Biomedical Engineering Reference
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(
,
,
,
)
number of particles lost during a time step is
W
v
v
2
v
3
v
4
and due to symmetry
(
,
,
,
)=
(
,
,
,
)
the number of particles added to
.Takinginto
account the fact that
W
is proportional to the number of particles with the relevant
velocity
f
μ
is
W
v
3
v
4
v
2
v
W
v
v
2
v
3
v
4
(
)
v
i
one ends up with
∂
∂
)
∇
v
f
d
3
v
2
d
3
v
3
1
m
f
d
3
v
4
W
t
+
v
∇
x
+
(
x
(
x
,
v
,
t
)=
(
v
,
v
2
,
v
3
,
v
4
)
×
[
f
(
x
,
v
3
,
t
)
f
(
x
,
v
4
,
t
)
−
f
(
x
,
v
2
,
t
)
f
(
x
,
v
,
t
)]
(15)
which is a nonlinear integrodifferential equation. Since the Boltzmann equation (
15
)
cannot be solved analytically, different approximations have been proposed to find
solutions like the one by Bhatnagar, [
11
]. Their idea was that the system relaxes
towards a local equilibrium distribution
f
eq
. Further, it can be assumed that in
case the system is not far from this equilibrium it will relax with a single relaxation
rate 1
(
x
,
v
)
/
τ
. Thus, the Boltzmann equation with BGK collision operator reads as
∂
∂
)
∇
v
f
f
eq
1
m
f
f
(
x
,
v
,
t
)
−
t
+
v
∇
x
+
(
x
(
x
,
v
,
t
)=
(16)
τ
Here, it should be noted that one can now “choose” the equilibrium function in
order to obtain the desired physical properties. However, one needs to fulfill some
requirements such as isotropy and conservation of mass and momentum. Usually,
the Maxwell distribution or some expansions of it are used. By performing a
Chapman Enskog procedure it is possible to show that such a model reproduces
the Navier-Stokes equation, if the distribution function is chosen right. It should be
noted that this is a two-way process. One can either construct the method in such a
way that it simulates a given macroscopic equation or one can show what equation
is solved by a given method.
A.2
Concept of Lattice Boltzmann
The core concept of the lattice-Boltzmann method is to discretize the Boltzmann
equation in time, velocity and real space [
58
]. This means instead of having
infinitely small phase space elements
μ
one takes into account the single particle
distribution function
f
only on a lattice site
x
j
, with velocities
c
i
that point
to the nearest and next-nearest neighbors. The lattice-Boltzmann equation [
79
]then
reads as
(
x
j
,
c
i
,
t
)
f
(
x
+
c
i
,
c
i
,
t
+
δ
t
)
−
f
(
x
,
c
i
,
t
)=
Ω
(17)
In two-dimensions a D2Q9 model is commonly used. Other lattice types in two
and three-dimensions can be used as well but will not be discussed here. The
article of Qian et al. provides an overview on different lattice schemes [
70
]. In his
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