2011, 4(1): 317-331. doi: 10.3934/krm.2011.4.317

Fourteen moment theory for granular gases

1. 

Departamento de Física, Universidade Federal do Paraná, Curitiba, Brazil

2. 

Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19044, 81531-990 Curitiba, Brazil

Received  August 2010 Revised  December 2010 Published  January 2011

A fourteen moment theory for a granular gas is developed within the framework of the Boltzmann equation where the full contracted moment of fourth order is added to the thirteen moments of mass density, velocity, pressure tensor and heat flux vector. The spatially homogeneous solutions of the fourteen moment theory implied into a time decay of the temperature field which follows closely Haff's law, besides the more accentuated time decays of the pressure deviator, heat flux vector and fourth moment. The requirement that the fourth moment remains constant in time inferred into its identification with the coefficient $a_2$ in the Chapman-Enskog solution of the Boltzmann equation. The laws of Navier-Stokes and Fourier are obtained by restricting to a five field theory and using a method akin to the Maxwellian procedure. The dependence of the heat flux vector on the gradient of the particle number density was obtained thanks to the inclusion of the forth moment. The analysis of the dynamic behavior of small local disturbances from the spatially homogeneous solutions caused by spontaneous internal fluctuations is performed by considering a thirteen field theory and it is shown that for the longitudinal disturbances there exist one hydrodynamic and four kinetic modes, while for the transverse disturbances one hydrodynamic and two kinetic modes are present.
Citation: Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic & Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317
References:
[1]

C. K. K. Lun, S. B. Savage, D. J. Jeffrey and N. Chepurniy, Kinetic theories for granular flow: Inelastic particles in Couette-flow and slightly inelastic particles in a general flowfield,, J. Fluid Mech., 140 (1984), 223. doi: 10.1017/S0022112084000586.

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J. T. Jenkins and M. W. Richman, Grad's 13-Moment system for a dense gas of inelastic spheres,, Arch. Ration. Mech. Anal., 87 (1985), 355. doi: 10.1007/BF00250919.

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J. T. Jenkins and M. W. Richman, Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks,, Phys. Fluids, 28 (1985), 3485. doi: 10.1063/1.865302.

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A. Goldshtein and M. Shapiro, Mechanics of collisional motion of granular-materials. 1. General hydrodynamic equations,, J. Fluid Mech., 282 (1995), 75. doi: 10.1017/S0022112095000048.

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J. J. Brey, J. W. Dufty, C. S. Kim and A. Santos, Hydrodynamics for granular flow at low density,, Phys. Rev. E, 58 (1998), 4638. doi: 10.1103/PhysRevE.58.4638.

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T. Pöschel and S. Luding (Editors), "Granular Gases,'', Springer-Verlag, (2001).

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T. Pöschel and N. V. Brilliantov (Editors), "Granular Gas Dynamics,'', Springer-Verlag, (2003).

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N. V. Brilliantov and T. Pöschel, "Kinetic Theory of Granular Gases,'', Oxford University Press, (2004). doi: 10.1093/acprof:oso/9780198530381.001.0001.

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M. Bisi, G. Spiga and G. Toscani, Grad's equations and hydrodynamics for weakly inelastic granular flows,, Phys. Fluids, 16 (2004), 4235. doi: 10.1063/1.1805371.

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G. M. Kremer, Extended thermodynamics of ideal gases with 14 fields,, Ann. Inst. Henri Poincaré, 45 (1986), 419.

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D. Risso and P. Cordero, Dynamics of rarefied granular gases,, Phys. Rev. E, 65 (2002), 021304. doi: 10.1103/PhysRevE.65.021304.

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E. Ikenberry and C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, I.,, J. Rational Mech. Anal., 5 (1956), 1.

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J. J. Brey, F. Moreno and J. W. Dufty, Model kinetic equation for low-density granular flow,, Phys. Rev. E, 54 (1996), 445. doi: 10.1103/PhysRevE.54.445.

show all references

References:
[1]

C. K. K. Lun, S. B. Savage, D. J. Jeffrey and N. Chepurniy, Kinetic theories for granular flow: Inelastic particles in Couette-flow and slightly inelastic particles in a general flowfield,, J. Fluid Mech., 140 (1984), 223. doi: 10.1017/S0022112084000586.

[2]

J. T. Jenkins and M. W. Richman, Grad's 13-Moment system for a dense gas of inelastic spheres,, Arch. Ration. Mech. Anal., 87 (1985), 355. doi: 10.1007/BF00250919.

[3]

J. T. Jenkins and M. W. Richman, Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks,, Phys. Fluids, 28 (1985), 3485. doi: 10.1063/1.865302.

[4]

A. Goldshtein and M. Shapiro, Mechanics of collisional motion of granular-materials. 1. General hydrodynamic equations,, J. Fluid Mech., 282 (1995), 75. doi: 10.1017/S0022112095000048.

[5]

J. J. Brey, J. W. Dufty, C. S. Kim and A. Santos, Hydrodynamics for granular flow at low density,, Phys. Rev. E, 58 (1998), 4638. doi: 10.1103/PhysRevE.58.4638.

[6]

T. Pöschel and S. Luding (Editors), "Granular Gases,'', Springer-Verlag, (2001).

[7]

T. Pöschel and N. V. Brilliantov (Editors), "Granular Gas Dynamics,'', Springer-Verlag, (2003).

[8]

N. V. Brilliantov and T. Pöschel, "Kinetic Theory of Granular Gases,'', Oxford University Press, (2004). doi: 10.1093/acprof:oso/9780198530381.001.0001.

[9]

M. Bisi, G. Spiga and G. Toscani, Grad's equations and hydrodynamics for weakly inelastic granular flows,, Phys. Fluids, 16 (2004), 4235. doi: 10.1063/1.1805371.

[10]

G. M. Kremer, Extended thermodynamics of ideal gases with 14 fields,, Ann. Inst. Henri Poincaré, 45 (1986), 419.

[11]

D. Risso and P. Cordero, Dynamics of rarefied granular gases,, Phys. Rev. E, 65 (2002), 021304. doi: 10.1103/PhysRevE.65.021304.

[12]

E. Ikenberry and C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, I.,, J. Rational Mech. Anal., 5 (1956), 1.

[13]

J. J. Brey, F. Moreno and J. W. Dufty, Model kinetic equation for low-density granular flow,, Phys. Rev. E, 54 (1996), 445. doi: 10.1103/PhysRevE.54.445.

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