Boltzmann equation and hydrodynamics at the Burnett level

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June 2012, 5(2): 237-260. doi: 10.3934/krm.2012.5.237

Boltzmann equation and hydrodynamics at the Burnett level

Alexander Bobylev ^1, and Åsa Windfäll ^1,

1.	Department of Mathematics, Karlstad University, SE-651 88 Karlstad

Received October 2011 Revised November 2011 Published April 2012

Abstract
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The hydrodynamics at the Burnett level is discussed in detail. First we explain the shortest way to derive the classical Burnett equations from the Boltzmann equation. Then we sketch all the computations needed for details of these equations. It is well known that the classical Burnett equations are ill-posed. We therefore explain how to make a regularization of these equations and derive the well-posed generalized Burnett equations (GBEs). We discuss briefly an optimal choice of free parameters in GBEs and consider a specific version of these equations. It is remarkable that this version of GBEs is even simpler than the original Burnett equations, it contains only third derivatives of density. Finally we prove a linear stability for GBEs. We also present some numerical results on the sound propagation based on GBEs and compare them with the Navier-Stokes results and experimental data.

Keywords: Hydrodynamics, regularized Burnett equations, Stability, sound propagation..

Mathematics Subject Classification: 35Q35, 76P0.

Citation: Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic and Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237

References:

[1]	M. Bisi, M. P. Cassinari and M. Groppi, Qualitative analysis of the generalized Burnett equations and applications to half-space problems, Kinet. Relat. Models, 1 (2008), 295-312. doi: 10.3934/krm.2008.1.295.
[2]	A. V. Bobylev, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27 (1982), 29-31.
[3]	A. V. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys., 124 (2006), 371-399. doi: 10.1007/s10955-005-8087-6.
[4]	A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys., 132 (2008), 569-580.
[5]	A. V. Bobylev, M. Bisi, M. P. Cassinari and G. Spiga, Shock wave structure for generalized Burnett equations, Phys. of Fluids, 23 (2011).
[6]	D. Burnett, The distribution of velocities in a slightly non-uniform gas, Proc. London. Math. Soc., 39 (1935), 385-430. doi: 10.1112/plms/s2-39.1.385.
[7]	D. Burnett, The distribution of molecular velocities and the mean motion in a non-uniform gas, Proc. London. Math. Soc., 40 (1935), 382-435. doi: 10.1112/plms/s2-40.1.382.
[8]	C. Cercignani, "The Boltzmann Equation and its Applications,'' Springer-Verlag, 1988.
[9]	S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," Cambrige University Press, 1990.
[10]	J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases," North-Holland, 1972.
[11]	S. Jin and M. Slemrod, Regularization of the Burnett equations via relaxation, J. Stat. Phys., 103 (2001), 1009-1033. doi: 10.1023/A:1010365123288.
[12]	M. N. Kogan, "Rarefied Gas Dynamics," Plenum Press, 1969.
[13]	M. M. Postnikov, "Stable Polynomials," Nauka, Moscow, 1981 (in Russian).
[14]	M. Slemrod, A normalization method for the Chapman-Enskog expansion, Physica D, 109 (1997), 257-273. doi: 10.1016/S0167-2789(97)00068-7.
[15]	M. Slemrod, Constitutive relations for monoatomic gases based on a generalized rational approximation to the sum of the Chapman-Enskog expansion, Arch. Rat. Mech. Anal, 150 (1999), 1-22. doi: 10.1007/s002050050178.
[16]	M. Slemrod, In the Chapman-Enskog expansion the Burnett coefficients satisfy the universal relation $\omega_3+\omega_4+\theta_3=0$, Arch. Rat. Mech. Anal., 161 (2002), 339-344. doi: 10.1007/s002050100180.
[17]	M. Torrilon and H. Struchtrup, Regularized 13 moment equations: Shock structure calculations and comparison to Burnett models, J. Fluid Mech., 513 (2004), 171-198. doi: 10.1017/S0022112004009917.
[18]	C. Truesdell and R. G. Muncaster, "Fundamental of Maxwell's Kinetic Theory of a Simple Monoatomic Gas," Academic Press, New York, 1980.

show all references

References:

[1]	M. Bisi, M. P. Cassinari and M. Groppi, Qualitative analysis of the generalized Burnett equations and applications to half-space problems, Kinet. Relat. Models, 1 (2008), 295-312. doi: 10.3934/krm.2008.1.295.
[2]	A. V. Bobylev, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27 (1982), 29-31.
[3]	A. V. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys., 124 (2006), 371-399. doi: 10.1007/s10955-005-8087-6.
[4]	A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys., 132 (2008), 569-580.
[5]	A. V. Bobylev, M. Bisi, M. P. Cassinari and G. Spiga, Shock wave structure for generalized Burnett equations, Phys. of Fluids, 23 (2011).
[6]	D. Burnett, The distribution of velocities in a slightly non-uniform gas, Proc. London. Math. Soc., 39 (1935), 385-430. doi: 10.1112/plms/s2-39.1.385.
[7]	D. Burnett, The distribution of molecular velocities and the mean motion in a non-uniform gas, Proc. London. Math. Soc., 40 (1935), 382-435. doi: 10.1112/plms/s2-40.1.382.
[8]	C. Cercignani, "The Boltzmann Equation and its Applications,'' Springer-Verlag, 1988.
[9]	S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," Cambrige University Press, 1990.
[10]	J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases," North-Holland, 1972.
[11]	S. Jin and M. Slemrod, Regularization of the Burnett equations via relaxation, J. Stat. Phys., 103 (2001), 1009-1033. doi: 10.1023/A:1010365123288.
[12]	M. N. Kogan, "Rarefied Gas Dynamics," Plenum Press, 1969.
[13]	M. M. Postnikov, "Stable Polynomials," Nauka, Moscow, 1981 (in Russian).
[14]	M. Slemrod, A normalization method for the Chapman-Enskog expansion, Physica D, 109 (1997), 257-273. doi: 10.1016/S0167-2789(97)00068-7.
[15]	M. Slemrod, Constitutive relations for monoatomic gases based on a generalized rational approximation to the sum of the Chapman-Enskog expansion, Arch. Rat. Mech. Anal, 150 (1999), 1-22. doi: 10.1007/s002050050178.
[16]	M. Slemrod, In the Chapman-Enskog expansion the Burnett coefficients satisfy the universal relation $\omega_3+\omega_4+\theta_3=0$, Arch. Rat. Mech. Anal., 161 (2002), 339-344. doi: 10.1007/s002050100180.
[17]	M. Torrilon and H. Struchtrup, Regularized 13 moment equations: Shock structure calculations and comparison to Burnett models, J. Fluid Mech., 513 (2004), 171-198. doi: 10.1017/S0022112004009917.
[18]	C. Truesdell and R. G. Muncaster, "Fundamental of Maxwell's Kinetic Theory of a Simple Monoatomic Gas," Academic Press, New York, 1980.

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