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PRIFYSGOL CYMRU / UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS & PHYSICS
SEMESTER 2 EXAMINATIONS, MAY/JUNE 2009
MA25610 – HYDRODYNAMICS I
Time allowed – 2hours
All questions may be attempted. Full marks will be given for complete answers to
all questions in Section A and to two questions in Section B. In Section B, credit will
be given for the BEST TWO answers.
Marks gained from questions in Section B will be given greater consideration in
assessing a first class performance.
Calculators are permitted, provided they are silent, self-powered, without
communications facilities, and incapable of holding text or other material that could
be used to give a candidate an unfair advantage. They must be made available on
request for inspection by invigilators, who are authorised to remove any suspect
calculators.
Useful formulae
In the usual notation:
3 3 3
2 2 2
1 1 1 h x
h x
h x
grad e e e ∂
3
1 2 3
2
1 3 2
1
2 3 1
1 2 3 x
hhu
x
hhu
x
hhu
hhh
div u
3 2
1 1
1
2 2
1 2
2 1
3 3
3
1 1
1 3
1 3
2 2
2
3 3
2 3 x
hu
x
hu
hh
x
hu
x
hu
hh
x
hu
x
hu
hh
curl u e e e
Alternatively:
11 2 2 3 3
1 2 3
1 1 2 2 3 3
12 3 hu hu hu
x x x
h h h
hhh
curl ∂
e e e
u
3 3
1 2
2 2 3
1 3
1 1 2
2 3
1 2 3 1
2
h x
hh
h x x
hh
h x x
hh
hhh x
1 φ φ φ φ
Section B
- a) A point source has a velocity potential given by
4 r
m
π
φ = in terms of standard
spherical polar coordinates. Determine the velocity field and show that the source
strength m is equal to the volumetric flow rate emitted from the source. [6]
b) Derive the velocity potential corresponding to a point doublet. [12]
c) A point source of strength m is placed in a uniform stream of speed U. Show
that the flow contains a single stagnation point that occurs on the axis of
symmetry, at a distance
4 U
m
π
upstream of the source. [12]
- A spherical bubble of gas is initially of radius a, and at a pressure P 0. It expands
in an infinite expanse of incompressible liquid of density ρ in which the pressure
at infinity is zero. The liquid flow is irrotational. The gas is initially at rest and its
pressure, p and volume, V are governed by the equation
4 pV 3 = constant. Prove
that the bubble doubles its radius in time:
0
28 2
15
a
P
ρ
. [30]
- Explain, briefly, the method of images in the solution of hydrodynamic flows. [5]
An infinite flat barrier is located at z = 0. An incompressible, inviscid fluid of
density ρ, occupies the semi-infinite space z > 0. The pressure in the fluid is p 0
when it is undisturbed. A point doublet of strength μ , placed parallel to Oz, is
introduced at the point (a,0,0) relative to a standard set of spherical polar
coordinates with origin at O. Construct an image system for this flow and show
that the pressure on the barrier is minimum at r = a/2. [25]
