Untitled - Aerobib - Universidad Politécnica de Madrid

5.2. WAVE EQUATIONS 109
2) Burnt gases,
x → +∞ : p → p f , ρ → ρ f , T → T f , v → v f , Y → 1, ε → 1. (5.8)
Such conditions imply, naturally, the following ones,
x → ±∞ :
dp
dx → 0, dρ
dx → 0, dT
dx → 0, dY
dx → 0,
dε
→ 0. (5.9)
dx
Herein, the possibility of satisfying the previous boundary conditions, will not
be discussed since it is subjected to a careful study in the chapter dedicated to deflagrations.
Therein, it will be seen that the boundary conditions relative to the cold
boundary of the flame give rise to a problem which, so far, has not been satisfactorily
solved.
In the present study it will be assumed that the boundary conditions can be
satisfied and, therefore, that the problem is completely determined.
By taking the boundary conditions (5.7), (5.8) and (5.9) into the system of
equations (5.1.a), (5.4.a)) and (5.4.a), the following laws of conservation are obtained,
which agree with the invariants deduced in the preceding chapter
ρ 0 v 0 = ρ f v f , (5.10)
p 0 + ρ 0 v 2 0 = p f + ρ f v 2 f , (5.11)
1
2 v2 0 + c p T 0 + q = 1 2 v2 f + c p T f . (5.12)
The boundary conditions relative, for instance, to the burnt gases, make possible
the elimination of constants i, e and f from equations (5.3.a), (5.4.a) and (5.5.a),
thus obtaining
p + mv − 4 3 µ dv
dx = p f + mv f , (5.13)
1
)
m(
2 v2 + c p T − qε − λ dT
dx − 4 dv
( 1
)
µv
3 dx = m 2 v2 f + c p T f − q , (5.14)
ρD dY
− Y + ε = 0. (5.15)
m dx
The discussion and analysis of the possible solutions of the previous system are
rather complicated. An example of a discussion for a similar system (but in which the
diffusion neglected) can be found in Friedrichs’s work [3]. As a result of his analysis,
Friedrichs concludes that except in the case where the reaction rate w is exceptionally
high (and takes a well defined value) weak detonations are impossible. Consequently,
the only possible detonations with a normal reaction rate are strong detonations and