Mass flux vector:(1.8)

Momentum flux tensor:(1.9)

Heat flux vector:

(1.10)

Notice that the molecular fluxes have the same general structure as the convective fluxes. Each of these molecular-flux expressions contains the distribution function, which must be obtained by solving the Boltzmann equation.

1d. Solution of the Boltzmann equation

When a gas mixture is at rest, the distribution function is given by the Maxwell-Boltzmann distribution function, obtainable from equilibrium statistical mechanics. In the presence of gradients, however, we have to multiply the equilibrium solution by a correction factor to account for the influence of these gradients:

(1.11)

in which is the first "correction term," which is taken to be of the form:

(1.12)