To perform the Chapman-Enskog expansion we must first Taylor expand equation (4.42):
Expanding the population functions and the time and space derivatives in terms of the Knudsen number,
and using equation (4.50) we can perform an Chapman-Enskog expansion. Substituting equation (4.51) into equations (4.42), (4.43) and (4.50) gives
The notation
has
been used.
The zeroth-order approximation, 
,
is taken to be the equilibrium distribution:
. The conservation of mass and momentum
require that
.
To first-order in 
equation (4.52) is
Summing equation (4.53), using equations (4.46) and (4.47), gives
Multiply equation (4.53) by 
we get
Summing this, using equations (4.47) and (4.48), gives
To second-order in we get
Summing equation (4.57) over i, we see that terms two and three are zero due to the conservation of mass and momentum and terms four and five are zero by equations (4.54) and (4.56). This leaves
Multiplying equation (4.57) by 
and summing over i
gives
The second term is zero by the definition
of 
while the fourth term is zero
by equations (4.56). The fifth term is given by equations (4.48)
and (4.49). The third term can be found by considering
equations (4.53) multiplied by 
and summed over i giving, to order O(u),
So we get, using equation (4.54) to convert time derivatives into spatial derivatives,
where
and 
are the kinematic shear and bulk viscosities.
Summing the first- and second-order density and momentum equations and
recombining the derivatives gives the continuity equation,
and the Navier-Stokes equation
