SESM3004 Fluid Mechanics
Dr Anatoliy Vorobev
Office: 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)
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.
http://www.claymath.org/millennium/
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.
Lectures: Tuesday, 9-11am, 54/5025.
Tutorials (weeks 3-11): Monday, 1-3pm,
07/3027.
Grading
Homework (10 assignments, assignments and solutions will
be uploaded to the course web-site): 20%
Final Exam (closed-book, written): 80%
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 (r ),

velocity (v), force (f ))
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(r )). Similarly, vector and tensor
fields.
Cartesian coordinates
( x, y , z )
- coordinates
(i , j , k )
- unit vectors
v  (v x , v y , v z )  v x i  v y j  v z k
Cylindrical coordinates
z
(r,θ, z)
y
r
x
θ
(r ,  , z )  coordinate s
  
(er ,eθ ,e z )  unit vecto rs (basis)




v  (v r ,v ,v z )  v r er  v eθ  v z e z
Scalar and Vector Products
• Scalar product A  B  Ax Bx  Ay By  Az Bz
• Vector product (not commutative)
i
A  B  Ax
j
Ay
k
Az
Bx
By
Bz
 ( Ay Bz  Az By )i  ( Az Bx  Ax Bz ) j  ( Ax By  Ay Bx )k
• Triple products
     
A  B  C   B AC   C  A B
A B  C  B  C  A  C  A B
Differentiation
   
 
 i
j
k
x
y
z
-- del operator (nabla)
 a  a  a
a  i
j
k
 grad a
x
y
z
-- gradient
 v v v

 v 


 div v ;
x y z
-- divergence
y
x
i

 v  
v
x
x
j

v
z
k
 A A 
  i 

 
z 
 y
v
z
y
y
z
z
y
 A A   A A 

j


  curl v -- curl
  k 
x 
y 
 z
 x
x
z
y
x
Double differentiation



    


x
y z
2
2
2
2
2
2
a a a
a 


;
x y z



 A A A
A


x
y
z
2
2
2
2
2
2
2
2
2
2
2
2
2
  a   0

    A   0
2
2
-- Laplacian operator
Cylindrical coordinates
   1   
  er  e
 ez
-- del
r
r 
z
 a  1 a  a
a  er  e
 ez
-- gradient
r
r 
z
 1  rAr  1 A Az
-- divergence
 A 


r r
r 
z
   1 Az A    Ar Az   1   rA  Ar 
  A  er 



  e 

  ez 
r  r
 
 z r 
 r  z 
-- curl
Useful identities
     
   aA  a    A    a   A
  aA  a   A  A   a

 A2 
A A     A   A
 2 


     
     A      A   A


 

  A B  A   B  B  A  B   A  A B
2
Sample proof:





  aA     a A     aA  A grad a  a div A