Discrete models for complex fluids

If we add Eq. (2.29) to the first order equation (2.15), we get the energy
equation up to second order with respect to ε [8, 9, 10]
∂t
[ D
2
nkBT +
1
2
ρu2
]
+ ∂β
[( D
2
nkBT +
1
2
ρu2
)
uβ
]
−
− ∂γ
[( D + 2
2
τn
kBT
m
)
∂γ(kBT )
]
= ρaγuγ (2.35)
2.3 Phase space discretization
The starting point of the Lattice Boltzmann models is the Boltzmann transport
equation, where the collision term has been linearized using the BGK approxima-
tion [1, 3, 4, 5, 11]
(
∂
∂t
+ v · ∇ + Fm · ∇v
)
f(x,v, t) = − 1
τ
[ f(x,v, t) − f eq(x,v, t) ] (2.36)
As mentioned in the Introduction, the equilibrium distribution function f eq(x,v, t)
for an ideal gas is the well-known Maxwell - Boltzmann distribution (1.18). If the
system is not too far from the equilibrium state, we may assume
∇vf(x,v, t) ' ∇vf eq(x,v, t) = − m
kBT
[v − u(x, t) ] f eq(x,v, t) (2.37)
After introducing the expression (2.37) in (2.36), we get the following form of the
Boltzmann equation:
∂tf(x,v, t) + v · ∇f(x,v, t) = 1
kBT
F · [v − u(x, t) ] f eq(x,v, t)
− 1
τ
[ f(x,v, t) − f eq(x,v, t) ] (2.38)
The independent variables x and v generate the phase space and belong to
the continuum spectrum. The basic ideea of Lattice Boltzmann models is the
discretization of the phase space [5, 16, 17, 18]. According to the discretization
procedure, the distribution functions are now defined only in the nodes of a lattice
L, which belongs to the D-dimensional space (D ∈ { 1, 2, 3 }). The velocity space
is reduced to a finite set of vectors { ei }, i = 0, 1, . . . N . After discretization of
the phase space, the distribution function f(x,v, t) is replaced by a finite set of N
distribution functions fi(x, t) ≡ f(x, ei, t). Thus, the Boltzmann equation (2.38)
is replaced by a system of N evolution equations:
∂tfi(x, t) + ei · ∇fi(x, t) =
1
kBT
F · [ ei − u(x, t) ] f eqi (x, t)
10