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MATHEMATICAL TRIPOS Part III
Tuesday 14 June, 2005 9 to 12
PAPER 71
ASTROPHYSICAL FLUID DYNAMICS
Attempt THREE questions. There are FOUR questions in total. The questions cary equal weight.
Candidates may bring their notebooks into the examination. The following equations may be assumed.
Dρ Dt
ρ
Du Dt
= −∇p − ρ∇Φ + j ∧ B
ρ
De Dt
p ρ
Dρ Dt
div B = 0; j = μ− 0 1 curl B ∇^2 Φ = 4πGρ ∂B ∂t
= curl (u ∧ B)
p = (γ − 1)ρe =
R
μ
ρT
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Cover sheet None Treasury Tag Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
1 A plane shock wave lies (in the frame of the shock) in the plane x = 0. The flow velocity is in the x-direction and of magnitude UL (UR) to the left (right) of the shock, where left (right) corresponds to the half-space x < 0 (x > 0). In the same notation, the densities are ρL (ρR), the pressures pL (pR) and the energy densities eL (eR). Assuming that the perfect gas law p = (γ − 1)ρe applies on each side of the shock use the Rankine- Hugoniot relations to show that
ρR ρL
(γ + 1)M (^) L^2 (γ − 1)M (^) L^2 + 2
where ML is the Mach number in x < 0.
Deduce that ρR > ρL ⇐⇒ M (^) L^2 > 1, and hence that the flow must be supersonic on one side of the shock and subsonic on the other.
Show further that ( 2 γ + 1
u^2 L + uL (uR − uL) −
2 γ γ + 1
pL ρL
and that (^) ( 2 γ + 1
ρLu^2 L pL
pR pL
γ − 1 γ + 1
Now consider a plane shock lying in the plane x = X(t) < 0 and impinging on a stationary solid wall at x = 0. Prior to the passage of the shock the gas is at rest with pressure p 0 and density ρ 0. As the shock moves towards the wall with steady velocity dX/dt = U+ > 0, the gas behind the shock has velocity us (0 < us < U+), pressure ps and density ρs. After the shock has rebounded from the wall it moves with velocity dx/dt = −U− < 0, into the already once-shocked gas. The gas between the shock and the wall is now stationary and has pressure p 1 and density ρ 1. Use (*) to both the pre- and post-rebound configurations to show that (us + U−) and (us − U+) satisfy the same quadratic equation.
Deduce that (us − U+) (us + U−) = −γps/ρs. (†)
Similarly apply (**) to both pre- and post-rebound configurations, and hence, using (†) obtain a relationship between p 1 /ps and p 0 /ps, independent of the velocities.
In the case of a strong shock (p 0 ps) show that
p 1 ps
3 γ − 1 γ − 1
Paper 71
3 The interstellar medium is modelled as a perfect gas subject to cooling per unit volume at the rate (ρ, T ) = −ρ^2 Λ(T ), and with thermal conductivity λ(T ) = λ 0 T α, where λ 0 is a constant and α > 0. Gravity is neglected. Explain briefly the circumstances for which it is reasonable to assume that the pressure remains uniform, i.e.∇p = 0.
In this case show that a planar one-dimensional flow obeys the equation
1 γ − 1
∂p ∂t
γ γ − 1
p
∂v ∂x
∂x
λ
∂T
∂x
where v is the velocity in the x-direction.
Show further that if the flow remains at constant pressure then
∂T ∂t
∂T
∂x
γ − 1 γ
μ R
p
Λ(T )
T
(γ − 1) γ
λ 0 T p
∂x
T α^
∂T
∂x
Using the Lagrangian variable
m(x, t) =
∫ (^) x
0
ρ(x, t)dx,
and an appropriately scaled time τ = Ct, where constant C is to be determined, show that this equation can be written in the form
∂T ∂τ
Λ(T )
T
− λ 0
∂m
T α−^1
∂T
∂m
At time t = 0, gas fills the half space x > 0 and has uniform temperature T = T 0. The region x < 0 contains cold (T = 0) infinitely dense gas which does not move but cools infinitely fast. The gas in x > 0 cools only by thermal conduction (i.e. Λ = 0 if T > 0). Explain why it is reasonable to seek a similarity solution of the form
T (m, τ ) = T 0 f (ξ),
with similarity variable ξ = m/
λ 0 T 0 α −^1 τ
, and write down appropriate boundary conditions for f (ξ) at ξ = 0 and as ξ → ∞.
If λ(T ) = λ 0 T , where λ 0 is a constant, find the function f (ξ) in terms of the
function erf (z) = √^2 π
∫ (^) z 0 e
−s^2 ds, and sketch the resulting solution T (m, τ ), indicating the
behaviour as τ increases.
Show that the rate L at which energy is radiated by the gas at x 6 0 varies as L ∝ t−k, where k is to be determined.
[Hint: You may assume erf (∞) = 1].
Paper 71
4 An infinite cylinder (0 6 R 6 R 0 ) of incompressible fluid with uniform density ρ 0 rotates about the R = 0 axis with velocity u 0 = (0, RΩ(R), 0), with Ω(R) = kR where k is a constant.
The fluid is self-gravitating. Show that if the central pressure p(R = 0) = π^2 G^2 ρ^30 /k^2 then the radius is R 0 = (2πGρ 0 )^1 /^2 /k, and the effective surface gravity is zero.
The fluid is subject to small perturbations so that the velocity is u 0 + u, where u is of the form u ∝ (uR(R), uφ(R), 0) exp(iωt + imφ).
Show that the perturbation equations are
iσuR − 2Ωuφ = −
∂W
∂R
3ΩuR + iσuφ = −
imW R
duR dR
uR R
uφ R
where σ = ω + mΩ(R) and W =
p′ ρ
Show that these equations can be reduced to
d^2 uR dR^2
R
duR dR
uR R^2
1 − m^2 −
3 mΩ σ
Now consider the case m = 1. Show that a solution to this equation is
uR = 1 +
kR ω
Assuming that this is the only solution which is regular at R = 0, show that the oscillation frequencies obey the equation ω^2 = 0. Give a physical explanation of this result.
[You may assume that in cylindrical polars
∇^2 Φ =
R
∂R
R
∂R
R^2
∂^2 Φ
∂φ^2
∂^2 Φ
∂z^2
and that the equations of motion are
∂vR ∂t
v φ^2 R
ρ
∂p ∂R
∂R
∂vφ ∂t
vRvφ R
ρR
∂p ∂φ
R
∂φ
.]
END OF PAPER
Paper 71
