Gravity can be introduced into the model following a method
similar to the local interaction method of section 4.4.3
but considering the momentum change to be caused by a
body force rather than an inter-particle force [66].
If a gravitational force
F is acting then at every time-step there is a
change of momentum 
.
To incorporate this into the model we let the equilibrium distribution
be given by
where
Here 
is defined, as before, by 
the sum of the
product of the
distribution function before a collision and the lattice vector.
Combining equations (4.42) and (4.43) and summing over i gives
Multiplying by 
before summing gives
as in equation (4.99).
Finally we define the fluid momentum 
to be the
average of the momentum before the collision, 
,
and the momentum after the collision, 
:
Now we can perform a Chapman-Enskog expansion of the left-hand sides of
equations (5.10) and (5.11)
using 
as the first-order approximation 
.
Now 
and 
so
we require
and
This gives the left hand sides of
equations (4.53), (4.55), (4.57) and (4.59)
as before. The right hand sides are 
and 0 respectively.
Summing these equations (where now 
) we get
and
for the first two equations.
Summing the left hand side of equation (4.57) we see that the second
and fourth terms
are zero as before, the third term is 
by equation (5.14) and the fifth term
is 
by equation (5.16). This gives
Summing the left hand side of equation (4.59) we similarly see
that the second and fourth terms cancel and the remaining terms give
equation (4.61), the second-order momentum equation,
with replaced by
.
Note that the third term of equation (4.59) is still given by
equation (4.60) with replaced by
. Consider the first-order expansion of
equations (4.42) and (4.43):
where 
and
.
Multiplying equation (5.18) by 
and
summing over i and noting that to first-order in the velocity
we get
as before. Combining the first- and second-order equations we get the
continuity equation (4.62) in terms of the fluid velocity
and the Navier-Stokes
equation
This is the same as equation (4.63) in terms of the fluid velocity
with the additional force term 
which
comes from equation (5.16).
