SESM3004 Fluid Mechanics

Dr Anatoliy VorobevOffice: 25/2055,

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

Aim

• 1st and 2nd 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, 9-11am, 54/5025.Tutorials (weeks 3-11): Monday, 1-3pm, 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.

v

f r

r

Cartesian coordinates

( , , )

( , , )

( , , )x y z x y z

x y z

i j k

v v v v v i v j v k

- coordinates

- unit vectors

Cylindrical coordinates

(r,θ, z)

θr

zzθrrzr

zθr

evevevvvvv

)e,e,e(

zr

),,(

(basis) rsunit vecto

scoordinate),,(

z

x

y

Scalar and Vector Products• Scalar product• Vector product (not commutative)

• Triple products

x x y y z zA B A B A B A B

( ) ( ) ( )

x y z

x y z

y z z y z x x z x y y x

i j k

A B A A A

B B B

A B A B i A B A B j A B A B k

A B C B C A C A B

A B C B A C C A B

Differentiation

zk

yj

xi

aza

kya

jxa

ia grad

vyA

x

Ak

xA

zA

jz

A

yA

i

vvv

kji

v

vzv

y

v

xv

v

xyzxyz

zyx

zyx

zyx

curl

;div

-- del operator (nabla)

-- gradient

-- curl

-- divergence

Double differentiation

2

2

2

2

2

2

2

zyx

2

2

2

2

2

2

2

2

2

2

2

2

2

2

zA

yA

xA

A

za

ya

xa

a

;

-- Laplacian operator

0

0

A

a

rz

zrzr

zr

zr

zr

ArrA

re

rA

zA

ezAA

reA

zAA

rrrA

rA

za

ea

re

ra

ea

ze

re

re

11

11

1

1

Cylindrical coordinates

-- del

-- gradient

-- divergence

-- curl

Useful identities

2

2

2

aA a A A a

aA a A a A

AA A A A

A B A B B A B A A B

A A A

Sample proof:

AaaAAaAaAa

divgrad