Untitled - Aerobib - Universidad Politécnica de Madrid

5.4. LIMITING FORM OF THE WAVE EQUATIONS 113
Equation (5.27) reduces to
Y = ε. (5.30)
Therefore in this case, the effects of viscosity, thermal conductivity and
diffusion can be neglected, and the mixture can be considered as an ideal
gas whose composition varies due to chemical reactions.
2) α ≪ 1, α/M 2 ∼ 1.
Will be designated case B. As aforesaid, in this case M 2
(5.28) is reduced to
≪ 1, and Eq
p = i, (5.31)
that is to say, the combustion occurs in first approximation at constant pressure.
In Eq. (5.26) the second and last terms of the left hand side can be
neglected. Then, the following simplified equation is obtained
(
c p T 1 − q
c p T ε − 1 )
α 1 dT
P r M 2 = e, (5.32)
T dε
which can be written in the following form, returning to the old variable x
by means of equation (5.2.a)
c p T − qε − λ m
dT
dx = e.
(5.32.a)
All the terms of equation (5.27) must be preserved. By returning to variable
x, equation (5.15) is obtained.
Of the two limiting cases determined herein, the former is represented by the
system of equations (5.1.a), (5.2.a), (5.28.a) and (5.30), that is, by
Case A: α ≪ 1, M 2 ∼ 1 .
ρv = m,
m dε
dx = w,
p + ρv 2 = i,
c p T + 1 2 v2 − qε = e,
(5.1.a)
(5.2.a)
(5.28.a)
(5.29.a)
Y = ε. (5.30)
As aforesaid, in this system, the influence of viscosity, thermal conductivity
and diffusion has disappeared. As will presently be seen, the solutions corresponding
to this system represent detonation waves.