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June  2012, 5(2): 417-440. doi: 10.3934/krm.2012.5.417

Unique moment set from the order of magnitude method

Henning Struchtrup 1,

1. 

Department of Mechanical Engineering, University of Victoria, Victoria BC V8W 3P6, Canada

Received  November 2011 Revised  February 2012 Published  April 2012

The order of magnitude method [Struchtrup, Phys. Fluids 16, 3921-3934 (2004)] is used to construct a unique moment set for 1-D transport with scattering. Simply speaking, the method uses a series of leading order Chapman-Enskog expansions in the Knudsen number to construct the moments such that the number of moments at a given Chapman-Enskog order is minimal. For isotropic scattering, when one begins with monomials for the moments, the method constructs step by step moments of the Legendre polynomials. For anisotropic scattering, however, it constructs moments of new polynomials relevant for the particular scattering mechanism. All terms in the final moment equations are scaled by powers of the Knudsen number, which gives an easy handle to model reduction.
Keywords: order of magnitude method., moment method, Knudsen number approximation, Chapman-Enskog method, Kinetic theory of gases.
Mathematics Subject Classification: Primary: 76P05, 82B40; Secondary: 53C3.
Citation: Henning Struchtrup. Unique moment set from the order of magnitude method. Kinetic & Related Models, 2012, 5 (2) : 417-440. doi: 10.3934/krm.2012.5.417
References:
[1]

A. V. Bobylëv, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27 (1982), 29-31. Google Scholar

[2]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases,'' Third edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.  Google Scholar

[3]

H. Grad, Principles of the Kinetic Theory of Gases, in "1958 Handbuch der Physik" (ed. S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205-294.  Google Scholar

[4]

P. Kauf, M. Torrilhon and M. Junk, Scale-induced closure for approximations of kinetic equations, J. Stat. Phys., 141 (2010), 848-888. doi: 10.1007/s10955-010-0073-y.  Google Scholar

[5]

M. Frank and B. Seibold, Optimal prediction for radiative transfer: A new perspective on moment closure, Kinetic and Related Models, 4 (2011), 717-733. doi: 10.3934/krm.2011.4.717.  Google Scholar

[6]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002.  Google Scholar

[7]

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.  Google Scholar

[8]

H. Struchtrup, Stable transport equations for rarefied gases at high orders in the Knudsen number, Phys. Fluids, 16 (2004), 3921-3934. doi: 10.1063/1.1782751.  Google Scholar

[9]

H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials,, Multiscale Model. Simul., 3 (): 211.   Google Scholar

[10]

H. Struchtrup, Failures of the Burnett and super-Burnett equations in steady state processes, Cont. Mech. Thermodyn., 17 (2005), 43-50. doi: 10.1007/s00161-004-0186-0.  Google Scholar

[11]

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

[12]

H. Struchtrup, Linear kinetic heat transfer: Moment equations, boundary conditions, and Knudsen layers, Physica A, 387 (2008), 1750-1766. doi: 10.1016/j.physa.2007.11.044.  Google Scholar

[13]

H. Struchtrup and P. Taheri, Macroscopic transport models for rarefied gas flows: A brief review, IMA J. Appl. Math., 76 (2011), 672-697. doi: 10.1093/imamat/hxr004.  Google Scholar

[14]

M. Schäfer, M. Frank and C. D. Levermore, Diffusive correction to $P_N-$ approximations, Multiscale. Model. Simul., 9 (2011), 1-28. doi: 10.1137/090764542.  Google Scholar

[15]

Y. Zheng and H. Struchtrup, Burnett equations for the ellipsoidal statistical BGK Model, Cont. Mech. Thermodyn., 16 (2004), 97-108. doi: 10.1007/s00161-003-0143-3.  Google Scholar

show all references

References:
[1]

A. V. Bobylëv, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27 (1982), 29-31. Google Scholar

[2]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases,'' Third edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.  Google Scholar

[3]

H. Grad, Principles of the Kinetic Theory of Gases, in "1958 Handbuch der Physik" (ed. S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205-294.  Google Scholar

[4]

P. Kauf, M. Torrilhon and M. Junk, Scale-induced closure for approximations of kinetic equations, J. Stat. Phys., 141 (2010), 848-888. doi: 10.1007/s10955-010-0073-y.  Google Scholar

[5]

M. Frank and B. Seibold, Optimal prediction for radiative transfer: A new perspective on moment closure, Kinetic and Related Models, 4 (2011), 717-733. doi: 10.3934/krm.2011.4.717.  Google Scholar

[6]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002.  Google Scholar

[7]

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.  Google Scholar

[8]

H. Struchtrup, Stable transport equations for rarefied gases at high orders in the Knudsen number, Phys. Fluids, 16 (2004), 3921-3934. doi: 10.1063/1.1782751.  Google Scholar

[9]

H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials,, Multiscale Model. Simul., 3 (): 211.   Google Scholar

[10]

H. Struchtrup, Failures of the Burnett and super-Burnett equations in steady state processes, Cont. Mech. Thermodyn., 17 (2005), 43-50. doi: 10.1007/s00161-004-0186-0.  Google Scholar

[11]

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

[12]

H. Struchtrup, Linear kinetic heat transfer: Moment equations, boundary conditions, and Knudsen layers, Physica A, 387 (2008), 1750-1766. doi: 10.1016/j.physa.2007.11.044.  Google Scholar

[13]

H. Struchtrup and P. Taheri, Macroscopic transport models for rarefied gas flows: A brief review, IMA J. Appl. Math., 76 (2011), 672-697. doi: 10.1093/imamat/hxr004.  Google Scholar

[14]

M. Schäfer, M. Frank and C. D. Levermore, Diffusive correction to $P_N-$ approximations, Multiscale. Model. Simul., 9 (2011), 1-28. doi: 10.1137/090764542.  Google Scholar

[15]

Y. Zheng and H. Struchtrup, Burnett equations for the ellipsoidal statistical BGK Model, Cont. Mech. Thermodyn., 16 (2004), 97-108. doi: 10.1007/s00161-003-0143-3.  Google Scholar

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