Geoscience Reference
In-Depth Information
Let us assume that the number concentration of the particles of each sort
is conserved in collisions. Then
∂f
α
∂t
d
V
=0
.
(1.16)
coll
Substitution
Q
(
α
)
(
V
) = 1 into (1.15) yields the equation of conservation of
the particle numbers
∂N
α
∂t
∇
·
(
N
α
v
(
α
)
)=0
,
+
(1.17)
and
Q
(
α
)
(
V
)=
m
α
V
leads to the momentum equation
m
α
N
α
∂v
(
α
)
j
∂t
)
v
(
α
)
j
=
N
α
F
(
α
)
j
−
∂
∂x
k
P
(
α
)
jk
+
R
(
α
)
j
.
+(
v
(
α
)
∇
(1.18)
Here
F
(
α
)
=
Z
α
e
E
+
1
B
]
+
m
α
g
,
c
[
v
×
where
g
is the acceleration in non-electromagnetic fields (e.g., accelera-
tion of gravity), for neutrals
Z
α
=0.
P
(
α
)
jk
is the pressure tensor with
elements
P
(
α
)
jk
=
P
(
α
)
kj
=
m
α
N
α
(
V
j
−
v
(
α
)
j
)(
V
k
−
v
(
α
)
k
)
.
(1.19)
The last term in the right-hand side of (1.18) is the frictional drag
R
(
α
)
=
m
α
V
v
(
α
)
∂f
α
∂t
−
d
V
.
(1.20)
coll
Equation (1.18) is the equation of the momentum conservation and it is
an analogue of the corresponding hydrodynamic equation. An equation simi-
lar to the hydrodynamic energy conservation law can be obtained by taking
Q
(
α
)
(
V
)=
m
α
V
2
/
2 in (1.15). These equations become completed just with
additional assumptions about the pressure tensor
P
(
α
)
jk
and the friction force
R
(
α
)
. Strictly speaking, they should be obtained from the initial kinetic equa-
tions [2].
In the simplest approximation, pressure tensors are supposed to be isotropic
P
(
α
)
jk
=
P
α
δ
jk
,
where
P
α
is pressure (scalar),
δ
jk
is the unit tensor (
δ
jk
is the
Kronecker symbol ). A more common approximation is often used in plasma
applications with diagonal tensors
P
(
α
)
jk
but non-equal components
P
(
α
)
xx
,
P
(
α
)
yy
,
P
(
α
)
zz
.
Finally, it is possible to take into account viscous stresses for
the pressure tensor. Then
P
(
α
)
jk
=
P
α
δ
jk
−
τ
jk
,
where
τ
jk
is a tensor of vis-
cous stresses. A 'minus' sign reflects the fact that the viscous stresses cause
momentum losses caused by an internal friction.
Search WWH ::
Custom Search