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 & 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.  doi: 10.1073/pnas.68.8.1686.  Google Scholar

[2]

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

[3]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics,", North Holland, (1962).   Google Scholar

[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.  doi: 10.1016/j.jcp.2006.11.032.  Google Scholar

[5]

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

[6]

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

[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, (1998).   Google Scholar

[8]

H. C. Öttinger, "Beyond Equilibrium Thermodynamics,", Wiley, (2005).   Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory,", Interaction of Mechanics and Mathematics, (2005).   Google Scholar

[13]

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

[14]

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

[15]

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

[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.  doi: 10.1007/s00161-009-0115-3.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

M. Torrilhon and H. Struchtrup, Regularized 13-moment-equations: Shock structure calculations and comparison to Burnett models,, J. Fluid Mech., 513 (2004), 171.  doi: 10.1017/S0022112004009917.  Google Scholar

[22]

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

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.  doi: 10.1073/pnas.68.8.1686.  Google Scholar

[2]

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

[3]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics,", North Holland, (1962).   Google Scholar

[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.  doi: 10.1016/j.jcp.2006.11.032.  Google Scholar

[5]

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

[6]

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

[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, (1998).   Google Scholar

[8]

H. C. Öttinger, "Beyond Equilibrium Thermodynamics,", Wiley, (2005).   Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory,", Interaction of Mechanics and Mathematics, (2005).   Google Scholar

[13]

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

[14]

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

[15]

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

[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.  doi: 10.1007/s00161-009-0115-3.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

M. Torrilhon and H. Struchtrup, Regularized 13-moment-equations: Shock structure calculations and comparison to Burnett models,, J. Fluid Mech., 513 (2004), 171.  doi: 10.1017/S0022112004009917.  Google Scholar

[22]

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

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