Untitled - Aerobib - Universidad Politécnica de Madrid

5.4. LIMITING FORM OF THE WAVE EQUATIONS 111
Now, by substituting the expressions of ¯v and l deduced from (5.17) and (5.18) into
the ratio l/¯v, it results that this ratio is proportional to τ t as defined by (5.16).
As a characteristic time τ c of the chemical transformations, the following ratio
is adopted
τ c = ρ w , (5.19)
that measures the average time between those molecular collisions successful in producing
chemical reaction. 4
Let
α = τ t
= µw
τ c ρp
(5.20)
be the ratio of the thermodynamic time τ t to the chemical time τ c . The assumption
that chemical transformations are very slow when compared to the thermodynamic
transformations is expressed by the condition that parameter α is much smaller than
unity
α ≪ 1. (5.21)
The limiting form of the wave equations, obtained when taking assumption (5.21) into
the system deduced in the preceding paragraph, is given in the paragraph that follows.
5.4 Limiting form of the wave equations
In order to discuss the relative values of the various terms of the differential system
of the wave, we shall start by showing the influence of α and the flow Mach number
M = v/a = v/ √ γp/ρ.
For this purpose, let us first eliminate x from (5.3.a), (5.4.a) and (5.5.a), making
use of equation (5.2.a), then
p + mv − 4 µw dv
= i,
3 m dε
(5.22)
1
2 v2 + c p T − qε − λw dT
m 2 dε − 4 µw
3 m 2 v dv = e,
dε
(5.23)
ρDw
m 2
dY
dε
− Y + ε = 0. (5.24)
4 See chapter 1.