SESM3004 Fluid Mechanics

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24 Οκτ 2013 (πριν από 4 χρόνια και 2 μήνες)

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SESM3004 Fluid Mechanics

Dr Anatoliy Vorobev

Office: 25/2055,

Tel: 28383, E
-
mail: A.Vorobev@soton.ac.uk

Aim


1
st

and 2
nd

year Fluid Mechanics: Introduction and basic
equations.



SESM3004: use of equations to particular problems,
such as steady and non
-
steady plane
-
parallel flows,
water waves, convection, capillary flows and sound
waves + the concepts of the hydrodynamic instability.

-

The concept of physical modelling used to understand the
fluid flow aspects in existent applications.


Fluid Mechanics vs Hydraulics


Hydraulics

is a topic in applied science and engineering
dealing with the mechanical properties of liquids.



Fluid mechanics

provides the theoretical foundation for
hydraulics, which focuses on the engineering uses of
fluid properties.


Admittedly, as useful a matter as the motion of fluid and
related sciences has always been an object of thought.
Yet until this day neither our knowledge of pure
mathematics nor our command of the mathematical
principles of nature have permitted a successful
treatment
’ (Daniel Bernoulli, Sept. 1734)

http://www.claymath.org/millennium/

Mathematicians and physicists believe that an
explanation for and the prediction of both the breeze and
the turbulence can be found through an understanding
of solutions to the Navier
-
Stokes equations.
Although
these equations were written down in the 19th Century,
our understanding of them remains minimal. The
challenge is to make substantial progress toward a
mathematical theory which will unlock the secrets
hidden in the Navier
-
Stokes equations.


Syllabus


Revision: vector algebra & calculus; fundamental
equations of fluid mechanics.



Isothermal flows: steady and non
-
steady plane
parallel flows; laminar boundary layers; water
waves; capillary flows.



Non
-
isothermal flows: convection; sound waves.



Hydrodynamic stability: convective instability;
transition to turbulence.

Grading

Homework

(10 assignments, assignments and solutions will
be uploaded to the course web
-
site): 20%

Final Exam
(closed
-
book, written): 80%

Lectures
:

Tuesday,

9
-
11
am,

54
/
5025
.

Tutorials

(
weeks

3
-
11
)
:

Monday,

1
-
3
pm,

07
/
3027
.


Text books


Paterson A.R., 1983.
A first course in fluid dynamics.

Cambridge
University Press.


Landau L.D., Lifshitz E.M., 1959.
Course of Theoretical Physics.
Volume 6: Fluid Mechanics.

Pergamon Press.


G.K. Batchelor, 1967.
An Introduction to Fluid Dynamics.

Cambridge
University Press.


W.F. Hughes, J.A. Brighton, 1999.
Schaum's outline of theory and
problems of fluid dynamics
. New York: McGraw Hill.


James A. Fay, 1994.
Introduction to Fluid Mechanics.

MIT Press.


Tritton D.J., 1988.
Physical Fluid Dynamics.

Clarendon Press.


R.F. Probstein, 1989.
Physicochemical Hydrodynamics.

Butterworths.


P.G. Drazin, W.H., 2004.
Hydrodynamic Stability.

Cambridge University
Press.


Lecture notes + problem worksheet on Blackboard web
-
site

L1
-
2
: Vector Algebra & Calculus


Scalar, vector and tensor fields


Systems of coordinates: Cartesian and
cylindrical coordinates


Scalar and vector products. Triple
products

Scalars, Vectors, Tensors

scalar



single element (e.g. length (
L
)
, mass (
m
)
,
temperature (
T
)
)

vector



1D array of elements (position vector (

),
velocity ( ), force ( ))

tensor



n
-
dimensional array of elements, but we
are interested in tensors of rank 2 (stress tensor
(
σ
))

Scalar field

associates a scalar value to any point
in space (e.g. T( )). Similarly,
vector and tensor
fields.


Cartesian coordinates

-

coordinates

-

unit vectors

Cylindrical coordinates

(
r,
θ
, z
)

θ

r

z

x

y

Scalar and Vector Products


Scalar product


Vector product (not commutative)







Triple products

Differentiation

--

del operator (nabla)

--

gradient

--

curl

--

divergence

Double differentiation

--

Laplacian operator

Cylindrical coordinates

--

del

--

gradient

--

divergence

--

curl

Useful identities

Sample proof: