Knudsen number Chapman-Enskog expansion

Hydrodynamic models are derived from the mesoscale model (e.g. the Boltzmann equation) using a Chapman-Enskog expansion in powers of the Knudsen number (Bardos et al., 1991 Cercignani et al, 1994 Chapman Cowling, 1961 Ferziger Kaper, 1972 Jenkins Mancini, 1989). The basic idea is that the collision term will drive the velocity distribution n towards an equilibrium function eq (i-e. the solution to C( eq) = 0), and thus the deviation from equilibrium can be approximated by n -i- Knui. From the... 

In the limit of small Knudsen number, the Chapman-Enskog expansion (Chapman, 1916 Enksog, 1921) of the elastic Boltzmann equation yields a first-order term for CTp of the form... 

For show that ERB can use for describing the fluid s behavior, NS equations are derivate by process are named Chapman-Enskog s expansion or multi-scale analysis. It depends of Knudsen s number it was mentioned at the first part of this chapter it is the relation between the free mean trajectory and the characteristic length. 

As mentioned previously, the Chapman-Enskog solution is based on a small Knudsen number expansion of the Boltzmann equation. Under the conditions of a small Knudsen number, we substitute the following expansion... 

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