Chapman-Enskog Solution to the Boltzmann Transport Equation

5 Chapman-Enskog Solution to the Boltzmann Transport Equation 

In dimensionless form, the Boltzmann transport equation, Eq. (3.37), including the possibility of an external force, can be written as 

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 

The solution to Eq. (6.83) follows from Boltzmann s i -theorem (Chap. 3), written in dimensionless form as 

Equations (6.85) and (6.86) refiect the local equilibrium nature of the solution to Eq. (6.83). It is, therefore, seen that the so-called Chapman-Enskog method of the solution is based on an expansion about local equilibrium conditions. Pitfalls of this approach have been previously noted. Writing, without loss of generality. 

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