The Navier-Stokes equation
can be written
where 
, 
, 
and 
.
If 
then equation (5.3) is the same as
equation (5.2) without a body force but with a modified pressure.
Following this approach we can re-define the equilibrium distribution:
Using this expression for the equilibrium distribution and performing the Chapman-Enskog expansion, as in section 4.2.3, equations (4.46), (4.47) and (4.49) remain unchanged and equation (4.48) becomes
Equations (4.54) and (4.58), the first- and second-order
density equations, will therefore remain unchanged.
Equation (4.56), the first-order momentum equation,
has the same form except the pressure term on the right
hand side, 
, has an additional term,
, which can remain where it is to give a modified pressure or
be separated from the pressure term to give a separate force term,
.
Equation (4.61), the second-order momentum equation,
has the same form as before except now 
giving an altered bulk
viscosity.
This method has the disadvantage that it requires the density
to be constant, that is . Thus it could only be applied if 
is small enough that there
is only a negligible change in 
with depth.
