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
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.
2017, Vol. 34 
