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Hydrodynamics I Exam - Univ. of Wales, Aberystwyth, May/June 2008 (75 characters), Exams of Dynamics

The exam questions for the hydrodynamics i module (ma256110) at the university of wales, aberystwyth, held in may/june 2008. The exam covers topics such as conservation of mass, incompressible flow, velocity potential, and bernoulli's equation. Students are required to solve problems using various hydrodynamic concepts and equations.

Typology: Exams

2012/2013

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PRIFYSGOL CYMRU / UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICAL & PHYSICAL SCIENCES
SEMESTER 2 EXAMINATIONS, MAY/JUNE 2008
MA256110 – 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.
pf3
pf4

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Download Hydrodynamics I Exam - Univ. of Wales, Aberystwyth, May/June 2008 (75 characters) and more Exams Dynamics in PDF only on Docsity!

PRIFYSGOL CYMRU / UNIVERSITY OF WALES

ABERYSTWYTH

INSTITUTE OF MATHEMATICAL & PHYSICAL SCIENCES

SEMESTER 2 EXAMINATIONS, MAY/JUNE 2008

MA256110 – 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

φ

Section B

  1. Consider small oscillatory disturbances on the surface of an incompressible, inviscid liquid that is unbounded in the x and y directions. The depth of the liquid is assumed to be large relative to both the amplitude and wavelength of the disturbance. Assume that the flow is 2-dimensional and irrotational. Determine the velocity potential for the flow and derive the dispersion relation:

 

λ h

tanh gh h

c 2 2

2

, relating the wave speed c and wavelength λ of the

disturbance. [30]

  1. The gas pressure p 1 , inside a spherical bubble in an infinite mass of liquid of

constant density ρ, can be varied in a prescribed way. Initially the bubble is at rest

in equilibrium with a radius a. At the instant t = 0, p 1 is suddenly increased to (5/3) p ∞ where p ∞ is the constant pressure at infinity, and p 1 subsequently varies in direct proportion to the square of the radius of the bubble, as this expands. Prove that:

a

t 3

2 p R acosh

2 1

[30]

  1. 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]