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Note on Divergence of the Chapman–Enskog Expansion for Solving Boltzmann Equation
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| Nan-Xian Chen1**, Bo-Hua Sun2 |
1State Key Laboratory of Low-dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084
2Department of Mechanical Engineering, Cape Peninsula University of Technology, Cape Town, South Africa |
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| Cite this article: |
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Nan-Xian Chen, Bo-Hua Sun 2017 Chin. Phys. Lett. 34 020502 |
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Abstract Within about a year (1916–1917) Chapman and Enskog independently proposed an important expansion for solving the Boltzmann equation. However, the expansion is divergent or indeterminant in the case of relaxation time $\tau \geq 1$. Since then, this divergence problem has puzzled researchers for a century. Using a modified Möbius series inversion formula, we propose a modified Chapman–Enskog expansion with a variable upper limit of the summation. The new expansion can give not only a convergent summation but also the best-so-far explanation on some unbelievable scenarios occurring in previous practice.
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Received: 16 January 2017
Published: 25 January 2017
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| PACS: |
05.20.Dd
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(Kinetic theory)
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05.70.Ln
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(Nonequilibrium and irreversible thermodynamics)
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02.30.Mv
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(Approximations and expansions)
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| Fund:
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| [1] | Pitaevskii L P and Lifshitz E M 1981 Physical Kinetics (Oxford: Butterworth-Heinemann) | | [2] | Cercignani C 1988 The Boltzmann Equation and Its Applications (New York: Springer-Verlag) | | [3] | Chapman S and Cowling T G 1990 The Mathematical Theory of Non-uniform Gases (Cambridge: Cambridge University Press) | | [4] | Bhatnagar P L, Gross E P and Krook M 1954 Phys. Rev. 94 511 | | [5] | McLannan A 1965 Phys. Fluids 8 1580 | | [6] | Santos A, Brey J J and Dufty J W 1986 Phys. Rev. Lett. 56 1571 | | [7] | Hardy G H and Wright E M 1979 An Introduction to the Theory of Numbers (Oxford: Clarendon) | | [8] | Chen N X 1990 Phys. Rev. Lett. 64 1193 | | [9] | Maddox J 1990 Nature 344 29 | | [10] | Chen N X 2010 Möbius Inversion in Physics (Singapore: World Scientific) |
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