# Theory of Dilute Binary Granular Gas Mixtures

D. Serero; S. H. Noskowicz; I. Goldhirsch

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 6, Issue: 1, page 17-47
- ISSN: 0973-5348

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topSerero, D., Noskowicz, S. H., and Goldhirsch, I.. "Theory of Dilute Binary Granular Gas Mixtures." Mathematical Modelling of Natural Phenomena 6.1 (2010): 17-47. <http://eudml.org/doc/197699>.

@article{Serero2010,

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.},

author = {Serero, D., Noskowicz, S. H., Goldhirsch, I.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {kinetic theory; granular gases; mixtures; segregation; Chapman-Enskog expansion; hydrodynamics},

language = {eng},

month = {6},

number = {1},

pages = {17-47},

publisher = {EDP Sciences},

title = {Theory of Dilute Binary Granular Gas Mixtures},

url = {http://eudml.org/doc/197699},

volume = {6},

year = {2010},

}

TY - JOUR

AU - Serero, D.

AU - Noskowicz, S. H.

AU - Goldhirsch, I.

TI - Theory of Dilute Binary Granular Gas Mixtures

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/6//

PB - EDP Sciences

VL - 6

IS - 1

SP - 17

EP - 47

AB - 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.

LA - eng

KW - kinetic theory; granular gases; mixtures; segregation; Chapman-Enskog expansion; hydrodynamics

UR - http://eudml.org/doc/197699

ER -

## References

top- B. Ö. Arnarson, J. T. Willits. Thermal diffusion in binary mixtures of smooth, nearly elastic spheres with and without gravity. Phys. Fluids, 10 (1998), No. 1, 1324–1328.
- M. Bose, P. R. Nott, V. Kumaran. Excluded-volume attraction in vibrated granular mixtures. Europhys. Lett., 68 (2004), No. 4, 508–514.
- J. J. Brey, M. J. Ruiz-Montero, F. Moreno. Hydrodynamics of an open vibrated granular system. Phys. Rev. E, 63 (2001), No. 6, 061305.
- J. J. Brey, M. J. Ruiz-Montero, F. Moreno. Energy partition and segregation for an intruder in a vibrated granular system under gravity. Phys. Rev. Lett., 95 (2005), No. 9, 098001.
- N. V. Brillantov, T. Pöschel. Breakdown of the Sonine expansion for the velocity distribution of granular gases. Europhys. Lett., 74 (2006), No. 3, 424–430.
- R. Brito, H. Enriquez, S. Godoy, R. Soto. Segregation induced by inelasticity in a vibrofluidized granular mixture. Phys. Rev. E, 77 (2008), No. 6, 061301.
- D. Brone, F. J. Muzzio. Size segregation in vibrated granular systems: A reversible process. Phys. Rev. E, 56 (1997), No. 1, 1059–1063.
- S. Chapman and T. G. Cowling. The mathematical Theory of Nonuniform Gases. Cambridge Univ. Press, London, 1970. Zbl0049.26102
- W. Cooke, S. Warr, J. M. Huntley, R. C. Ball. Particle size segregation in a two-dimensional bed undergoing vertical vibration. Phys. Rev. E, 53 (1996), No. 3, 2812–2822.
- S. R. de Groot and P. Mazur. Non-Equilibrium Thermodynamics. North-Holland, Amsterdam, 1969. Zbl0041.58401
- S. E. Esipov, T. Pöschel. The granular phase diagram. J. Stat. Phys., 86 (1997), No. 5-6, 1385–1395. Zbl0935.82036
- Z. Farkas, F. Szalai, D. E. Wolf, T. Vicsek. Segregation of binary mixtures by a ratchet mechanism. Phys. Rev. E, 65 (2002), No. 2, 022301.
- V. Garzó. Segregation in granular binary mixtures: Thermal diffusion. Europhys. Lett., 75 (2006), No. 4, 521–527.
- V. Garzó. Brazil-nut effect versus reverse Brazil-nut effect in a moderately dense granular fluid. Phys. Rev. E, 78 (2008), No. 2, 020301.
- V. Garzó, J. W. Dufty. Hydrodynamics for a granular binary mixture at low density. Phys Fluids, 14 (2002), No. 4, 1476–14902. Zbl1185.76138
- V. Garzó, F. V. Reyes, J. M. Montanero. Modified Sonine approximation for granular binary mixtures. J Fluid Mech., 623 (2009), 387–411. Zbl1157.76043
- I. Goldhirsch. Rapid granular flows. Annu Rev Fluid Mech., 35 (2003), 267–293. Zbl1125.76406
- I. Goldhirsch, D. Ronis. Theory of thermophoresis I: General considerations and mode coupling analysis. Phys Rev. A, 27 (1983), No. 3, 1616–1634.
- I. Goldhirsch, D. Ronis. Theory of thermophoresis II: Low-density behavior. Phys Rev. A, 27 (1983), No. 3, 1635–1656.
- D. C. Hong, P. V. Quinn, S. Luding. The reverse Brazil nut problem: Competition between percolation and condensation. Phys Rev Lett., 86 (2001), No. 15, 3423–3426.
- S. S. Hsiau, M. L. Hunt. Granular thermal diffusion in flows of binary-sized mixtures. Acta Mech., 114 (1996), No. 1-4, 121–137. Zbl0858.76079
- H. M. Jaeger, S. R. Nagel and R. P. Behringer. Granular solids, liquids, and gases. Rev Mod Phys., 68 (1996), No. 4, 1259–1273.
- J. T. Jenkins, F. Mancini. Kinetic theory for binary mixtures of smooth nearly elastic spheres. Phys Fluids A, 1 (1989), No. 12, 2050–2059. Zbl0696.76093
- J. T. Jenkins, D. K. Yoon. Segregation in binary mixture under gravity. Phys Rev Lett., 88 (2002), No. 19, 194304.
- J. M. Kincaid, E. G. D. Cohen, M. Lopez de Haro. The Enskog theory for multicomponent mixtures. iv. thermal diffusion. J Chem Phys., 86 (1987), No. 2, 963–975.
- J. B. Knight, E. E. Ehrlich, V. Y. Kuperman, J. K. Flint, H. M. Jaeger, S. R. Nagel. Experimental study of granular convection. Phys Rev. E, 54 (1996), No. 5, 5726–5738.
- J. B. Knight, H. M. Jaeger, S. R. Nagel. Vibration-induced size separation in granular media, No. 4, The convection connection. Phys Rev Lett., 70 (1993), No. 24, 3728–3731.
- L. Kondic, R. R. Hartley, S. G. K. Tennakoon, B. Painter, R. P. Behringer. Segregation by friction. Europhys Lett., 61 (2003), No. 6, 742–748.
- A. Kudrolli. Size separation in vibrated granular matter. Reports on Progress in Physics., 67 (2004), No. 3, 209–247.
- L. D. Landau, E. M. Lifshitz. Fluid Mechanics. Pergamon, London, 1959. Zbl0146.22405
- M. E. Mobius, X. Cheng, P. Eshuis, S. R. Karczmar, G. S. Nagel, H. M. Jaeger. Effect of air on granular size separation in a vibrated granular bed. Phys Rev. E, 72 (2005), No. 1, 011304.
- S. H. Noskowicz, O. Bar-Lev, D. Serero, I. Goldhirsch. Computer-aided kinetic theory and granular gases. Europhys Lett., 79 (2007), No. 6, 60001. Zbl1200.76206
- J. M. Ottino, D. V. Khakhar. Mixing and segregation of granular materials. Annu Rev Fluid Mech., 32 (2000), 55–91. Zbl0989.76087
- T. Pöschel, N. V. Brillantov, A. Formella. Impact of high-energy tails on granular gas properties. Phys Rev. E, 74 (2006), No. 4, 041302.
- T. Pöschel, H. J. Herrmann. Size segregation and convection. Europhys Lett., 29 (1995), No. 2, 123–128.
- D. C. Rapaport. Mechanism for granular segregation. Phys Rev. E, 64 (2001), No. 6, 061304.
- P. M. Reis, T. Mullin. Granular segregation as a critical phenomenon. Phys Rev Lett., 89 (2002), No. 24, 244301.
- A. Rosato, K. J. Strandburg, F. Prinz, R. H. Swendsen. Why the Brazil nuts are on top: size segregation of particulate matter by shaking. Phys Rev Lett., 58 (1987), No. 10, 1038–1040.
- M. Schröter, S. Ulrich, J. Keft, J. B. Swift, H. L. Swinney. Mechanism in the size segregation of a binary granular mixture. Phys Rev. E, 74 (2006), No. 1, 011307.
- N. Sela, I. Goldhirsch. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J Fluid Mech., 361 (1998), 41–74. Zbl0927.76008
- 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.
- D. Serero, I. Goldhirsch, S. H. Noskowicz, M. L. Tan. Hydrodynamics of granular gases and granular gas mixtures. J Fluid Mech., 554 (2006), 237–258. Zbl1091.76062
- D. Serero, S. H. Noskowicz, I. Goldhirsch. Exact versus mean field solutions for granular gas mixtures. Gran. Matt., 10 (2007), No. 1, 37–46. Zbl1200.76206
- T. Shinbrot, F. J. Muzzio. Reverse buoyancy in shaken granular beds. Phys Rev Lett., 81 (1998), No. 20, 4365–4368.
- T. Shinbrot, F. J. Muzzio. Nonequilibrium patterns in granular mixing and segregation. "Physics Today", 53 (2000), No. 3, 25–30.
- L. Trujillo, M. Alam, H. J. Herrmann. Segregation in a fluidized binary granular mixture: competition between buoyancy and geometric force. Europhys Lett., 64 (2003), No. 2, 190–196.
- S. Ulrich, M. Schröter, H. L. Swinney. Influence of friction on granular segregation. Phys Rev. E, 76 (2007), No. 4, 042301.
- H. Viswanathan, R. D. Wildman, J. M. Huntley, T. W. Martin. Comparison of kinetic theory predictions with experimental results for a vibrated three-dimensional granular bed. Phys Fluids, 18 (2006), No. 11, 113302. Zbl1146.76560
- R. D. Wildman, J. T. Jenkins, P. E. Krouskop, J. Talbot. 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.
- D. K. Yoon, J. T. Jenkins. The influence of different species’ granular temperatures on segregation in a binary mixture of dissipative grains. Phys Fluids, 18 (2006), No. 7, 073303.

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