Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 65 / 1 / b

Solution

Write

V = \dot{R}

a = 2/ (5 - β)

and

ξ = \frac{r}{R}, u (r, t) = V U (ξ), ρ (r, t) = C R^{- β} D (ξ), p (r, t) = C R^{- β} V^{2} P (ξ) .

Here

u

is the outward radial velocity;

U, D, P

are dimensionless. The relations

\partial_{t} ξ = - \frac{V}{R} ξ, \partial_{r} ξ = \frac{1}{R}, q \equiv \frac{R V ˙}{V ^{2}} = \frac{a - 1}{a} = \frac{β - 3}{2}

are sufficient for the reduction. In smooth post-shock gas the spherical continuity equation, Euler equations and adiabatic pressure equation are

\partial_{t} ρ + u \partial_{r} ρ = - ρ (\partial_{r} u + 2 u / r), \partial_{t} u + u \partial_{r} u = - \frac{1}{ρ} \partial_{r} p,

\partial_{t} p + u \partial_{r} p = - γ p (\partial_{r} u + 2 u / r) .

C R^{- β} (V / R) [(U - ξ) D^{'} - β D]

(V^{2} / R) [(U - ξ) U^{'} + q U]

. After cancellation of the dimensional factors the spherical blast wave in a power-law ambient density obeys

(U - ξ) D^{'} + D (U^{'} + \frac{2 U}{ξ} - β) = 0,

(U - ξ) U^{'} + \frac{β - 3}{2} U = - \frac{P ^{'}}{D},

(U - ξ) P^{'} + [γ (U^{'} + \frac{2 U}{ξ}) - 3] P = 0.

Primes here mean

d / d ξ

2 q - β = - 3

explains the constant term. Once the profiles are found, their finite energy integral fixes

A

through

E = 4 π C R^{3 - β} V^{2} \int_{0}^{1} [\frac{1}{2} D U^{2} + \frac{P}{γ - 1}] ξ^{2} d ξ .

No global constant in

p / ρ^{γ}

is assumed: the shock wave generates different entropies in shells swept up at different times.

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