Computational Physics

454 The lattice Boltzmann method for fluid dynamics

The second term in the square brackets of(14.16)can be written, using the quantity
wα=vα−uα(see alsoEq. (14.10)and(14.19b)), in the form
∫
(uα+wα)(uβ+wβ)(uγ+wγ)∂γneq(r,v)d^3 v

=∂γ(ρuαuβuγ+uαPδβγ+uβPδαγ+uγPδαβ). (14.21)

This term can now be worked out to yield

∑

γ

[uαuβuγ∂γρ+ρuβuγ(∂γuα)+ρuαuγ(∂γuβ)+ρuαuβ(∂γuγ)

+∂γ(Puγ)δαβ+∂γ(Puα)δβγ+∂γ(Puβ)δαγ]. (14.22)

Adding the two terms of Eq. (14.16), many terms occurring in (14.20) and (14.22)
cancel. The ones that remain are

P(∂βuα+∂αuβ)+δαβ







P ̇+

∑

γ

[uγ(∂γP)+P∂γuγ]







. (14.23)

The terms

P ̇+

∑

γ

uγ(∂γP) (14.24)

can be calculated using (14.19b) and the equilibrium distribution. When we write
this out, we obtain, again withwα=vα−uα:

∂

∂t

∫

mw^2 nd^3 w+

∑

γ

uγ∂γ

∫

mw^2 nd^3 w

=

∫

mw^2



∂n

∂t

+

∑

γ

uγ∂γn



d^3 w=−^1

τ

∫

mw^2 nnoneqd^3 w. (14.25)

This is the trace of the tensor

−

1

τ

∫

mwαwβnnoneqd^3 v. (14.26)

Now we use the fact that Trnoneqvanishes. This can only happen when the
trace occurring in the last equation cancels the trace of the remaining terms in the
expression fornoneq. This tensor must therefore be

noneq=−Pτ

(

∂αuβ+∂βuα−^23 δαβ∂γuγ

)

. (14.27)

Using this, we can formulate the momentum conservation equation, withν=
τkBT/m,as

∂u

∂t

+u·∇u=−

1

ρ

∇P+ν∇^2 u+

1

3

ν∇(∇·u). (14.28)