Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Granular Gaseous Flows

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

Abstract

The approximate expressions obtained in Chap. 5 for the Navier–Stokes transport coefficients of granular mixtures are compared first in this chapter with controlled numerical simulations of certain specific situations. In particular, the tracer diffusion and shear viscosity coefficients are obtained by numerically solving the Boltzmann and Enskog kinetic equations by means of the Direct Simulation Monte Carlo method. As in the case of monocomponent granular fluids, comparison between theory and simulations shows a good agreement over a wide range of values of the coefficients of restitution, density, and the parameters of the mixture (masses and sizes). Once the reliability of the theoretical results is assessed, some interesting applications of the Navier–Stokes granular hydrodynamic equations will be considered. First, the violation of the Einstein relation between the diffusion and mobility coefficients in granular fluids is quantified. Analysis indicates that this violation is essentially due to two independent reasons: the cooling of the reference homogeneous cooling state and the occurrence of different temperatures for the particle and surrounding fluid. Since the constitutive equations for mass and heat fluxes in granular mixtures are different from those obtained for ordinary mixtures, the (possible) violation of Onsager’s reciprocal relations among various transport coefficients is also assessed. Additionally, as with single granular fluids, a linear stability analysis of the Navier–Stokes equations with respect to homogeneous cooling state is performed to identify the unstable hydrodynamic modes. Theoretical predictions for instability associated with transversal shear modes (velocity vortices) are compared against MD simulations for conditions of practical interest. Excellent agreement between theory and simulation is found when mechanical properties of particles are relatively similar, while only good agreement occurs for disparate-mass binary mixtures. Finally, the chapter ends with an analysis of thermal diffusion segregation. Special attention is paid to the tracer limit situation where a segregation criterion is explicitly derived to explain the transition between Brazil-nut effect \(\Leftrightarrow \) reverse Brazil-nut effect.

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

Access this chapter

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

Institutional subscriptions

References

  1. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows. Clarendon, Oxford (1994)

    Google Scholar 

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

    MATH  Google Scholar 

  3. Santos, A., Dufty, J.W.: Dynamics of a hard sphere granular impurity. Phys. Rev. Lett. 97, 058001 (2006)

    Article  ADS  Google Scholar 

  4. Brey, J.J., Ruiz-Montero, M.J., Cubero, D., García-Rojo, R.: Self-diffusion in freely evolving granular gases. Phys. Fluids 12, 876–883 (2000)

    Article  ADS  Google Scholar 

  5. McLennan, J.A.: Introduction to Nonequilibrium Statistical Mechanics. Prentice-Hall, New Jersey (1989)

    Google Scholar 

  6. Garzó, V., Montanero, J.M.: Diffusion of impurities in a granular gas. Phys. Rev. E 69, 021301 (2004)

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  8. Mason, E.A.: Transport properties of gases obeying a modified Buckingham potential. J. Chem. Phys. 22, 169–192 (1954)

    Article  ADS  Google Scholar 

  9. López de Haro, M., Cohen, E.G.D.: The Enskog theory for multicomponent mixtures. III. Transport properties of dense binary mixtures with one tracer component. J. Chem. Phys. 80, 408–415 (1984)

    Article  ADS  Google Scholar 

  10. Garzó, V., Vega Reyes, F.: Mass transport of impurities in a moderately dense granular gas. Phys. Rev. E 79, 041303 (2009)

    Article  ADS  Google Scholar 

  11. Brilliantov, N.V., Pöschel, T.: Self-diffusion in granular gases. Phys. Rev. E 61, 1716–1721 (2000)

    Article  ADS  Google Scholar 

  12. Garzó, V., Montanero, J.M.: Navier-Stokes transport coefficients of \(d\)-dimensional granular binary mixtures at low-density. J. Stat. Phys. 129, 27–58 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  13. Montanero, J.M., Garzó, V.: Shear viscosity for a heated granular binary mixture at low density. Phys. Rev. E 67, 021308 (2003)

    Article  ADS  Google Scholar 

  14. Garzó, V., Montanero, J.M.: Shear viscosity for a moderately dense granular binary mixture. Phys. Rev. E 68, 041302 (2003)

    Article  ADS  Google Scholar 

  15. 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 

  16. 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 

  17. Dufty, J.W., Brey, J.J., Lutsko, J.F.: Diffusion in a granular fluid I. Theory. Phys. Rev. E 65, 051303 (2002)

    Article  ADS  Google Scholar 

  18. Dufty, J.W., Garzó, V.: Mobility and diffusion in granular fluids. J. Stat. Phys. 105, 723–744 (2001)

    Article  MathSciNet  Google Scholar 

  19. Garzó, V., Hrenya, C.M., Dufty, J.W.: Enskog theory for polydisperse granular mixtures. II. Sonine polynomial approximation. Phys. Rev. E 76, 031304 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  20. Garzó, V.: On the Einstein relation in a heated granular gas. Physica A 343, 105–126 (2004)

    Google Scholar 

  21. Garzó, V.: A note on the violation of the Einstein relation in a driven moderately dense granular gas. J. Stat. Mech. P05007 (2008)

    Google Scholar 

  22. Barrat, A., Loreto, V., Puglisi, A.: Temperature probes in binary granular gases. Physica A 66, 513–523 (2004)

    Google Scholar 

  23. Puglisi, A., Baldasarri, A., Vulpiani, A.: Violation of the Einstein relation in granular fluids: the role of correlations. J. Stat. Mech. P08016 (2007)

    Google Scholar 

  24. de Groot, S.R., Mazur, P.: Nonequilibrium Thermodynamics. Dover, New York (1984)

    MATH  Google Scholar 

  25. Mitrano, P.P., Garzó, V., Hrenya, C.M.: Instabilities in granular binary mixtures at moderate densities. Phys. Rev. E 89, 020201(R) (2014)

    Article  ADS  Google Scholar 

  26. Garzó, V., Montanero, J.M., Dufty, J.W.: Mass and heat fluxes for a binary granular mixture at low density. Phys. Fluids 18, 083305 (2006)

    Article  ADS  Google Scholar 

  27. Brey, J.J., Ruiz-Montero, M.J.: Shearing instability of a dilute granular mixture. Phys. Rev. E 87, 022210 (2013)

    Article  ADS  Google Scholar 

  28. Garzó, V.: Stability of freely cooling granular mixtures at moderate densities. Chaos Solitons Fractals 81, 497–509 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  29. Kudrolli, A.: Size separation in vibrated granular matter. Rep. Prog. Phys. 67, 209–247 (2004)

    Article  ADS  Google Scholar 

  30. Daniels, K.E., Schröter, M.: Focus on granular segregation. New J. Phys. 15, 035017 (2013)

    Article  ADS  Google Scholar 

  31. Rosato, A., Strandburg, K.J., Prinz, F., Swendsen, R.H.: Why the Brazil nuts are on top: size segregation of particulate matter by shaking. Phys. Rev. Lett. 58, 1038–1040 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  32. Knight, J.B., Jaeger, H.M., Nagel, S.R.: Vibration-induced size separation in granular media: the convection connection. Phys. Rev. Lett. 70, 3728–3731 (1993)

    Article  ADS  Google Scholar 

  33. Duran, J., Rajchenbach, J., Clément, E.: Arching effect model for particle size segregation. Phys. Rev. Lett. 70, 2431–2434 (1993)

    Article  ADS  Google Scholar 

  34. Shinbrot, T., Muzzio, F.J.: Reverse buoyancy in shaken granular beds. Phys. Rev. Lett. 81, 4365–4368 (1998)

    Article  ADS  Google Scholar 

  35. Hong, D.C., Quinn, P.V., Luding, S.: Reverse Brazil nut problem: competition between percolation and condensation. Phys. Rev. Lett. 86, 3423–3426 (2001)

    Article  ADS  Google Scholar 

  36. Luding, S., Clément, E., Blumen, A., Rajchenbach, J., Duran, J.: Onset of convection in molecular dynamics simulations of grains. Phys. Rev. E 50, R1762–R1765 (1994)

    Article  ADS  Google Scholar 

  37. Möbius, M.E., Lauderdale, B.E., Nagel, S.R., Jaeger, H.M.: Brazil-nut effect: size separation of granular particles. Nature 414, 270 (2001)

    Article  ADS  Google Scholar 

  38. Goldhirsch, I., Ronis, D.: Theory of thermophoresis. I. General considerations and mode-coupling analysis. Phys. Rev. A 27, 1616–1634 (1983)

    Article  ADS  Google Scholar 

  39. Goldhirsch, I., Ronis, D.: Theory of thermophoresis. II. Low-density behavior. Phys. Rev. A 27, 1635–1656 (1983)

    Article  ADS  Google Scholar 

  40. Grew, K.E., Ibbs, T.L.: Thermal Diffusion in Gases. Cambridge University Press, Cambridge (1952)

    MATH  Google Scholar 

  41. Maitland, G.C., Rigby, M., Smith, E.B., Wakeham, W.A.: Intermolecular Forces: Their Origin and Determination. Clarendon, Oxford (1981)

    Google Scholar 

  42. Jenkins, J.T., Yoon, D.K.: Segregation in binary mixtures under gravity. Phys. Rev. Lett. 88, 194301 (2002)

    Article  ADS  Google Scholar 

  43. Garzó, V.: Thermal diffusion segregation in granular binary mixtures described by the Enskog equation. New J. Phys. 13, 055020 (2011)

    Article  ADS  Google Scholar 

  44. Serero, D., Goldhirsch, I., Noskowicz, S.H., Tan, M.L.: Hydrodynamics of granular gases and granular gas mixtures. J. Fluid Mech. 554, 237–258 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  45. Serero, D., Noskowicz, S.H., Tan, M.L., Goldhirsch, I.: Binary granular gas mixtures: theory, layering effects and some open questions. Eur. Phys. J. Spec. Top. 179, 221–247 (2009)

    Article  Google Scholar 

  46. Brito, R., Enríquez, H., Godoy, S., Soto, R.: Segregation induced by inelasticity in a vibrofluidized granular mixture. Phys. Rev. E 77, 061301 (2008)

    Article  ADS  Google Scholar 

  47. Brito, R., Soto, R.: Competition of Brazil nut effect, buoyancy, and inelasticity induced segregation in a granular mixture. Eur. Phys. J. Spec. Top. 179, 207–219 (2009)

    Article  Google Scholar 

  48. 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)

    Article  ADS  Google Scholar 

  49. 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)

    Article  ADS  Google Scholar 

  50. Garzó, V.: Segregation in granular binary mixtures: thermal diffusion. Europhys. Lett. 75, 521–527 (2006)

    Article  ADS  Google Scholar 

  51. 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)

    Article  ADS  Google Scholar 

  52. Yoon, D.K., Jenkins, J.T.: The influence of different species’ granular temperatures on segregation in a binary mixture of dissipative grains. Phys. Fluids 18, 073303 (2006)

    Article  ADS  Google Scholar 

  53. Garzó, V.: Segregation by thermal diffusion in moderately dense granular mixtures. Eur. Phys. J. E 29, 261–274 (2009)

    Article  Google Scholar 

  54. 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)

    Article  ADS  Google Scholar 

  55. Galvin, J.E., Dahl, S.R., Hrenya, C.M.: On the role of non-equipartition in the dynamics of rapidly flowing granular mixtures. J. Fluid Mech. 528, 207–232 (2005)

    Article  ADS  MathSciNet  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 Multicomponent 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_6

Download citation

Publish with us

Policies and ethics

Search

Navigation