# Stream Function _ Velocity Potential

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```					                   Stream Function & Velocity Potential
    Stream lines/ Stream Function (Y)
 Concept
 Relevant Formulas
 Examples
 Rotation, vorticity

    Velocity Potential(f)
 Concept
 Relevant  Formulas
 Examples
 Relationship between stream function and velocity
potential
 Complex velocity potential
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Lines
    Consider 2D incompressible flow
    Continuity Eqn

                 
 Vx   Vy   Vz   0
t x       y       z

                                    Vx   
Vx   Vy   0          Vy         dy
x        y                          x    
    Vx and Vy are related
    Can you write a common function for both?
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Function
       Assume                   
Vx 
y
   Then
 Vx              2 
Vy           dy     xy dy
 x                     
  2         
       dy       
 yx        x 
   Instead of two functions, Vx and Vy, we need to solve
for only one function   Stream Function
   Order of differential eqn increased by one

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Function
   What does Stream Function  mean?

   Equation for streamlines in 2D are given by
 = constant
   Streamlines may exist in 3D also, but stream function
does not
 Why?    (When we work with velocity potential, we may
get a perspective)
 In 3D, streamlines follow the equation

dx dy dz
   
Vx V y Vz

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation
   Definition of rotation


Vx    y Dy

y                        Dy            Time=t


Vx     y                   Dx
Vy                      Vy
x                       x Dx

x
d    
ROTATION   z 
Assume Vy|x < Vy|x+Dx
     
and Vx|y > Vx|y+Dy                                                       dt  2 
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation
   To Calculate Rotation

Dy1
tan  
Dx

Dy1  Vy          x Dx
 
Dt  Vy
x
Dt       
Dy1

Dx

  arctan
V               y x Dx    Vy
x
 Dt
Dx
   Similarly
  arctan

 Vx y Dy Vx              y    Dt
Dy
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation
    To Calculate Rotation

d      1                       t 
ROTATION   z              lim                                 
t Dt
       2 Dt 0              Dt              
dt  2                                          

V
arctan 

 y x Dx Vy        x
 Dt 




V
 x y Dy Vx
arctan 
y    Dt 



        Dx                   1                       Dy             
 1                                                                                     
   lim                                               Dt 0
lim
 2  Dxt 0 0
D                 Dt                          2                  Dt

Dx  0
Dy  0
Dy  0
    For very small time and very small element, Dx, Dy
and Dt are close to zero

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation
   To Calculate Rotation
   For very small , i.e. ~ 0
sin                      cos  1                       tan   

 arctan   
V
 y
 arctan 
      x Dx
Vy
x
 Dt   V
           y x Dx   Vy
x
 Dt

                      Dx                                      Dx
                                            

 V
 y
arctan 
     x Dx
Vy
x
 Dt 


 V
 y

    x Dx
Vy
x
 Dt 


                Dx                                      Dx
lim                                            lim 
                                 

Dt 0                   Dt                           Dt 0                Dt
Dx  0                                               Dx  0
Dy  0                                               Dy  0
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation
   To Calculate Rotation

lim
V   y x Dx   Vy
x
  V   y
Dx 0          Dx                 x

V       
 y x Dx Vy
arctan                        x
 Dt 



V
 x y Dy Vx
arctan 
y    Dt 


        Dx                          1                        Dy             
 1                                                                                              
z    lim                                                       Dt 0
lim
 2 DxDt 00
               Dt                                  2 Dx  0           Dt
Dy  0                                                  Dy  0

Simplifies to
 1   Vy Vx 
z              
 2   x   y 
1
z     V 
2
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation in terms of Stream Function
   To write rotation in terms of stream functions
                        
Vx                        Vy       
y                         x 

 1   Vy Vx   1       
2   2
z                   2  2 
 2   x   y   2   x  y 
1
     2
2
   That is
 2  2 z  0

   For irrotational flow (z=0)

 2  0

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Rotation and Potential

    For irrotational flow (z=0)
1
2

z    V  0      
 V  0

Vy       Vx
       0
x        y

    This equation is “similar” to continuity equation
    Vx and Vy are related
    Can we find a common function to relate both Vx
and Vy ?

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Vy        Vx
       0
Velocity Potential                                        x         y

    Assume
f                 Then             Vy V       2f       2f
Vx                
 x             
x                                  x   y     y x     x y

f
Vy 
y

    In 3D, similarly it can be shown that
f
Vz 
z
    f is the velocity potential

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Velocity Potential vs Stream
Function

Stream Function () Velocity Potential (f)
only 2D flow                all flows
Irrotational (i.e. Inviscid or
viscous or non-viscous flows zero viscosity) flow
Exists
for

    In 2D inviscid flow (incompressible flow OR steady
state compressible flow), both functions exist
    What is the relationship between them?

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Stream Function- Physical
meaning
              2D (viscous or inviscid) flow
Statement: In
(incompressible flow OR steady state compressible
flow),  = constant represents the streamline.
    Proof

   If  = constant, then d0
           
d       dx          dy
 x         y 
  Vy  dx  Vx  dy
0                                  Vy
   If  = constant, then
dy Vy                   Vx

dx Vx
IIT-Madras, Momentum Transfer: July 2005-Dec 2005

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