This is the equation of motion in terms of f, ϕ and ψ which are all functions of the
dimensionless parameter; η ¼ r/R.
5.4.2 Continuity Equation
Let us now consider the continuity equation (see Eq. (1.63), Chap. 1);
∂ρ
∂t
þ u
∂ρ
∂r
þ ρ
∂u
∂r
þ
2u
r
¼ 0,
and using Eqs. (5.7a), (5.7b) and (5.7c) we have
∂ρ
∂t
¼ ρ 0
∂ψ
∂t
¼ ρ 0
∂ψ
∂η
∂η
∂R
∂R
∂t
¼ Àρ 0 ψ
0
ηAR
À
5
2 ,
where ψ
0 denotes differentiation with respect to η. Similarly,
∂ρ
∂r
¼ ρ 0
∂ψ
∂η
∂η
∂r
¼
ρ 0 ψ
0
R
, hence,
u
∂ρ
∂r
¼ AR
À
3
2 ρ 0
ϕψ
0
R
¼ AR
À
5
2 ρ 0 ϕψ
0
and as before we have
∂u
∂r
¼ AR
À
5
2 ϕ
0
:
Collecting all of the various terms above and substituting them in the continuity
equation, we obtain
Àρ 0 ψ
0
ηAR
À
5
2 þ AR
À
5
2 ρ 0 ϕψ
0
þ ρ 0 ψ AR
À
5
2 ϕ
0
þ 2
AR
À
5
2 ϕ
η
¼ 0,
and after cancelling some common terms this latter equation reduces to
Àψ
0
η þ ϕψ
0
þ ψ ϕ
0
þ 2
ϕ
η
¼ 0
so that
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5 Spherical Shock Waves: The Self-similar Solution