Linear transport equations valid for arbitrary collisionality: comparison with the Chapman-Enskog expansion

Recently we proposed a method to solve the perturbed Boltzmann equation modeled by the Bhatnagar-Gross-Krook operator [Phys. Rev. E 74, 041204 (2006)]. In this work we use this method to derive linear transport equations in the whole collisionality range. A comparison of the closure relations derived up to the third order in the Knudsen number (super-Burnett) yields the same results as the Chapman-Enskog expansion. The contribution of the projection operators to the transport is investigated. It is pointed out that their contributions are not negligible in the super-Burnett equations and very significant in the collisionless range. The test of stability of the super-Burnett equations is also performed. It is shown that the stability problem can be related to the positivity of the generalized transport coefficients. Using the Padé approximants, nonlocal transport coefficients are proposed which present the desirable stability properties.