997)

    Article  ADS  Google Scholar 

  43. Brey, J.J., Ruiz-Montero, M.J., Moreno, F.: Boundary conditions and normal state for a vibrated granular fluid. Phys. Rev. E 62, 5339–5346 (2000)

    Article  ADS  Google Scholar 

  44. Brey, J.J., Ruiz-Montero, M.J., Moreno, F.: Hydrodynamics of an open vibrated granular system. Phys. Rev. E 63, 061305 (2001)

    Article  ADS  Google Scholar 

  45. McNamara, S., Luding, S.: Energy nonequipartition in systems of inelastic rough spheres. Phys. Rev. E 58, 2247–2250 (1998)

    Article  ADS  Google Scholar 

  46. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)

    MATH  Google Scholar 

  47. 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)

    Article  ADS  Google Scholar 

  48. 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)

    Article  ADS  Google Scholar 

  49. 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)

    Article  MathSciNet  Google Scholar 

  50. Goldhirsch, I., Noskowicz, S.H., Bar-Lev, O.: Nearly smooth granular gases. Phys. Rev. Lett. 95, 068002 (2005)

    Article  ADS  Google Scholar 

  51. Goldhirsch, I., Noskowicz, S.H., Bar-Lev, O.: Hydrodynamics of nearly smooth granular gases. J. Phys. Chem. B 109, 21449–21470 (2005)

    Article  Google Scholar 

  52. Lun, C.K.K.: Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. J. Fluid Mech. 233, 539–559 (1991)

    Article  ADS  Google Scholar 

  53. Kremer, G., Santos, A., Garzó, V.: Transport coefficients of a granular gas of inelastic rough hard spheres. Phys. Rev. E 90, 022205 (2014)

    Article  ADS  Google Scholar 

  54. Chapman, S., Cowling, T.G.: The Mathematical Theory of Nonuniform Gases. Cambridge University Press, Cambridge (1970)

    MATH  Google Scholar 

  55. Bridges, F.G., Hatzes, A., Lin, D.N.C.: Structure, stability and evolution of Saturn’s rings. Nature (London) 309, 333–335 (1984)

    Article  ADS  Google Scholar 

  56. Sorace, C.M., Louge, M.Y., Crozier, M.D., Law, V.H.C.: High apparent adhesion energy in the breakdown of normal restitution for binary impacts of small spheres at low speed. Mech. Res. Comm. 36, 364–368 (2009)

    Article  Google Scholar 

  57. Grasselli, Y., Bossis, G., Goutallier, G.: Velocity-dependent restitution coefficient and granular cooling in microgravity. Europhys. Lett. 86, 60007 (2009)

    Article  ADS  Google Scholar 

  58. Brilliantov, N.V., Pöschel, T.: Hydrodynamics and transport coefficients for dilute granular gases. Phys. Rev. E 67, 061304 (2003)

    Article  ADS  Google Scholar 

  59. Brilliantov, N., Pöschel, T.: Kinetic Theory of Granular Gases. Oxford University Press, Oxford (2004)

    Book  Google Scholar 

  60. Brilliantov, N.V., Pöschel, T.: Velocity distribution in granular gases of viscoelastic particles. Phys. Rev. E 61, 5573–5587 (2000)

    Article  ADS  Google Scholar 

  61. Brilliantov, N.V., Salueña, C., Schwager, T., Pöschel, T.: Transient structures in a granular gas. Phys. Rev. Lett. 93, 134301 (2004)

    Article  ADS  Google Scholar 

  62. Hummel, M., Mazza, M.G.: Declustering in a granular gas as a finite-size effect. Phys. Rev. E 93, 022905 (2016)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, Badajoz, Spain

    Vicente Garzó

Authors
  1. Vicente Garzó
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vicente Garzó .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Garzó, V. (2019). Navier–Stokes Transport Coefficients for Monocomponent Granular Gases. II. Simulations and Applications. In: Granular Gaseous Flows. Soft and Biological Matter. Springer, Cham. https://doi.org/10.1007/978-3-030-04444-2_4

Download citation

Publish with us

Policies and ethics

Search

Navigation