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Theory of Dilute Binary Granular Gas Mixtures

Published online by Cambridge University Press:  09 June 2010

D. Serero ,
S. H. Noskowicz  and
I. Goldhirsch
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Abstract

A computer-aided method for accurately carrying out the Chapman-Enskog expansion of the Boltzmann equation, including its inelastic variant, is presented and employed to derive a hydrodynamic description of a dilute binary mixture of smooth inelastic spheres. Constitutive relations, formally valid for all physical values of the coefficients of restitution, are calculated by carrying out the pertinent Chapman-Enskog expansion to sufficient high orders in the Sonine polynomials to ensure numerical convergence. The resulting hydrodynamic description is applied to the analysis of a vertically vibrated binary mixture of particles (under gravity) differing only in their respective coefficients of restitution. It is shown that even with this “minor”difference the mixture partly segregates, its steady state exhibiting a sandwich-like configuration.

Keywords

kinetic theorygranular gasesmixturessegregationChapman-Enskog expansionhydrodynamics
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 1: Instability and patterns. Issue dedicated to the memory of A. Golovin , 2011 , pp. 17 - 47
Copyright
© EDP Sciences, 2010

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References

Arnarson, B. Ö., Willits, J. T.. Thermal diffusion in binary mixtures of smooth, nearly elastic spheres with and without gravity . Phys. Fluids, 10 (1998), No. 1, 1324–1328.CrossRefGoogle Scholar
Bose, M., Nott, P. R., Kumaran, V.. Excluded-volume attraction in vibrated granular mixtures . Europhys. Lett., 68 (2004), No. 4, 508–514.CrossRefGoogle Scholar
Brey, J. J., Ruiz-Montero, M. J., Moreno, F.. Hydrodynamics of an open vibrated granular system . Phys. Rev. E, 63 (2001), No. 6, 061305.CrossRefGoogle ScholarPubMed
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 (2005), No. 9, 098001.CrossRefGoogle Scholar
Brillantov, N. V., Pöschel, T.. Breakdown of the Sonine expansion for the velocity distribution of granular gases . Europhys. Lett., 74 (2006), No. 3, 424–430.CrossRefGoogle Scholar
Brito, R., Enriquez, H., Godoy, S., Soto, R.. Segregation induced by inelasticity in a vibrofluidized granular mixture . Phys. Rev. E, 77 (2008), No. 6, 061301.CrossRefGoogle Scholar
Brone, D., Muzzio, F. J.. Size segregation in vibrated granular systems: A reversible process . Phys. Rev. E, 56 (1997), No. 1, 1059–1063.CrossRefGoogle Scholar
S. Chapman and T. G. Cowling. The mathematical Theory of Nonuniform Gases. Cambridge Univ. Press, London, 1970.
Cooke, W., Warr, S., Huntley, J. M., Ball, R. C.. Particle size segregation in a two-dimensional bed undergoing vertical vibration . Phys. Rev. E, 53 (1996), No. 3, 2812–2822.CrossRefGoogle Scholar
S. R. de Groot and P. Mazur. Non-Equilibrium Thermodynamics. North-Holland, Amsterdam, 1969.
Esipov, S. E., Pöschel, T.. The granular phase diagram . J. Stat. Phys., 86 (1997), No. 5-6, 1385–1395.CrossRefGoogle Scholar
Farkas, Z., Szalai, F., Wolf, D. E., Vicsek, T.. Segregation of binary mixtures by a ratchet mechanism . Phys. Rev. E, 65 (2002), No. 2, 022301.CrossRefGoogle ScholarPubMed
Garzó, V.. Segregation in granular binary mixtures: Thermal diffusion . Europhys. Lett., 75 (2006), No. 4, 521–527.CrossRefGoogle Scholar
Garzó, V.. Brazil-nut effect versus reverse Brazil-nut effect in a moderately dense granular fluid . Phys. Rev. E, 78 (2008), No. 2, 020301.CrossRefGoogle Scholar
Garzó, V., Dufty, J. W.. Hydrodynamics for a granular binary mixture at low density . Phys Fluids, 14 (2002), No. 4, 1476–14902.CrossRefGoogle Scholar
Garzó, V., Reyes, F. V., Montanero, J. M.. Modified Sonine approximation for granular binary mixtures . J Fluid Mech., 623 (2009), 387–411.CrossRefGoogle Scholar
Goldhirsch, I.. Rapid granular flows . Annu Rev Fluid Mech., 35 (2003), 267–293.CrossRefGoogle Scholar
Goldhirsch, I., Ronis, D.. Theory of thermophoresis I: General considerations and mode coupling analysis . Phys Rev. A, 27 (1983), No. 3, 1616–1634.CrossRefGoogle Scholar
Goldhirsch, I., Ronis, D.. Theory of thermophoresis II: Low-density behavior . Phys Rev. A, 27 (1983), No. 3, 1635–1656.CrossRefGoogle Scholar
Hong, D. C., Quinn, P. V., Luding, S.. The reverse Brazil nut problem: Competition between percolation and condensation . Phys Rev Lett., 86 (2001), No. 15, 3423–3426.CrossRefGoogle ScholarPubMed
Hsiau, S. S., Hunt, M. L.. Granular thermal diffusion in flows of binary-sized mixtures . Acta Mech., 114 (1996), No. 1-4, 121–137.CrossRefGoogle Scholar
Jaeger, H. M., Nagel, S. R. and Behringer, R. P.. Granular solids, liquids, and gases . Rev Mod Phys., 68 (1996), No. 4, 1259–1273.CrossRefGoogle Scholar
Jenkins, J. T., Mancini, F.. Kinetic theory for binary mixtures of smooth nearly elastic spheres . Phys Fluids A, 1 (1989), No. 12, 2050–2059.CrossRefGoogle Scholar
Jenkins, J. T., Yoon, D. K.. Segregation in binary mixture under gravity . Phys Rev Lett., 88 (2002), No. 19, 194304.CrossRefGoogle ScholarPubMed
Kincaid, J. M., Cohen, E. G. D., Lopez de Haro, M.. The Enskog theory for multicomponent mixtures. iv. thermal diffusion . J Chem Phys., 86 (1987), No. 2, 963–975.CrossRefGoogle Scholar
Knight, J. B., Ehrlich, E. E., Kuperman, V. Y., Flint, J. K., Jaeger, H. M., Nagel, S. R.. Experimental study of granular convection . Phys Rev. E, 54 (1996), No. 5, 5726–5738.CrossRefGoogle ScholarPubMed
Knight, J. B., Jaeger, H. M., Nagel, S. R.. Vibration-induced size separation in granular media, No. 4, The convection connection . Phys Rev Lett., 70 (1993), No. 24, 3728–3731.CrossRefGoogle Scholar
Kondic, L., Hartley, R. R., Tennakoon, S. G. K., Painter, B., Behringer, R. P.. Segregation by friction . Europhys Lett., 61 (2003), No. 6, 742–748.CrossRefGoogle Scholar
Kudrolli, A.. Size separation in vibrated granular matter . Reports on Progress in Physics., 67 (2004), No. 3, 209–247.CrossRefGoogle Scholar
L. D. Landau, E. M. Lifshitz. Fluid Mechanics. Pergamon, London, 1959.
Mobius, M. E., Cheng, X., Eshuis, P., Karczmar, S. R., Nagel, G. S., Jaeger, H. M.. Effect of air on granular size separation in a vibrated granular bed . Phys Rev. E, 72 (2005), No. 1, 011304.CrossRefGoogle Scholar
Noskowicz, S. H., Bar-Lev, O., Serero, D., Goldhirsch, I.. Computer-aided kinetic theory and granular gases . Europhys Lett., 79 (2007), No. 6, 60001.CrossRefGoogle Scholar
Ottino, J. M., Khakhar, D. V.. Mixing and segregation of granular materials . Annu Rev Fluid Mech., 32 (2000), 55–91.CrossRefGoogle Scholar
Pöschel, T., Brillantov, N. V., Formella, A.. Impact of high-energy tails on granular gas properties . Phys Rev. E, 74 (2006), No. 4, 041302.CrossRefGoogle ScholarPubMed
Pöschel, T., Herrmann, H. J.. Size segregation and convection . Europhys Lett., 29 (1995), No. 2, 123–128.CrossRefGoogle Scholar
Rapaport, D. C.. Mechanism for granular segregation . Phys Rev. E, 64 (2001), No. 6, 061304.CrossRefGoogle ScholarPubMed
Reis, P. M., Mullin, T.. Granular segregation as a critical phenomenon . Phys Rev Lett., 89 (2002), No. 24, 244301.CrossRefGoogle ScholarPubMed
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 (1987), No. 10, 1038–1040.CrossRefGoogle Scholar
Schröter, M., Ulrich, S., Keft, J., Swift, J. B., Swinney, H. L.. Mechanism in the size segregation of a binary granular mixture . Phys Rev. E, 74 (2006), No. 1, 011307.CrossRefGoogle ScholarPubMed
Sela, N., Goldhirsch, I.. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order . J Fluid Mech., 361 (1998), 41–74.CrossRefGoogle Scholar
D. Serero, S. H. Noskowicz, and I. Tan, M. L. Goldhirsch. Layering effects in vertically vibrated systems., Eur. Phys. J. E (2009).
D. Serero. Kinetic Theory of Granular Gas Mixtures. PhD thesis, Tel Aviv University, 2009.
Serero, D., Goldhirsch, I., Noskowicz, S. H., Tan, M. L.. Hydrodynamics of granular gases and granular gas mixtures . J Fluid Mech., 554 (2006), 237–258.CrossRefGoogle Scholar
Serero, D., Noskowicz, S. H., Goldhirsch, I.. Exact versus mean field solutions for granular gas mixtures . Gran. Matt., 10 (2007), No. 1, 37–46.CrossRefGoogle Scholar
Shinbrot, T., Muzzio, F. J.. Reverse buoyancy in shaken granular beds . Phys Rev Lett., 81 (1998), No. 20, 4365–4368.CrossRefGoogle Scholar
Shinbrot, T., Muzzio, F. J.. Nonequilibrium patterns in granular mixing and segregation . "Physics Today", 53 (2000), No. 3, 25–30. CrossRefGoogle Scholar
Trujillo, L., Alam, M., Herrmann, H. J.. Segregation in a fluidized binary granular mixture: competition between buoyancy and geometric force . Europhys Lett., 64 (2003), No. 2, 190–196.CrossRefGoogle Scholar
Ulrich, S., Schröter, M., Swinney, H. L.. Influence of friction on granular segregation . Phys Rev. E, 76 (2007), No. 4, 042301.CrossRefGoogle ScholarPubMed
Viswanathan, H., Wildman, R. D., Huntley, J. M., Martin, T. W.. Comparison of kinetic theory predictions with experimental results for a vibrated three-dimensional granular bed . Phys Fluids, 18 (2006), No. 11, 113302.CrossRefGoogle Scholar
Wildman, R. D., Jenkins, J. T., Krouskop, P. E., Talbot, J.. A comparison of the predictions of a simple kinetic theory with experimental and numerical results for a vibrated granular bed consisting of nearly elastic particles . Phys Fluids, 18 (2006), No. 7, 073301.CrossRefGoogle Scholar
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 (2006), No. 7, 073303.CrossRefGoogle Scholar