The Navier-Stokes equation can be derived from the Boltzmann equation using a Chapman-Enskog expansion. Only a brief outline of the method will be given here since the same procedure can be applied, in a much more straightforward manner, to the simplified lattice Boltzmann equation in which we are mainly interested. A full derivation of the Navier-Stokes equation using the Chapman-Enskog expansion will be given in section 4.2.3 for the lattice Boltzmann equation. Details of the Chapman-Enskog method for the classical Boltzmann equation can be found in [11].
The Chapman Enskog expansion parameter is the Knudsen number, 
,
defined as
where 
is the mean free path of the molecules and l is a
typical macroscopic length. The derived equations will only be valid
if the Knudsen number is small. By analysing the time and length scales
involved in the Boltzmann equation [13], can be introduced
into the Boltzmann equation:
Setting
we look for solutions of equation (2.37) such that
The zeroth order them 
is taken to be
the local Maxwell-Boltzmann distribution 
and 
, for 
, are chosen so they have no contribution
to the moments expressed in equation (2.39).
The first-order solution can be found by considering
which gives the
Euler equation of section 2.1.1 [11, 13]. The second-order
solution, found by considering 
, requires a knowledge of
the collision operator and can be shown [11] to give the
Navier-Stokes equation when the binary collision function is used.
