450 The lattice Boltzmann method for fluid dynamics
in cases where the substance is not so dilute, such as a fluid. The Boltzmann equation
reads:
∂n
∂t
+v·∇rn=(
dn
dt)
collisions. (14.1)
The second term describes the change due to particle flow, and the right hand side
is the change due to collisions.
If the particles were simply to flow according to their initial velocity, without
interaction, equilibrium would never be reached: the role of the collisions is to
establishlocal equilibrium, that is, a distribution which is in equilibrium in a small
cell with fixed volume, constant temperature, density and average velocityu.We
know this equilibrium distribution; it is given as
neq(r,v)=(
mπ
2 kBT) 3 / 2
n(r)exp{−m[v−u(r)]^2 /( 2 kBT)}, (14.2)which holds for cells small enough to justify a constant potential. Once the liquid
is in (local) equilibrium, the collisions will not push it away from equilibrium. It
can be shown that the collisions have the effect of increasing the entropy – hence
they generate heat.
Before we continue, we note that the mass mustalwaysbe conserved, whether
there are collisions or not. The mass density is found as
ρ(r,t)=∫
mn(r,v,t)d^3 v=mn(r). (14.3)Its evolution can be calculated by integrating equation(14.1), multiplied by the
single particle massm, over the velocity:
∂ρ(r,t)
∂t+
∫
mv·∇rn(r,v,t)d^3 v=∫ (
mdn
dt)
collisionsd^3 v. (14.4)The second term of this equation can be written as∇·j(r,t)wherejdenotes the
mass flux, or momentum density, of the fluid:
j(r,t)=∫
mvn(r,v,t)d^3 v=ρu, (14.5)whereuis the average local velocity. This equation can be checked using (14.2).
The collisions change the velocity distribution, but not the mass density of the
particles. Hence the right hand side of(14.4)vanishes and we obtain the familiar
continuity equation:
∂ρ(r,t)
∂t
+∇·j(r,t)=0. (14.6)
Another interesting equation describes the conservation of momentum. We would
like to know howj(r,t)changes with time. This is again evaluated straightforwardly