March  2012, 5(1): 185-201. doi: 10.3934/krm.2012.5.185

H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory

1. 

Center for Computational Engineering Science, RWTH Aachen University, Schinkelstr.2, 52062 Aachen, Germany

Received  May 2011 Revised  July 2011 Published  January 2012

The regularized 13-moment equations (R13) are a successful macroscopic model to describe non-equilibrium gas flows in rarefied or micro situations. Even though the equations have been derived for the nonlinear case and many examples demonstrate the usefulness of the equations, sofar, the important property of an accompanying entropy law could only be shown for the linearized equations [Struchtrup&Torrilhon, Phys. Rev. Lett. 99, (2007), 014502]. Based on an approach suggested by Öttinger [Phys. Rev. Lett. 104, (2010), 120601], this paper presents a nonlinear entropy law for the R13 system. In the derivation the variables and equations of the R13 system are nonlinearily extended such that an entropy law with non-negative production can be formulated. It is then demonstrated that the original R13 system is included in the new equations.
Citation: Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic and Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185
References:
[1]

K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686.

[2]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.

[3]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics," North Holland, Amsterdam, 1962.

[4]

X.-J. Gu and D. Emerson, A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions, J. Comput. Phys., 225 (2007), 263-283. doi: 10.1016/j.jcp.2006.11.032.

[5]

G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases," Springer, Berlin, 2010. doi: 10.1007/978-3-642-11696-4.

[6]

C. D. Levermore and W. J. Morokoff, The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1999), 72-96.

[7]

I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," Second edition, With supplementary chapters by H. Struchtrup and Wolf Weiss, Springer Tracts in Natural Philosophy, 37, Springer-Verlag, New York, 1998.

[8]

H. C. Öttinger, "Beyond Equilibrium Thermodynamics," Wiley, Hoboken, 2005.

[9]

H. C. Öttinger, Reply to the comment on 'Thermodynamically admissible 13 moment equations from the Boltzmann equation', Phys. Rev. Lett., 105 (2010), 128902.

[10]

H. C. Öttinger, Thermodynamically admissible 13 moment equations from the Boltzmann equation, Phys. Rev. Lett., 104 (2010), 120601. doi: 10.1103/PhysRevLett.104.120601.

[11]

H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials,, Multiscale Model. Simul., 3 (): 221.  doi: 10.1137/040603115.

[12]

H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory," Interaction of Mechanics and Mathematics, Springer, Berlin, 2005.

[13]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13-moment-equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680. doi: 10.1063/1.1597472.

[14]

H. Struchtrup and M. Torrilhon, H-theorem, regularization, and boundary conditions for linearized 13 moment equations, Phys. Rev. Letters, 99 (2007), 014502. doi: 10.1103/PhysRevLett.99.014502.

[15]

H. Struchtrup and M. Torrilhon, Comment on 'Thermodynamically admissible 13 moment equations from the Boltzmann equation', Phys. Rev. Letters, 105 (2010) 128901. doi: 10.1103/PhysRevLett.105.128901.

[16]

P. Taheri, A. S. Rana, M. Torrilhon and H. Struchtrup, Macroscopic presentation of steady and unsteady rarefaction effects in the fundamental boundary value problems of gas dynamics, Continuum Mech. Thermodyn., 21 (2009), 423-443. doi: 10.1007/s00161-009-0115-3.

[17]

P. Taheri, M. Torrilhon and H. Struchtrup, Couette and poiseuille microflows: Analytical solutions for regularized 13-moment equations, Phys. Fluids, 21 (2009), 017102. doi: 10.1063/1.3064123.

[18]

M. Torrilhon, Two-dimensional bulk microflow simulations based on regularized Grad's 13-moment-equations, Multiscale Model. Simul., 5 (2006), 695-728. doi: 10.1137/050635444.

[19]

M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Comm. Comput. Phys., 7 (2010), 639-673.

[20]

M. Torrilhon, Slow rarefied flow past a sphere: Analytical solutions based on moment equations, Phys. Fluids, 22 (2010), 072001. doi: 10.1063/1.3453707.

[21]

M. Torrilhon 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.

[22]

M. Torrilhon and H. Struchtrup, Boundary conditions for regularized 13-moment-equations for micro-channel-flows, J. Comput. Phys., 227 (2008), 1982-2011. doi: 10.1016/j.jcp.2007.10.006.

show all references

References:
[1]

K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686.

[2]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.

[3]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics," North Holland, Amsterdam, 1962.

[4]

X.-J. Gu and D. Emerson, A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions, J. Comput. Phys., 225 (2007), 263-283. doi: 10.1016/j.jcp.2006.11.032.

[5]

G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases," Springer, Berlin, 2010. doi: 10.1007/978-3-642-11696-4.

[6]

C. D. Levermore and W. J. Morokoff, The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1999), 72-96.

[7]

I. Müller and T. Ruggeri, "Rational Extended Thermodynamics," Second edition, With supplementary chapters by H. Struchtrup and Wolf Weiss, Springer Tracts in Natural Philosophy, 37, Springer-Verlag, New York, 1998.

[8]

H. C. Öttinger, "Beyond Equilibrium Thermodynamics," Wiley, Hoboken, 2005.

[9]

H. C. Öttinger, Reply to the comment on 'Thermodynamically admissible 13 moment equations from the Boltzmann equation', Phys. Rev. Lett., 105 (2010), 128902.

[10]

H. C. Öttinger, Thermodynamically admissible 13 moment equations from the Boltzmann equation, Phys. Rev. Lett., 104 (2010), 120601. doi: 10.1103/PhysRevLett.104.120601.

[11]

H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials,, Multiscale Model. Simul., 3 (): 221.  doi: 10.1137/040603115.

[12]

H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory," Interaction of Mechanics and Mathematics, Springer, Berlin, 2005.

[13]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13-moment-equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680. doi: 10.1063/1.1597472.

[14]

H. Struchtrup and M. Torrilhon, H-theorem, regularization, and boundary conditions for linearized 13 moment equations, Phys. Rev. Letters, 99 (2007), 014502. doi: 10.1103/PhysRevLett.99.014502.

[15]

H. Struchtrup and M. Torrilhon, Comment on 'Thermodynamically admissible 13 moment equations from the Boltzmann equation', Phys. Rev. Letters, 105 (2010) 128901. doi: 10.1103/PhysRevLett.105.128901.

[16]

P. Taheri, A. S. Rana, M. Torrilhon and H. Struchtrup, Macroscopic presentation of steady and unsteady rarefaction effects in the fundamental boundary value problems of gas dynamics, Continuum Mech. Thermodyn., 21 (2009), 423-443. doi: 10.1007/s00161-009-0115-3.

[17]

P. Taheri, M. Torrilhon and H. Struchtrup, Couette and poiseuille microflows: Analytical solutions for regularized 13-moment equations, Phys. Fluids, 21 (2009), 017102. doi: 10.1063/1.3064123.

[18]

M. Torrilhon, Two-dimensional bulk microflow simulations based on regularized Grad's 13-moment-equations, Multiscale Model. Simul., 5 (2006), 695-728. doi: 10.1137/050635444.

[19]

M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Comm. Comput. Phys., 7 (2010), 639-673.

[20]

M. Torrilhon, Slow rarefied flow past a sphere: Analytical solutions based on moment equations, Phys. Fluids, 22 (2010), 072001. doi: 10.1063/1.3453707.

[21]

M. Torrilhon 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.

[22]

M. Torrilhon and H. Struchtrup, Boundary conditions for regularized 13-moment-equations for micro-channel-flows, J. Comput. Phys., 227 (2008), 1982-2011. doi: 10.1016/j.jcp.2007.10.006.

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