Lecture 3: Tensor Analysis
a – scalar
Ai – vector, i=1,2,3
σij – tensor, i=1,2,3; j=1,2,3
Rules for Tensor Manipulation:
1. A subscript occurring twice is a repeated (or dummy) index and is
summed over 1,2, and 3.
2. A subscript occurring once in a term is called a free index, can take
on the range 1,2,3 but not summed.
3. No index can appear in a term more than twice.
Scalar and Vector Products
Scalar product:
A  B   Ae
i i    Bk ek   Ai Bk (ei  ek )  Ai Bk  ik  Ai Bi
 
A  B  Ai Bi
vector
notations
tensor
notations
Vector product:
A  B  ( Ae
i i )  ( B j e j )  Ai B j (ei  e j )  Ai B j eijk ek
 
A  B  eijk A j Bk
Kronecker delta
1 0 0
 ik   0 1 0 
0 0 1


-- Kronecker delta (or,
delta-symbol)
Manipulations involving delta-symbol, Ai  ik  Ak
Levi-Civita symbol
 1; (ijk )  (123)  (231)  (312) -- even permutation

eijk  1;(ijk )  (213)  (132)  (321) -- odd permutation
 0; if i  j or j  k or i  k

Properties:
eijk  e jki  ekij (even permutation)
eijk e pqk   ip jq   iq  jp
eijk  e jki (odd permutation)
Calculus
grad a i  ia

div A  i Ai

curl A i  eijk  j Ak
-- gradient
-- divergence
-- curl
Example of use of tensor notations:




v  curlv  v  curlv i  eijkv j ekpq  pv q  eijk e pqkv j  pv q
  ip  jq   iq  jp v j  pv q  v j iv j  v j  jv i
2
v jv j
v  

 i
 v j  jv i     v   v
2
2
Levi-Civita
Tullio Levi-Civita (29 March 1873 – 29
December 1941) (pronounced /'levi ʧivita/)
was an Italian mathematician, most famous
for his work on tensor calculus and its
applications to the theory of relativity
Lecture 4: Hydrodynamic approach
• Lagrangian and Eulerian Description
• Material Derivative
• Number of governing equations
Fluid velocity
1. Hydrodynamics is a macroscopic approach when the
random motion of single molecules is averaged over a
macroscopic volume.
2. The smallest examined object in hydrodynamics is a ‘fluid
particle’. A fluid particle consists of ‘many’ randomly moving
molecules.
3. Fluid velocity is defined as the velocity of the centre mass
of a fluid particle.
Lagrangian and Eulerian Description.
Material Derivative
• Lagrangian: tracing the position and velocity of chosen fluid particles.
All variables are function of time.
• Eulerian: tracing the fluid velocity (and other quantities) at a chosen
point. All variables are functions of time and coordinate.
• Material derivative: as an example, consider the rate of change of
density ρ of a fluid particle,




d  dt  dx  dy  dz 
t
x
y
z



        


dt   i 
j  k   i dx  j dy  kdz   dt    dR 
t
y
z 
t
 x

 


dt  vdt       v   dt


t
full or material d
 t

  v   
derivative:
dt t
Number of unknowns/governing
equations
• To define the thermodynamic state of a single-phase fluid, two
variables are needed, e.g. temperature and pressure. (only pressure
would be required for isothermal flow.)
• + three components of the velocity field.
• On the whole, 5 variables (unknowns) are required to describe a
single-phase fluid flow.
• As the number of equations should be equal to the number of
unknowns, 5 governing equations should be provided.
• These are the continuity equation, the Navier-Stokes equation (this
is the vector equation, hence, it gives 3 scalar equations), and the
equation for the energy transport.