p s ¼
2ρ 0
γ þ 1
ð
Þ
A
2 R
À3 ,
ð5:6aÞ
u s ¼
2
γ þ 1
AR
À
3
2
ð5:6bÞ
and
ρ s ¼ ρ 0
γ þ 1
γ À 1
:
ð5:6cÞ
The self-similarity of the flow implies that the pressure, gas velocity and density
can be expressed using the dimensionless similarity variable η and we write,
p ¼
ρ 0 A
2
γ
R
À3 f η
ð Þ
ð5:7aÞ
u ¼ AR
À3=2
ϕ η
ð Þ
ð5:7bÞ
ρ ¼ ρ 0 ψ η
ð Þ,
ð5:7cÞ
where the quantities, f(η), ϕ(η) and ψ(η) are functions only of the dimensionless
variable η, (the notation for these distributions follows that in the paper by Taylor).
At the shock front where η ¼ 1 it is clear by comparing Eqs. (5.6a), (5.6b), (5.6c),
(5.7a), (5.7b) and (5.7c) that
f 1
ð Þ ¼
2γ
γ þ 1
:
pressure
ð
Þ
ϕ 1
ð Þ ¼
2
γ þ 1
:
velocity
ð
Þ
ψ 1
ð Þ ¼
γ þ 1
γ À 1
:
density
ð
Þ
ð5:8Þ
Let us now be substituted Eqs. (5.7a), (5.7b) and (5.7c) in the momentum,
continuity and energy equations as demonstrated below.
5.4.1 Momentum Equation
The equation of motion is (see Eq. (1.65), Chap. 1)
∂u
∂t
þ u
∂u
∂r
¼ À
1
ρ
∂p
∂r
,
hence, using Eqs. (5.7a), (5.7b) and (5.7c) above we have
222
5 Spherical Shock Waves: The Self-similar Solution