Waves, Instabilities, and Structures in Excited and Ionized Gases - Fundamentals of Ionized Gases - page 295

Substituting this into (5.28), we have

T 0 ) p 1

dZ 2
dT D

u 2 f ( T )

c p N D

2

e α ( T m T ) uZ ( T

h 1

e α ( T m T ) i

p 1

u 2
2

C α T 0 ) 2 e α ( T m T )

( T .

In the region 1, where the heat release is essential, the first term is

small compared with the second one. Then comparing this expression with the ap-
proximate dependence assumed above for f ( T ), we find the velocity of the thermal
wave

α ( T T 0 )

E a / T m to be α D

s 2

T m
E a ( T m

A
c p N .

u D

T 0 )

This result is in agreement with the Zeldovich formula (5.35) for the dependence
employed for f ( T ). The above analysis shows that the Zeldovich formula for the
velocity of a thermal wave is valid under the condition α ( T m T 0 ) 1.

5.1.8
Thermal Waves of Vibrational Relaxation

We shall apply the above results to the analysis of illustrative physical processes.
First we consider a thermal wave of vibrational relaxation that can propagate in an
excited molecular gas that is not in equilibrium. In particular, this process can oc-
cur inmolecular lasers, where it can result in the quenching of laser generation. We
consider the case where the number density of excited molecules is considerably
greater than the equilibrium density. Vibrational relaxation of excited molecules
causes the gas temperature to increase and the relaxation process accelerates. There
will be a level of excitation and a temperature at which thermal instability develops,
leading to the establishment of a new thermodynamic equilibrium between excited
and nonexcited molecules.
The balance equation for the number density of excited molecules N has the
form

@ N

@

D D Δ N NN k ( T ),

t

where N is the total number density of molecules, and where we assume N N .

D is the diffusion coefficient for excited molecules in a gas, and k ( T )istherate
constant for vibrational relaxation. Taking into account the usual dependence of
traveling wave parameters N ( x , t )

ut ), where u is the velocity of the
thermal wave, we transform the above equation into the form

D N ( x

D d 2 N
dx 2

u dN
dx

C N Nk ( T ) D 0 . (5.36)

In front of the thermal wave we have N

D N max , and after the wave we have

N D 0. That is, we are assuming the equilibrium number density of excited