Note on Divergence of the Chapman-Enskog Expansion for Solving Boltzmann Equation

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 $\tau \ge 1$. Even since this divergence problem has puzzled this subject for a century. By using a modified Möbius series inversion formula, this paper proposes 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 provide the best-so-far explanation on some unbelievable scenarios occurred in previous practice.

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