 S.N.P.I.T & R.C,UMRAKH
GUJRARAT TECHNICHAL UNIVERSITY
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SUBJECT: FLUID MECHANICS
(B.E III SEM-2014)
TOPIC NAME:
STREAM FUNCTION AND
FLOW NET
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Prepared By :
 PATEL FENIL M.:130490106075
 PATEL GHANSHYAM S.:130490106076
 PATEL HINAL K.:1304901069078
 PATEL KISHAN J.:130490106079
 PATEL KISHAN B.:130490106080
Guide by :
Bankim R. Joshi
Sarika G. Javiya
Kartila D. Uchdadiya
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 Stream line:
• A function is an imaginary line drawn through
the flow field in such a way that the velocity
vector of the fluid at each and every point on
the streamline is tangent to the streamline at
that instant.
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• Steady flow:
When the velocity at each location is constant,
the velocity field is invarient with time and the flow
is said to be steady.
• Uniform flow Uniform flow: occurs when the
magnitude and direction of velocity do not change
from point to point in the fluid. Flow of liquids
through long pipelines of constant diameter is
uniform whether flow is steady or unsteady.
• Non-uniform flow occurs when velocity, pressure
etc., change from point to point in the fluid.
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• Steady, unifrom flow:
• Conditions do not change with position or time.
e.g., Flow of liquid through a pipe of uniform
bore running completely full at constant velocity.
• Steady, non-unifrom flow: Conditions change
from point to point but do not with time. e.g.,
Flow of a liquid at constant flow rate through a
tapering pipe running completely full.
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• Unsteady, unifrom Flow: e.g. When a pump
starts-up.
• Unsteady, non-unifrom Flow: e.g. Conditions
of liquid during pipetting out of liquid.
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Uniform flow:
• A uniform flow consists of a velocity field
where ~V = uˆı + vˆj is a constant. In 2-D, this
velocity field is specified either by the
freestream velocity components u1, v1, or by
the freestream speed V1 and flow angle α.
u = u1 = V1 cos α
v = v1 = V1 sin α
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 Velocity potential function:
• We can define a potential function,! (x, z, t) ,
as a continuous function that satisfies the
basic laws of fluid mechanics: conservation of
mass and momentum, assuming
incompressible, inviscid and irrotational flow
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• Mathematically , the velocity potential is
defined as
• i = f(x,y,z) for steady flow such that
u=-di / dx
v=-di / dy
z=-di / dz
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• The continuity equation for an incompressible
steady flow is
du / dx + dv / dy + dw /dz.
Substituting the values of u , v and w we get
• “Laplace eqation”.
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 STREAM FUNCTION:
• Let A and be the two points lying on the
streamlines prescribed by some numerical
system is called stream function.
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• (a) We will show that if conservation of mass
(continuity) is:

1 D ui

 Dt xi
Then for an incompressible or slightly compressible fluid
D
0
Dt

ui
0
xi
or   u  0
~
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(b) Iff
u  f ( x3 ) or u  f ( z )
~
~
u1 u 2

0
x1 x2
or
u v

0
x y
(c) A function
 ( x, y , t )
can be defined such that
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Check
from
  
 
x  y
(d) Whenever
 u  0 
~
u v
 0
x y
B
    
   
0
 y  x 
can be defined
  f ( x, y )


 d 
dx 
dy
x
y
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
dx
dy




y
x


dx 
dy  0
x
y
 From
C
&
D
d  0
along a streamline
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(e) Finally, it can be shown (see Kundu Sec.) that
y

d


x
The volume flowrate between streamlines is numerically equal to the
difference in their  values.
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E.g. flow around a cylinder:
x
2
3
x
2
1

2 x32

1
 u1 
 1 2

2
x3
x3  x1
x32  x12

 

2
3
2
1
z
  x3 
x3
0
-1
-2
-3
-3
-2
-1
0
x
1
2
3
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Vortex flows:
(a) Vorticity is the curl of the velocity field
(b) Vorticity is also the circulation per unit
area
From Stokes Theorem
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- “Component of vorticity through a surface A
bounded by C equals the line integral of the
velocity around C.”
- If we define circulation
Then
    n dA
A ~
~
Circulation = Total amount of vorticity ┴ to a
given area; or flux of vorticity through a
given area.
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 FLOW NET:
• A grid obtained by drawing a series of
streamlines and equipotential lines is known
as a flow net.
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• From equation:
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Let two curves c= constant and s=constant,
intersect each other at any point. At the point
of intersect the slope are:
slope= d y / dx =( -dc / dx ) / (dc / dy)= -u/v
for the curve s=constant:
Slope= d y /d x=( -ds / dx ) / ( ds /dy)= v/u
now , product of the slope of these curves:
= -u / v . V / u= - 1
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 Typical flow nets:
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 Uses:
(1) The velocity at any point can be calculated if
the velocity at any reference point is known.
(2) It assists in determining efficient boundary
shape for which the flow does not separate
from boundary shape.
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(3) The net analysis developed for ideal fluids
can be used for the fluids particularly outside
boundary layer as the viscous forces diminish
rapidly outside boundary layer.
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