Environmental Engineering Reference
In-Depth Information
4
Transport Phenomena in Ionized Gases
4.1
Hydrodynamics of Ionized Gases
4.1.1
Macroscopic Gas Equations
The distribution function for atomic particles of a gaseous system gives detailed
information about this system. It yields a variety of macroscopic parameters of the
system by averaging the distribution function over particle velocities and internal
quantum numbers. This operation leads to a transition from a kinetic description
to a hydrodynamic description of the gaseous system and is accompanied by a
decrease of the amount of information about the system. It is clear from a gener-
al consideration that the kinetic description is useful for nonequilibrium gaseous
systems, whereas the hydrodynamic description is better for equilibrium systems
where average gaseous state parameters characterize the system in total. In this way
the macroscopic equations which follow from the kinetic equation are practically
the equations of mass and heat transfer [1-3].
Below we obtain macroscopic equations for an ionized gas from the kinetic equa-
tion, starting from integration of the kinetic equation (3.4) over atom velocities. The
right-hand side of the equation is the total variation of the density of particles per
unit time due to collisions. Assuming that particles are not formed or destroyed in-
side the system volume, the right-hand side of the resulting equation is zero, and
the kinetic equation becomes
Z
@
Z
Z
@
f
v
@
f
@
r
m
f
@
v
t
d
v
C
d
v
C
d
v
D
0.
@
We reverse the order of differentiation and integration in the first two terms, and
introduce the definitions
R
fd
v
D
N
and
R
v
fd
v
D
N
w
,where
N
is the particle
number density and
w
is the mean velocity of the particles or the drift velocity.
The third term is zero because the distribution function for infinite velocity is zero.
Thus, we obtain
@
N
@
t
C
div(
N
w
)
D
0.
(4.1)
