Chemistry Reference
In-Depth Information
Employing the summation of the particle velocity to the energy density (9.19) first,
it becomes obvious that it is connected with the sum of the components that are
associated with the directed and the random motion via
1
2
m
α
n
α
|
1
2
m
α
n
α
(
1
2
m
α
n
α
|
3
2
n
α
k
B
T
α
,
n
α
α
=
u
α
|
2
+
w
α
)
2
=
u
α
|
2
+
(9.26)
where the mean square of the thermal velocity is given by
m
α
with the temperature
T
α
and the Boltzmann constant
k
B
. The tensor (9.23) of the
momentum flux density receives the set of values
(
w
α
)
2
=
3
k
B
T
α
/
α
kl
=
m
α
n
α
u
k
u
l
+
m
α
n
α
w
k
w
l
=
m
α
n
α
u
k
u
l
+
p
kl
(9.27)
with the pressure tensor
. This tensor is generally divided into
the scalar pressure
p
α
and the so-called viscous stress tensor
p
α
ˇ
=
m
α
n
α
w
α
w
α
π
α
, which has the
ˇ
components
p
kl
=
p
α
δ
kl
+
π
kl
with π
kl
=
w
k
w
l
−
p
α
δ
kl
m
α
n
α
(9.28)
as is known from hydrodynamics. The scalar partial pressure
p
α
of the species α
is determined by the average energy of the thermal motion or by the temperature
according to
1
3
m
α
n
α
(
p
α
=
w
α
)
2
=
n
α
k
B
T
α
.
(9.29)
The tensor
π
α
represents the dissipative part of the pressure that arises from
deviations of the distribution function
f
α
from spherical symmetry with respect
to the thermal velocity, that is, from an isotropic distribution. Then, the combination
of equations (9.20), (9.21), (9.27), and (9.28) yields the vector equation
m
α
n
α
∂
ˇ
u
α
u
α
∂
u
α
t
+
(
·∇
x
)
u
α
G
coll
n
α
n
α
+
G
coll
j
α
n
α
F
a
−
L
coll
j
α
π
α
L
coll
=
−∇
x
p
−∇
x
ˇ
−
m
α
−
(9.30)
for the directed velocity describing the motion of the particle of species α.
When accordingly expressing the energy flux density (9.24) in dependence on
the directed and thermal velocity, the relation
1
2
m
α
n
α
|
π
α
1
2
m
α
n
α
(
Q
α
=
u
α
2
w
α
2
p
α
u
α
q
α
+
|
+
)
+
+ˇ
(9.31)
is obtained, where the first term
is the vector of the heat flux
density characterizing the energy transfer connected exclusively with the thermal
q
α
=
m
α
n
α
(
w
α
)
2
w
α
