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Modified Sonine approximation for granular binary mixtures

Published online by Cambridge University Press:  06 March 2009

VICENTE GARZÓ ,
FRANCISCO VEGA REYES  and
JOSÉ MARÍA MONTANERO
VICENTE GARZÓ*
Affiliation:
Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
FRANCISCO VEGA REYES
Affiliation:
Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
JOSÉ MARÍA MONTANERO
Affiliation:
Departamento de Ingeniería Mecánica, Energética y de los Materiales, Universidad de Extremadura, E-06071 Badajoz, Spain
*
†Email address for correspondence: [email protected]
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Abstract

We evaluate in this work the hydrodynamic transport coefficients of a granular binary mixture in d dimensions. In order to eliminate the observed disagreement (for strong dissipation) between computer simulations and previously calculated theoretical transport coefficients for a monocomponent gas, we obtain explicit expressions of the seven Navier–Stokes transport coefficients by the use of a new Sonine approach in the Chapman–Enskog (CE) theory. This new approach consists of replacing, where appropriate in the CE procedure, the Maxwell–Boltzmann distribution weight function (used in the standard first Sonine approximation) by the homogeneous cooling state distribution for each species. The rationale for doing this lies in the well-known fact that the non-Maxwellian contributions to the distribution function of the granular mixture are more important in the range of strong dissipation we are interested in. The form of the transport coefficients is quite common in both standard and modified Sonine approximations, the distinction appearing in the explicit form of the different collision frequencies associated with the transport coefficients. Additionally, we numerically solve by the direct simulation Monte Carlo method the inelastic Boltzmann equation to get the diffusion and the shear viscosity coefficients for two and three dimensions. As in the case of a monocomponent gas, the modified Sonine approximation improves the estimates of the standard one, showing again the reliability of this method at strong values of dissipation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 623 , 25 March 2009 , pp. 387 - 411
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Aranson, I. S. & Tsimring, L. V. 2006 Patterns and collective behaviour in granular media: theoretical concepts. Rev. Mod. Phys. 78, 641–692.CrossRefGoogle Scholar
Arnarson, B. & Willits, J. T. 1998 Thermal diffusion in binary mixtures of smooth, nearly elastic spheres with and without gravity. Phys. Fluids 10, 1324–1328.CrossRefGoogle Scholar
Barrat, A. & Trizac, E. 2002 Lack of energy equipartition in homogeneous heated binary granular mixtures. Gran. Matt. 4, 57–63.CrossRefGoogle Scholar
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows. Clarendon.CrossRefGoogle Scholar
Brey, J. J., Dufty, J. W. & Santos, A. 1997 Dissipative dynamics for hard spheres. J. Stat. Phys. 87, 1051–1066.CrossRefGoogle Scholar
Brey, J. J., Dufty, J. W., Santos, A. & Kim, C. S. 1998 Hydrodynamics for granular flows at low density. Phys. Rev. E 58, 4638–4653.CrossRefGoogle Scholar
Brey, J. J. & Ruiz-Montero, M. J. 2004 Simulation study of the Green–Kubo relations for dilute granular gases. Phys. Rev. E 70, 051301.CrossRefGoogle ScholarPubMed
Brey, J. J., Ruiz-Montero, M. J. & Cubero, D. 1999 On the validity of linear hydrodynamics for low-density granular flows described by the Boltzmann equation. Europhys. Lett. 48, 359–364.CrossRefGoogle Scholar
Brey, J. J., Ruiz-Montero, M. J., Cubero, D. & García-Rojo, R. 2000 Self-diffusion in freely evolving granular gases. Phys. Fluids 12, 876–883.CrossRefGoogle Scholar
Brey, J. J., Ruiz-Montero, M. J., Maynar, P. & García de Soria, I. 2005 a Hydrodynamic modes, Green–Kubo relations, and velocity correlations in dilute granular gases. J. Phys.: Condens. Matter 17 (S2502).Google Scholar
Brey, J. J., Ruiz-Montero, M. J. & Moreno, F. 2005 b Energy partition and segregation for an intruder in a vibrated granular system under gravity. Phys. Rev. Lett. 95, 098001.CrossRefGoogle Scholar
Brey, J. J., Ruiz-Montero, M. J. & Moreno, F. 2006 Hydrodynamic profiles for an impurity in an open vibrated granular gas. Phys. Rev. E 73, 031301.CrossRefGoogle Scholar
Brey, J. J., Ruiz-Montero, M. J., Moreno, F. & García-Rojo, R. 2002 Transversal inhomogeneities in dilute vibrofluidized granular fluids. Phys. Rev. E 65, 061302.CrossRefGoogle ScholarPubMed
Brilliantov, N. & Pöschel, T. 2004 Kinetic Theory of Granular Gases. Oxford University Press.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Nonuniform Gases. Cambridge University Press.Google Scholar
Clerc, M. G., Cordero, P., Dunstan, J., Huff, K., Mujica, N., Risso, D. & Varas, G. 2008 Liquid-solid-like transition in quasi-one-dimensional driven granular media. Nature Phys. 4, 249–254.CrossRefGoogle Scholar
Dahl, S. R., Hrenya, C. M., Garzó, V. & Dufty, J. W. 2002 Kinetic temperatures for a granular mixture. Phys. Rev. E 66, 041301.CrossRefGoogle ScholarPubMed
Feitosa, K. & Menon, N. 2002 Breakdown of energy equipartition in a 2d binary vibrated granular gas. Phys. Rev. Lett. 88, 198301.CrossRefGoogle Scholar
Garzó, V. 2005 Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, 021106.CrossRefGoogle Scholar
Garzó, V. & Dufty, J. W. 1999 a Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895–5911.CrossRefGoogle ScholarPubMed
Garzó, V. & Dufty, J. W. 1999 b Homogeneous cooling state for a granular mixture. Phys. Rev. E 60, 5706–5713.CrossRefGoogle ScholarPubMed
Garzó, V. & Dufty, J. W. 2002 Hydrodynamics for a granular binary mixture at low density. Phys. Fluids. 14, 1476–1490.CrossRefGoogle Scholar
Garzó, V., Dufty, J. W. & Hrenya, C. M. 2007 a Enskog theory for polydisperse granular mixtures. Part 1. Navier–Stokes order transport. Phys. Rev. E 76, 031303.CrossRefGoogle Scholar
Garzó, V., Hrenya, C. M. & Dufty, J. W. 2007 b Enskog theory for polydisperse granular mixtures. Part 2. Sonine polynomial approximation. Phys. Rev. E 76, 031304.CrossRefGoogle Scholar
Garzó, V. & Montanero, J. M. 2002 Transport coefficients of a heated granular gas. Physica A 313, 336–356.CrossRefGoogle Scholar
Garzó, V. & Montanero, J. M. 2004 Diffusion of impurities in a granular gas. Phys. Rev. E 69, 021301.CrossRefGoogle Scholar
Garzó, V. & Montanero, J. M. 2007 Navier–Stokes transport coefficients of d-dimensional granular binary mixtures at low density. J. Stat. Phys. 129, 27–58.CrossRefGoogle Scholar
Garzó, V., Montanero, J. M. & Dufty, J. W. 2006 Mass and heat fluxes for a binary granular mixture at low density. Phys. Fluids 18, 083305.CrossRefGoogle Scholar
Garzó, V. & Santos, A. 2003 Kinetic Theory of Gases in Shear Flows: Nonlinear Transport. Kluwer.CrossRefGoogle Scholar
Garzó, V., Santos, A. & Montanero, J. M. 2007 c Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas. Physica A 376, 94–107.CrossRefGoogle Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57–92.Google Scholar
Goldshtein, A. & Shapiro, M. 1995 Mechanics of collisional motion of granular materials. 1. General hydrodynamic equations. J. Fluid Mech. 282, 75–114.CrossRefGoogle Scholar
Hrenya, C. M., Galvin, J. E. & Wildman, R. D. 2008 Evidence of higher order effects in thermally driven granular flows. J. Fluid Mech. 598, 429–450.CrossRefGoogle Scholar
Huan, C., Yang, X., Candela, D., Mair, R. W. & Walsworth, R. L. 2004 NMR experiments on a three-dimensional vibrofluidized granular medium. Phys. Rev. E 69, 041302.CrossRefGoogle ScholarPubMed
Jenkins, J. T. & Mancini, F. 1989 Kinetic theory for binary mixtures of smooth, nearly elastic spheres. Phys. Fluids A 1, 2050–2057.CrossRefGoogle Scholar
Krouskop, P. & Talbot, J. 2003 Mass and size effects in three-dimensional vibrofluidized granular mixtures. Phys. Rev. E 68, 021304.CrossRefGoogle ScholarPubMed
Lois, G., Lemaître, A. & Carlson, J. M. 2007 Spatial force correlations in granular shear flow. Part 2. Theoretical implications. Phys. Rev. E 76, 021303.CrossRefGoogle Scholar
Lutsko, J. 2005 Transport properties of dense dissipative hard-sphere fluids for arbitrary energy loss models. Phys. Rev. E 72, 021306.CrossRefGoogle ScholarPubMed
Lutsko, J., Brey, J. J. & Dufty, J. W. 2002 Diffusion in a granular fluid. Part 2. Simulation. Phys. Rev. E 65, 051304.CrossRefGoogle Scholar
Montanero, J. M. & Garzó, V. 2002 Monte Carlo simulation of the homogeneous cooling state for a granular mixture. Gran. Matt. 4, 17–24.CrossRefGoogle Scholar
Montanero, J. M. & Garzó, V. 2003 Shear viscosity for a heated granular binary mixture at low density. Phys. Rev. E 67, 021308.CrossRefGoogle ScholarPubMed
Montanero, J. M., Santos, A. & Garzó, V. 2005 DSMC evaluation of the Navier–Stokes shear viscosity of a granular fluid. In Rarefied Gas Dynamics 24 (ed. Capitelli, M., AIP Conference Proceedings), vol. 762, pp. 797–802.Google Scholar
Montanero, J. M., Santos, A. & Garzó, V. 2007 First-order Chapman–Enskog velocity distribution function in a granular gas. Physica A 376, 75–93.CrossRefGoogle Scholar
Noskowicz, S. H., Bar-Lev, O., Serero, D. & Goldhirsch, I. 2007 Computer-aided kinetic theory and granular gases. Europhys. Lett. 79, 60001.CrossRefGoogle Scholar
Pagnani, R., Marconi, U. M. B. & Puglisi, A. 2002 Driven low density granular mixtures. Phys. Rev. E 66, 051304.CrossRefGoogle ScholarPubMed
Rericha, E. C., Bizon, C., Shattuck, M. D. & Swinney, H. L. 2002 Shocks in supersonic sand. Phys. Rev. Lett. 88, 014302.CrossRefGoogle ScholarPubMed
Santos, A., Garzó, V. & Dufty, J. W. 2004 Inherent rheology of a granular fluid in uniform shear flow. Phys. Rev. E 69, 061303.CrossRefGoogle ScholarPubMed
Schröter, M., Ulrich, S., Kreft, J., Swift, J. B. & Swinney, H. L. 2006 Mechanisms in the size segregation of a binary granular mixture. Phys. Rev. E 74, 011307.CrossRefGoogle ScholarPubMed
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres to Burnett order. J. Fluid Mech. 361, 41–74.CrossRefGoogle Scholar
Serero, D., Goldhirsch, I., Noskowicz, S. H. & Tan, M. L. 2006 Hydrodynamics of granular gases and granular gas mixtures. J. Fluid Mech. 554, 237–258.CrossRefGoogle Scholar
Vega Reyes, F. & Urbach, J. S. 2008 a The effect of inelasticity on the phase transitions of a thin vibrated granular layer. Phys. Rev. E 78, 051301.CrossRefGoogle Scholar
Vega Reyes, F. & Urbach, J. S. 2008 b Steady base states for Navier–Stokes granulodynamics with boundary heating and shear. J. Fluid. Mech. (submitted) (Preprint: arXiv: 0807.5125).Google Scholar
Wang, H., Jin, G. & Ma, Y. 2003 Simulation study on kinetic temperatures of vibrated binary granular mixtures. Phys. Rev. E 68, 031301.CrossRefGoogle Scholar
Wildman, R. D. & Parker, D. J. 2002 Coexistence of two granular temperatures in binary vibrofluidized beds. Phys. Rev. Lett. 88, 064301.CrossRefGoogle ScholarPubMed
Willits, J. T. & Arnarson, B. 1999 Kinetic theory of a binary mixture of nearly elastic disks. Phys. Fluids 11, 3116–3122.CrossRefGoogle Scholar
Yang, X., Huan, C., Candela, D., Mair, R. W. & Walsworth, R. L. 2002 Measurements of grain motion in a dense, three-dimensional granular fluid. Phys. Rev. Lett. 88, 044301.CrossRefGoogle Scholar
Zamankhan, Z. 1995 Kinetic theory for multicomponent dense mixtures of slightly inelastic spherical particles. Phys. Rev. E 52, 4877–4891.CrossRefGoogle ScholarPubMed
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