Note on Divergence of the Chapman–Enskog Expansion for Solving Boltzmann Equation1135

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.