Curvilinear Coordinates
Outline:
1. Orthogonal curvilinear coordinate systems
2. Differential operators in orthogonal curvilinear coordinate systems
3. Derivatives of the unit vectors in orthogonal curvilinear coordinate systems
4. Incompressible N-S equations in orthogonal curvilinear coordinate systems
5. Example: Incompressible N-S equations in cylindrical polar systems
The governing equations were derived using the most basic coordinate system, i.e,
Cartesian coordinates:
x  xˆi  yˆj  zkˆ
f
f
f
grad f  f  ˆi  ˆj  kˆ
x
y
z
F F F
div F    F  1  2  3
x y
z
ˆi
ˆj kˆ
curl f    F 

x
F1

y
F2

z
F3
2 f 2 f 2 f
Laplacian   f  2  2  2
x
y
z
2
Example: incompressible flow equations
 V  0
DV

   p   z    2 V
Dt
 V


 V V     p   z    2 V
 t

 V 1


   V  V   V  ω     p   z        V     ω 
 t 2

ω   V
   V   0 in the above equation, but retained to keep the complete vector identity
for  2 V in equation.
However, once the equations are expressed in vector invariant form (as above) they can
be transformed into any convenient coordinate system through the use of appropriate definitions
for the  ,  ,  , and  2 . Frequently, alternative coordinate systems are desirable which either
exploit certain features of the flow at hand or facilitate numerical procedures. The most general
coordinate system for fluid flow problems are nonorthogonal curvilinear coordinates. A special
case of these are orthogonal curvilinear coordinates. Here we shall derive the appropriate
relations for the latter using vector technique. It should be recognized that the derivation can
also be accomplished using tensor analysis
1. Orthogonal curvilinear coordinate systems
Suppose that the Cartesian coordinates  x, y, z  are expressed in terms of the new
coordinates  x1 , x2 , x3  by the equations
x  x  x1 , x2 , x3 
y  y  x1 , x2 , x3 
z  z  x1 , x2 , x3 
where it is assumed that the correspondence is unique and that the inverse mapping exists.
For example, circular cylindrical coordinates
x  r cos
y  r sin 
zz
i.e., at any point P , x1 curve is a straight line, x2 curve is a circle, and the x3 curve is a
straight line.
The position vector of a point P in space is
R  xˆi  yˆj  zkˆ
R   r cos   ˆi   r sin   ˆj   z  kˆ for cylindrical coordinates
By definition a vector tangent to the x1 curve is given by:
R x  xx ˆi  yx ˆj  z x kˆ (Subscript denotes partial differentiation)
1
1
1
1
So that the unit vectors tangent to the xi curve are
eˆ1 
R x1
h1
,
eˆ 2 
R x2
h2
,
eˆ 3 
R x3
h3
Where hi  R x are called the metric coefficients or scale factors
hr  1,
hz  1 for cylindrical coordinates
h  r ,
1
The arc length along a curve in any direction is given by
ds 2  dR  dR  h12 dx12  h22 dx22  h32 dx32
Since dR  R xi dxi  hi dxi eˆ i and R xi  hi eˆ i
0,
and since the xi are orthogonal: eˆ i  eˆ j  
1,
i j
i j
An element of volume is given by the triple product
d    h1dx1eˆ1  h2 dx2eˆ 2   h3dx3eˆ 3  h1h2 h3dx1dx2dx3
Where since the xi are orthogonal eˆ1  eˆ 2  eˆ 3
Finally, on the surface x1  constant, the vector element of surface area is given by
ds1  h2 dx2eˆ 2  h3dx3eˆ 3  eˆ1h2 h3dx2 dx3
With similar results for x2 and x3  constant
ds2  eˆ 2 h3h1dx3dx1
ds3  eˆ 3h1h2 dx1dx2
2. Differential operators in orthogonal curvilinear coordinate systems
With the above in hand, we now proceed to obtain the desired vector operators
1 f
1 f
1 f
eˆ1 
eˆ 2 
eˆ 3
h1 x1
h2 x2
h3 x3
By definition: df  f  dR  f xi dxi
2.1 Gradient f 
If we temporarily write f  1eˆ1  2eˆ 2  3eˆ 3
Then by comparison
df  f xi dxi  i hi dxi
1 f
hi xi
1 
1 
1 

eˆ1 
eˆ 2 
eˆ 3
h1 x1
h2 x2
h3 x3

1 

  eˆ r 
eˆ  eˆ z for cylindrical coordinates
r
r 
z
i 
eˆ i
hi
1 f
i 
hi xi
Note xi 
So that by definition ( curl  grad f   0 )
  xi   
Also
eˆ i
0
hi
eˆ1
eˆ eˆ
 2  3  x2  x3
h2 h3 h2 h3
So that by definition (   f g   0 )
 eˆ 
 eˆ 
 eˆ 
 1    2    3   0
 h1h2 
 h2 h3 
 h3h1 
2.2 Divergence   F 
1
h1h2 h3
 



 h3h1F2    h1h2 F3 
  h2 h3 F1  
x2
x3
 x1

 F    F1eˆ1     F2eˆ 2     F3eˆ 3 

 eˆ  
   F1eˆ1      h2 h3 F1  1   using  u    u  u 
 h2 h3  

 eˆ 
 eˆ 
 eˆ 
eˆ
 1   h2 h3 F1  using    1      2      3   0
h2 h3
 h1h2 
 h2 h3 
 h3h1 

1

 h2 h3 F1 
h1h2 h3 x1
Treating the other terms in a similar manner results in

1  


F 
 h3h1F2    h1h2 F3 
  h2 h3 F1  
h1h2 h3  x1
x2
x3

1 


  F    rF1  
 F2    rF3 
r  r

z

1 
1 


 rF1  
 F2    F3  for cylindrical coordinates
r r
r 
z
h1eˆ1
1

2.3 Curl   F 
h1h2 h3 x1
h2eˆ 2

x2
h3eˆ 3

x3
h1 F1
h2 F2
h3 F3
 F    F1eˆ1     F2eˆ 2     F3eˆ 3 

 eˆ  
   F1eˆ1      h1 F1   1  
 h1  


eˆ
eˆ1
   h1 F1  using  u    u    u and   i  0
h1
hi
eˆ1  1   h1F1 
1   h1F1 
1   h1F1  

eˆ1 
eˆ 2 
eˆ 3 
h1  h1 x1
h2 x2
h3 x3

eˆ 3 
eˆ 2 

 h1F1  
 h1F1 
h1h2 x2
h3 h1 x3

1 

 
 h3eˆ 3
 h2eˆ 2
  h1F1 
h1h2 h3 
x3
x2 
h1eˆ1 h2eˆ 2 h3eˆ 3
1



F 
h1h2 h3 x1 x2 x3

h1 F1 h2 F2 h3 F3
eˆ r reˆ eˆ z
1 


F 
for cylindrical coordinates
r r  z
F1 rF2 F3
2.4 Laplacian acting on a scalar  2 f 
1
h1h2 h3
   h2 h3     h3 h1     h1h2   
 





 x1  h1 x1  x2  h2 x2  x3  h3 x3  
 2   
   h2 h3     h3h1     h1h2   
 





 x1  h1 x1  x2  h2 x2  x3  h3 x3  
1        1      
2    r  

 r 
r  r  r    r   z  z  

1
h1h2 h3

1     1 
r 
r r  r  r 
1 

 r 
 1    

r 
 r z  z 
2.5 Laplacian acting on a vector 2F    F     F 
Using f 
and   F 
   F  
1 f
1 f
1 f
eˆ1 
eˆ 2 
eˆ 3
h1 x1
h2 x2
h3 x3
1
h1h2 h3
 



 h3h1F2    h1h2 F3 
  h2 h3 F1  
x2
x3
 x1


1   1  


 h3h1F2    h1h2 F3   eˆ1

  h2 h3 F1  
h1 x1  h1h2 h3  x1
x2
x3


1   1  



 h3h1F2    h1h2 F3   eˆ 2

  h2 h3 F1  
h2 x2  h1h2 h3  x1
x2
x3


1   1  



 h2 h3 F1    h3h1F2    h1h2 F3   eˆ 3


h3 x3  h1h2 h3  x1
x2
x3

h1eˆ1
1

Using   F 
h1h2 h3 x1
h2eˆ 2

x2
h3eˆ 3

x3
h1 F1
h2 F2
h3 F3
   F  

1
h2 h3
   h3


 x2  h1h2
 
   h

 h1F1     2
  h2 F2  
x2
 x1
  x3  h1h3
 
 

 h1F1    h3 F3   eˆ1

x1
 x3
  
   h1  
    h3  
 


ˆ
h
F

h
F

h
F

h
F













3 3
2 2 
2 2
1 1 


  e2
x3
x2
 x3  h2 h3  x2
  
  x1  h1h2  x1
   h  
 
1    h2  



 h1F1    h3 F3     1   h3 F3    h2 F2   eˆ 3
 

h1h2  x1  h1h3  x3
x1
x3
  x2  h2 h3  x2
  

1
h1h3
Combining those two terms gives
2F     F     F  

1   1  


 h2 h3 F1    h3h1F2    h1h2 F3    eˆ1


h1 x1  h1h2 h3  x1
x2
x3

 
    h2  
1    h3  


ˆ

h
F

h
F

h
F

h
F













2 2
1 1 
1 1
3 3 


  e1
h2 h3  x2  h1h2  x1
x2
x1
  x3  h1h3  x3
  


1   1  


 h3h1F2    h1h2 F3   eˆ 2

  h2 h3 F1  
h2 x2  h1h2 h3  x1
x2
x3

   h  
 
1    h1  



 h3 F3    h2 F2     3   h2 F2    h1F1   eˆ 2



h1h3  x3  h2 h3  x2
x3
x2
  
  x1  h1h2  x1


1   1  


 h2 h3 F1    h3h1F2    h1h2 F3   eˆ 3


h3 x3  h1h2 h3  x1
x2
x3

    h1  
 
1    h2  


ˆ

h
F

h
F

h
F

h
F








 




1 1
3 3 
3 3
2 2 

  e3
h1h2  x1  h1h3  x3
x1
x3
  x2  h2 h3  x2
  

For cylindrical coordinates  r, , z  , h1  hr  1, h2  h  r , h3  hz  1 , and use the definition
of Laplacian operator acting on a scalar 2 f
1        1      
2 f    r  

 r 
r  r  r    r   z  z  
1     1 2
2

r




r r  r  r 2  2 z 2
1  2
1 2
2

 2 2

r r r
r  2 z 2
F
1
2 F 
2 F 


 2 F  aeˆ r  beˆ  ceˆ z    2 F1  2 F1  2 2  eˆ r    2 F2  22  2 1  eˆ    2 F3  eˆ z
r
r  
r
r  


a

1   1  



 h3h1F2    h1h2 F3   

  h2 h3 F1  
h1 x1  h1h2 h3  x1
x2
x3

   h3


 x2  h1h2
 
   h

 h1F1     2
  h2 F2  
x2
 x1
  x3  h1h3

1
h2 h3

 1  



 rF1    F2    rF3  


r  r  r

z

 
 

 h1F1    h3 F3  

x1
 x3
  
1  1 

  


     rF2  
 F1     r   F1    F3  
r    r  r

r
  z   z
 
F  
F F
 1 
   F1  r 1  2  r 3  
r  r 
r 
z  
F  
F F     F
1  1
    F2  r 2  1     r 1  r 3  
r    r 
r    z  z
r  
F 1 F2 F3 
 1
  F1  1 

r  r
r r 
z 
 2 F3 
F 1 F1 
 2 F1
1  1
   F2  2 

r

r


r    r
r r  
z 2
zr 
2
2
2
 1  1 F1  F1  1 F2  1  F2  F3
  2 F1  
 2  2



r
 r r r
 r   r r rz
 2 F3 
 2 F1
1  1 F2  2 F2 1  2 F1
 



r

r

r  r  r r  2
z 2
zr 

 2 F3
1
1 F1  2 F1 1 F2 1  2 F2
F





1
r2
r r r 2 r 2  r r
rz
 1 F 1  2 F
1  2 F1  2 F1  2 F3 
2

  2 2 
 2
 2 
2
z
zr 
 r  r r r 

2
2
2
1
1 F1  F1 2 F2  1  F1  F1 
 2 F1 





r
r r r 2 r 2   r 2  2 z 2 
 1 F1  2 F1 1  2 F1  2 F1  1
2 F

 2  2
 2   2 F1  2 2
2
r 
z  r
r 
 r r r
 1   F1  1  2 F1  2 F1  1
2 F

 2   2 F1  2 2

 2
2
z  r
r 
 r r  r  r 
1
2 F
  2 F1  2 F1  2 2
r
r 
b

1   1  



 h3h1F2    h1h2 F3   

  h2 h3 F1  
h2 x2  h1h2 h3  x1
x2
x3

   h  
 
1    h1  



 h3 F3    h2 F2     3   h2 F2    h1F1  



h1h3  x3  h2 h3  x2
x3
x2
  
  x1  h1h2  x1
1  1  




 rF1    F2    rF3  


r   r  r

z

  1 



   1  
     F3    rF2       rF2  
 F1  
z

  r  r  r
 
 z  r  
F  
F F
1  1 

F1  r 1  2  r 3  


r   r 
r 
z  
   1 F3 F2    1 
F2 F1   
 



    F2  r
r    
 z  r  z  r  r 
F 1 F2 F3 
1  1
F1  1 


r   r
r r 
z 
 1  2 F3  2 F2   1
F 1 F1  

 2   F2  2 

r  r
r r   
 r z z

1  1 F1  2 F1 1  2 F2  2 F3 
 




r  r  r r  2 z 
 1  2 F3  2 F2   F2 1 F2  2 F2 1 F1 1  2 F1  

 2  2 
 2  2


r r
r
r  r r   
 r
 r z z

1 F1 1  2 F1
1  2 F2 1  2 F3



r 2  r r r 2  2 r z
 1  2 F3  2 F2 F2 1 F2  2 F2 1 F1 1  2 F1 
 
 2  2
 2  2


z
r
r r
r
r  r r 
 r z
 1 F2  2 F2 1  2 F2  2 F2  F2 2 F1

 2  2
 2  2  2
r
r  2
z  r
r 
 r r
 1   F2  1  2 F2  2 F2  F2 2 F1

 2  2  2
r
 2
2
z  r
r 
 r r  r  r 
F
2 F
  2 F2  22  2 1
r
r 
c

1   1  



 h2 h3 F1    h3h1F2    h1h2 F3  


h3 x3  h1h2 h3  x1
x2
x3

   h  
 
1    h2  



 h1F1    h3 F3     1   h3 F3    h2 F2   
 

h1h2  x1  h1h3  x3
x1
x3
  x2  h2 h3  x2
  
 1  



    rF1  
 F2    rF3  
z  r  r

z

1    



   1  
   r   F1    F3   
 F3    rF2  


r  r   z
r
z
    r  
 
F  
F F
 1 
   F1  r 1  2  r 3  
z  r 
r 
z  
F    1 F3 F2  
1    F
  r 1 r 3 



r  r  z
r    r 
z  
  F F 1 F2 F3 
  1 1

z  r
r r 
z 
 2 F   1  2 F3  2 F2  
 2 F1 F3
1  F
  1  r

 r 23   


r  z
r z r
r   r  2 z  

1 F1  2 F1 1  2 F2  2 F3


 2
r z
zr r z
z
 1 F1  2 F1 1 F3  2 F3 1  2 F3 1  2 F2
 


 2  2

rz r r
r
r  2 r z
 r z
1 F3  2 F3 1  2 F3  2 F3

 2  2

r r
r
r  2 z 2
1   F3   2 F3 1  2 F3  2 F3



r

r r  r  r 2 r 2  2 z 2
  2 F3



3. Derivatives of the unit vectors in orthogonal curvilinear coordinate systems
The last topic to be discussed concerning curvilinear coordinates is the procedure to obtain the

derivatives of the unit vectors, i.e.
eˆ i  eˆ ij
x j
Such quantities are required, for example, in obtaining the rate-of-strain and rotation tensor
eij  V
1
1
eij  eijT    V  V 

2
2
1
1
ij   eij  eijT    V  V 
2
2
To simplify the notation we define:
R xi  ri , R xi  hi eˆ i  ri
 ij 


R xi  rij and
hi  hij
x j
x j
Note that rij is symmetric, i.e. rij  r ji
r1  h1eˆ1
r2  h2eˆ 2
r3  h3eˆ 3
r11  aeˆ1  beˆ 2  ceˆ 3  h11eˆ1  h1eˆ11
r12  h12eˆ1  h1eˆ12
r13  h13eˆ1  h1eˆ13
h
h
3.1 Derivation of eˆ11   12 eˆ 2  13 eˆ 3
h2
r1  r1  h
2
1
r1  r11  h1h11
r1  r12  h1h12
r1  r13  h1h13
r1  r2  0
  r1  r2 
0
x1
 r11  r2  r1  r21  0

 r11  r2  r1  r12
h3
 r11  r2  h1h12
r1  r3  0
  r1  r3 
0
x1
 r11  r3  r1  r31  0
 r11  r3  r1  r13

 r11  r3  h1h13
hh
h1h12
eˆ 2  1 13 eˆ 3  h11eˆ1  h1eˆ11
h2
h3
h
h
 eˆ11   12 eˆ 2  13 eˆ 3
h2
h3
r11  h11eˆ1 
3.2 Derivation of eˆ 22  
h
h21
eˆ1  23 eˆ 3
h1
h3
r2  r2  h22
r2  r22  h2 h22
r2  r21  h2 h21
r2  r23  h2 h23
r1  r2  0
  r1  r2 
0
x2
 r12  r2  r1  r22  0

 r22  r1  r2  r21
 r22  r1  h2 h21
r2  r3  0
  r2  r3 
0
x2
 r22  r3  r2  r32  0

 r22  r3  r2  r23
 r22  r3  h2 h23
r22  
hh
h2 h21
eˆ1  h22eˆ 2  2 23 eˆ 3  h22eˆ 2  h2eˆ 22
h1
h3
 eˆ 22  
h
h21
eˆ1  23 eˆ 3
h1
h3
3.3 Derivation of eˆ 33  
h31
h
eˆ1  32 eˆ 2
h1
h2
r3  r3  h32
r3  r31  h3h31
r3  r32  h3h32
r3  r33  h3h33
r1  r3  0
  r1  r3 
0
x3
 r13  r3  r1  r33  0

 r33  r1  r3  r31
 r33  r1  h3h31
r2  r3  0
  r2  r3 
0
x3
 r23  r3  r2  r33  0

 r33  r2  r3  r32
 r33  r2  h3h32
h3 h31
hh
eˆ1  3 32 eˆ 2  h33eˆ 3  h33eˆ 3  h3eˆ 33
h1
h2
h
h
 eˆ 33   31 eˆ1  32 eˆ 2
h1
h2
r33  
3.4 Derivation of eˆ 32 
h23
h
eˆ 2 , eˆ 23  32 eˆ 3
h3
h2

 r1  r2   r13  r2  r1  r23  0
x3

r2  r3  0 
 r2  r3   r21  r3  r2  r31  0
x1

r3  r1  0 
 r3  r1   r32  r1  r3  r12  0
x2
r1  r2  0 
1
 2
 3
 3   2   0
 r32  r1  r2  r31  0
 r32  r1  r2  r31
 r32  r1  r1  r23
 r32  r1  0
r32  aeˆ1  beˆ 2  ceˆ 3  h32eˆ 3  h3eˆ 32
r32  aeˆ1  beˆ 2  ceˆ 3  h23eˆ 2  h32eˆ 3
h
eˆ 32  23 eˆ 2
h3
r23  h23eˆ 2  h2eˆ 23
r23  r32  aeˆ1  beˆ 2  ceˆ 3  h23eˆ 2  h32eˆ 3
h
eˆ 23  32 eˆ 3
h2
h
h
3.5 Derivation of eˆ 21  12 eˆ1 , eˆ12  21 eˆ 2
h2
 2  1  0
h1
 r21  r3  r1  r23  0
 r21  r3  r1  r23
 r21  r3  r3  r12
 r21  r3  0
r21  h21eˆ 2  h2eˆ 21
r21  h12eˆ1  h21eˆ 2
h
eˆ 21  12 eˆ1
h2
r12  h12eˆ1  h1eˆ12
r12  r21  h12eˆ1  h21eˆ 2
h
eˆ12  21 eˆ 2
h1
3.6 Derivation of eˆ13 
 3  1  0
h31
h
eˆ 3 , eˆ 31  13 eˆ1
h1
h3
 r3  r12  r13  r2  0
 r2  r13  r3  r12
 r2  r13  r2  r31
 r2  r13  0
r13  h13eˆ1  h1eˆ13
r13  h13eˆ1  h31eˆ 3
h
eˆ13  31 eˆ 3
h1
r31  h31eˆ 3  h3eˆ 31
r31  r13  h13eˆ1  h31eˆ 3
h
eˆ 31  13 eˆ1
h3
4. Incompressible N-S equations in orthogonal curvilinear coordinate systems
4.1 Continuity equation  V  0

1  


 h3h1F2    h1h2 F3 
  h2 h3 F1  
h1h2 h3  x1
x2
x3

and V  v1eˆ1  v2eˆ 2  v3eˆ 3
Since   F 
V 
1
h1h2 h3
 



 h3h1v2    h1h2v3   0
  h2 h3v1  
x2
x3
 x1

V
1
  V   V   p  2 V , (where p piezometric pressure)
t

Since V  v1eˆ1  v2eˆ 2  v3eˆ 3 , we can expand the momentum equation term by term
4.2 Momentum equation
Local derivative
v
v
V v1

eˆ1  2 eˆ 2  3 eˆ 3
t
t
t
t
Convective derivative  V   V
Since V  v1eˆ1  v2eˆ 2  v3eˆ 3 and V  
v1  v2  v3 


h1 x1 h2 x2 h3 x3
 V  V   V  v1eˆ1    V  v2eˆ 2    V   v3eˆ3 
 V  v1eˆ1  
v1   v1eˆ1  v2   v1eˆ1  v3   v1eˆ1 


h1 x1
h2 x2
h3 x3
v v eˆ
v v
v v eˆ v v
v v eˆ v v
 1 1 eˆ1  1 1 1  2 1 eˆ1  2 1 1  3 1 eˆ1  3 1 1
h1 x1
h1 x1 h2 x2
h2 x2 h3 x3
h3 x3

v
 1
 h1
v
 1
 h1

 v1v1

vv
v1 v2 v1 v3 v1 
vv


eˆ11  2 1 eˆ12  3 1 eˆ13 
 eˆ1  
x1 h2 x2 h3 x3 
h2
h3
 h1

h13 
v1 v2 v1 v3 v1 
v1v1  h12


 eˆ1 
  eˆ 2  eˆ 3 
x1 h2 x2 h3 x3 
h1  h2
h3 
v2v1  h21  v3v1  h31 
 eˆ 2  
 eˆ 3 
h2  h1  h3  h1 
 v v v v v v 
v vh
v vh
vvh 
vvh 
  1 1  2 1  3 1  eˆ1   2 1 21  1 1 12  eˆ 2   3 1 31  1 1 13  eˆ 3
h1h2 
h3h1 
 h1h2
 h1 x1 h2 x2 h3 x3 
 h3h1
 V  v2eˆ 2  
v   v2eˆ 2  v2   v2eˆ 2  v3   v2eˆ 2 
 1


h1 x1
h2 x2
h3 x3
v v
v v eˆ
v v
v v eˆ
v v
v v eˆ
 1 2 eˆ 2  1 2 2  2 2 eˆ 2  2 2 2  3 2 eˆ 2  3 2 2
h1 x1
h1 x1 h2 x2
h2 x2 h3 x3
h3 x3
v
 1
 h1
v
 1
 h1

vv
v2 v2 v2 v3 v2 
v1v2
vv


eˆ 21  2 2 eˆ 22  3 2 eˆ 23
 eˆ 2 
x1 h2 x2 h3 x3 
h1
h2
h3
v2 v2 v2 v3 v2 
v1v2  h12 


 eˆ 2 
 eˆ1 
x1 h2 x2 h3 x3 
h1  h2 
h23  v3v2  h32 
v2v2  h21
eˆ 3  
  eˆ1 
 eˆ 3 
h2  h1
h3  h3  h2 
 v v v v v v 
v v h
vv h
vvh 
vvh 
  1 2 12  2 2 21  eˆ1   1 2  2 2  3 2  eˆ 2   3 2 32  2 2 23  eˆ 3
h2 h1 
h2 h3 
 h1h2
 h1 x1 h2 x2 h3 x3 
 h2 h3
 V  v3eˆ 3  
v   v3eˆ 3  v2   v3eˆ 3  v3   v3eˆ 3 
 1


h1 x1
h2 x2
h3 x3
v v eˆ
v v eˆ v v
v v eˆ
v v
v v
 1 3 eˆ 3  1 3 3  2 3 eˆ 3  2 3 3  3 3 eˆ 3  3 3 3
h1 x1
h1 x1 h2 x2
h2 x2 h3 x3
h3 x3
 v v v v v v 
vv
vv
vv
  1 3  2 3  3 3  eˆ 3  1 3 eˆ 31  2 3 eˆ 32  3 3 eˆ 33
h1
h2
h3
 h1 x1 h2 x2 h3 x3 
 v v v v v v 
 v v h

vv h
  1 3  2 3  3 3  eˆ 3  1 3  13 eˆ1   2 3  23 eˆ 2 
h1  h3  h2  h3 
 h1 x1 h2 x2 h3 x3 

vv  h
h
 3 3   31 eˆ1  32 eˆ 2 
h3  h1
h2 
vv h
v v h
 v v v v v v 
vvh 
vvh 
  1 3 13  3 3 31  eˆ1   2 3 23  3 3 32  eˆ 2   1 3  2 3  3 3  eˆ 3
h3h1 
h3h2 
 h1h3
 h2 h3
 h1 x1 h2 x2 h3 x3 
Pressure gradient p 
1 p
1 p
1 p
eˆ1 
eˆ 2 
eˆ 3
h1 x1
h2 x2
h3 x3
Viscous term  2 V
2V 

1   1  



 h2 h3v1    h3h1v2    h1h2v3    eˆ1


h1 x1  h1h2 h3  x1
x2
x3

 
   h  
1    h3  



 h1v1     2   h1v1    h3v3   eˆ1


  h2v2  
h2 h3  x2  h1h2  x1
x2
x1
  x3  h1h3  x3
  

1   1  


 h2 h3v1    h3h1v2    h1h2v3   eˆ 2


h2 x2  h1h2 h3  x1
x2
x3

    h3  
 
1    h1  


ˆ

h
v

h
v

h
v

h
v













3 3
2 2 
2 2
1 1 


  e2
h1h3  x3  h2 h3  x2
x3
x2
  
  x1  h1h2  x1

1   1  



 h3h1v2    h1h2v3   eˆ 3

  h2 h3v1  
h3 x3  h1h2 h3  x1
x2
x3



1
h1h2
   h2
 
 x1  h1h3
 
   h

 h1v1    h3v3     1

x1
 x3
  x2  h2 h3
 
 

 h3v3    h2v2   eˆ 3

x3
 x2
  
4.2.1 Combine terms in ê1 direction to get momentum equation in ê1 direction
v1 v1 v1 v2 v1 v3 v1 v1v2 h12 v2v2 h21 v1v3h13 v3v3h31







t h1 x1 h2 x2 h3 x3
h1h2
h2 h1
h1h3
h3h1
1 1 p

 h1 x1

1   1  


 h2 h3v1    h3h1v2    h1h2v3   


h1 x1  h1h2 h3  x1
x2
x3

 
   h  
1    h3  



 h1v1     2   h1v1    h3v3  


  h2v2  
h2 h3  x2  h1h2  x1
x2
x1
  x3  h1h3  x3
  

4.2.2 Combine terms in ê2 direction to get momentum equation in ê2 direction
v2 v2 v1h21 v1v1h12 v1 v2 v2 v2 v3 v2 v2 v3h23 v3v3h32







t
h1h2
h1h2
h1 x1 h2 x2 h3 x3
h2 h3
h3h2
1 1 p

 h2 x2

1   1  


 h3h1v2    h1h2v3  

  h2 h3v1  
h2 x2  h1h2 h3  x1
x2
x3

    h3  
 
1    h1  



h
v

h
v

h
v

h
v













3 3
2 2 
2 2
1 1 



h1h3  x3  h2 h3  x2
x3
x2
  
  x1  h1h2  x1

4.2.3 Combine terms in ê3 direction to get momentum equation in ê3 direction
v3 v3v1h31 v1v1h13 v3v2 h32 v2v2 h23 v1 v3 v2 v3 v3 v3







t
h3h1
h3h1
h2 h3
h2 h3
h1 x1 h2 x2 h3 x3
1 1 p

 h3 x3

1   1  


 h2 h3v1    h3h1v2    h1h2v3  


h3 x3  h1h2 h3  x1
x2
x3

   h  
 
1    h2  



 h1v1    h3v3     1   h3v3    h2v2   
 

h1h2  x1  h1h3  x3
x1
x3
  x2  h2 h3  x2
  

5. Example: Incompressible N-S equations in cylindrical polar systems
5.1 Continuity equation  V  0 in cylindrical coordinates  r, , z 
For cylindrical coordinates  r, , z  , h1  hr  1, h2  h  r , h3  hz  1
1 


  V    rvr  
 v    rvz    0
r  r

z

1 
1 


 rvr  
 v    vz   0
r r
r 
z
V
1
  V   V   p  2 V in cylindrical coordinates  r, , z 
t

For cylindrical coordinates  r, , z  , h1  hr  1, h2  h  r , h3  hz  1 and only h21  h r  1 , all
5.2 Momentum equation
others are zero.
5.2.1 The r-momentum equation:
vr
v v v
v v 2
1 p
1
2 v 

 vr r   r  vz r    
   2vr  2 vr  2  
t
r r 
z r
 r
r
r  

5.2.2 The  -momentum equation:
v 1
v v v
v
F
1 p
2 F 

 vr v  vr      vz   
    2 F2  22  2 1 
t r
r
r 
z
 r 
r
r  

5.2.3 The z-momentum equation:
vz
v v v
v
1 p
 vr z   z  vz z  
 2vz
t
r r 
z
 z