We now have equation (3.15) giving an equilibrium solution for the mean occupation numbers of the links. Ensemble averaging the conservation equations (3.4) and (3.5) gives the conservation equations for the mean occupation numbers
In order to derive the macrodynamical equations it is assumed [8]
that the
actual mean population 
is close to
the equilibrium population, described in equation (3.15), which we
will now call 
. The population is expanded in terms of a small
parameter 
:
where we assume that 
and 
do not
contribute to the mean
density or momentum.
The derivatives are also expanded in terms of the same parameter [8]
and a Chapman-Enskog expansion performed up to second-order in
and 
.
To first-order in this gives
and
where 
is the first-order approximation to the momentum flux
tensor, and is given by
where
Expanding to the next order in and combining the second-order
solutions with equations (3.22) and (3.23)
we get the continuity equation
and
where
and the kinematic viscosity 
is a function of density.
The viscosity has been calculated by
Frisch [8] using the Boltzmann approximation
and also by
Hénon [23], by considering the particle motion on the grid.
The viscosity
is found to be model dependent as well as density dependent, the values
for the FHP-I, FHP-II and FHP-III models are
and
respectively. Here d is the mean density per link given by
for FHP-I and 
for FHP-II and FHP-III were rest particles are
allowed.
