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Elementary Singularities
We explore some special flows now, which satisfy the Laplace equation, but are
physically unrealistic. The interesting fact about these is, although they are singular
in nature, they can provide physically meaningful flows when they are combined
with other flows. We will use the flows mostly in cylindrical coordinates.
Source Flow: A source flow is defined at a point as the flow that creates new fluid
particles continuously. In 2-dimensions a source located at the origin will create
fluid streamlines as shown below:
y

r
(r,)

x
Source flow
y
Since the streamlines are all radial, the
source flow velocity components may be
written as Vr  0 , V  0 . We define the
E
x
strength of a source, q, as the volumetric flow
per unit depth through any closed circuit
enclosing the source.
Volumetric
flux by


definition is Q   V  dA .
A
d
W  Depth

 dA  (  d)  Wê r

 V  Vr êr  Vê


V  dA  W  d Vr
Let
Let us choose the circuit as a circle of radius,
E.
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 
Q   V  dA  WE  V d
2
Or,
 0
A
r
2
Q
q 
 E  Vr d
W
0
Because of the symmetry about the origin, Vr will not be a function of . Thus,
q
q through the circle of radius E. In general, Vr 
q  V (2 ) or,
V 
(2)r
(2 ) E
through a circle of radius r. Note that the appearance of Vr indicates the flow has an
infinite Vr as r   . Thus, a source flow is considered as singular at the origin. We
may define a sink flow in the same manner as a “negative source” (q < 0).
r
r
Let us find the stream function for a source (or, sink) flow.
q
1 
V 

2r r 
r
and
V  

0
r
It’s easy to show by integration:

source
( or , sink )

q
2
Note that we have omitted the constant of integration in the formula above. First, if
the constant is dropped it does not change the velocity field at all (velocity
components involve derivatives of ). Moreover, to plot streamlines, we must set the
q
  constant 
, and select different values of the constant.
2
Henceforth the remaining singularity
y
functions will be presented without additional
constants in their representation.
x
Vortex Flow: A counterclockwise vortex
located at the origin has circular streamlines
as shown.
In cylindrical coordinates, this means
Vortex Flow
Vr  0 , V  0 . Furthermore, we define the
strength of the vortex, , as the circulation around any closed curve enclosing the
vortex.
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 
As before, let the closed curve be a circle of radius E. Since circulation   V  d r :
C

d r  E d ê 
 
V  dr  V E d
2
 
    V  d r  V E  d
y
C
d
E
Vortex
x
0
 E V (2)
or, V   around circle of radius, E.

2E
[ V is not a function of , by symmetry]
C

, Vr  0 for a counterclockwise vortex located at the origin. It
2E
yields a stream function given by:

 vortex   ln r
2

ln r . Again, these are singular at the
Similarly, a clockwise vortex will give  
2
origin (as r  0 ).
In general, V 
Doublet Flow: A doublet (or, dipole) is like an electric magnet. It produces a
streamline pattern same as what you have seen in physics by spreading “iron dust”
on a piece of paper with a magnet underneath.
A doublet is obtained by bringing a
source and a sink close together.
Assume that a source and sink of
strength “q” and”-q” are placed at a
distance “l” apart. As the two
Axis of the doublet
singularities are brought closer to each
other (i.e., l  0), suppose we hold
x
ql    constant. Then we will create
the flow field given by the streamline
pattern to the left. A doublet (unlike
source or vortex) has no symmetry at
Doublet Flow
the origin. Also it has an axis as shown
(directed from the sink to the source inside the doublet). The relevant velocity
component and stream function representation is given below:
y
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Vr 
 cos 
2r 2
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V 
 sin 
2r 2
 doublet 
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 sin 
2r
The above formulae hold for a doublet axis along the positive x-axis.
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