Geoscience Reference
In-Depth Information
The mean velocity
v
(
α
)
of the
α
-th particles is
V
f
α
d
V
.
1
N
α
v
(
α
)
≡
V
(
α
)
=
(1.11)
The mean velocity
v
for the gas as a whole of the total density
ρ
m
=
α
N
α
m
α
is determined as
1
ρ
m
v
=
N
α
m
α
v
(
α
)
.
(1.12)
α
The total density of an electric current
j
is expressed in terms of mean
velocities as
eN
e
v
e
+
α
j
=
−
eZ
α
N
α
v
(
α
)
,
(1.13)
where
N
e
and
N
α
are electron and ion concentrations, respectively. The sum-
mation is taken over all ion species.
Moments of the Boltzmann's Equation
A velocity averaging of an arbitrary function
Q
(
α
)
(
V
) is given by
Q
(
α
)
(
V
)
f
α
(
r
,
V
,t
)d
V
,
1
N
α
(
r
,t
)
Q
(
α
)
(
r
,t
)=
(1.14)
where the concentration
N
α
(
r
,t
) is determined in (1.1).
The equations of conservation of momentum for mean quantities may be
called as multi-fluid hydrodynamics, which are close in form to the hydrody-
namic equations. They can be obtained by averaging the kinetic equations.
By multiplying the kinetic equation (1.7) by
Q
(
α
)
(
V
)d
V
and integrating over
velocities
V
, we obtain
∂x
k
N
α
∂
∂t
(
N
α
∂
N
α
m
α
∂
∂V
k
Q
(
α
)
Q
(
α
)
)+
Q
(
α
)
V
k
−
F
(
α
)
k
=
∂f
α
∂t
Q
(
α
)
(
V
)d
V
,
(1.15)
coll
where we have adopted the convention on tensor summation over index
k
=
1
,
2
,
3 (hereafter summation is adopted over all the double tensor index). The
third term is obtained through integration by parts.
Search WWH ::
Custom Search