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NPTEL Course Developer for Fluid Mechanics
Module 04; Lecture 29
Dr. Niranjan Sahoo
IIT-Guwahati
DYMAMICS OF FLUID FLOW
BASIC PLANE POTENTIAL FLOWS
One of the major advantages of Laplace equation is the linearity of partial differential
equation. Arithmetic operations (e.g. addition, subtraction etc.) can be performed for the
solutions of these equations. It has lot of practical implications that leads to interesting
solutions of complicated flow problems. Some of the basic potential flows are discussed
below.
Uniform Flow
It is the simplest type of flow in which the streamlines are straight and parallel. The
magnitude of the velocity is constant. Fig. 1 (a) and (b) shows the uniform flow in the
positive x-direction. Mathematically, the flow represented in Fig. 1(a) can be expressed
as,
u  U and v  0
y
U
 

(1)
y
 
U








x
(b)
(a)
x
Fig. 1: Schematic of a uniform flow: (a) positive x-direction; (b) any arbitrary direction.
In terms of velocity potential and stream function, we can write,

U
x

U
y

0
y

0
x
(2)
(3)
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NPTEL Course Developer for Fluid Mechanics
Module 04; Lecture 29
Dr. Niranjan Sahoo
IIT-Guwahati
The above two equations can be integrated (constants of integration may be discarded as
it does not affect the velocities in the flow) to yield,
  Ux and   Uy
(4)
If the uniform flow is at an angle  with respect to positive x-direction, then
  U  x cos   y sin  
  U  y cos   x sin  
(5)
Source and Sink
Source and sink are the hypothetical terms used in fluid flow where it is assumed that the
flow takes place radially (inward/outward) from origin. Consider a radial fluid flow
outward from a line through origin as shown in Fig. 2. If m is the mass flow rate (per
unit length) along the radial line from the origin, vr and v are the tangential and radial
velocities, then by conservation of mass principle, we can write,
m   2 r  .vr
constant
r
or vr 
m
2 r
(6)
constant

Fig. 2: Streamline and equipotential lines for source.
Since the flow is purely radial, so v  0 . By definition of velocity potential in streamline
coordinate system, we have

m

,
r 2 r
1 
0
r 
(7)
Integrating Eq. (7) and putting the constant of integration to zero,

m
ln r
2
(8)
In this expression, if m is positive, then the flow will be radially outward and is treated
as “source flow”. A “sink flow” will occur when the flow is towards origin ( m is
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NPTEL Course Developer for Fluid Mechanics
Module 04; Lecture 29
Dr. Niranjan Sahoo
IIT-Guwahati
negative). The radial velocity becomes infinite at r  0 which is practically impossible.
Thus, sources and sinks do not really exist in real flow fields rather some real flows can
be approximated at points away from the origin by using sources and sinks.
The stream function for the source can be defined such that

1 
m
 0,

r
r  2 r
m


2
(9)
It may be inferred from Eqs. (8) & (9) and Fig. 2 that the streamlines   constant  are
the radial lines and the equipotential lines   constant  are concentric circles centered
at the origin.
Vortex Flow
The streamline patterns in a vortex flow are the concentric circles and the equipotential
lines are along the direction of radial lines (Fig. 3).
constant

constant
r
Fig. 3: Streamline and equipotential lines for vortex.
Hence, the equation of motion for streamlines and velocity potentials can be written as,
  K
and
   K ln r
(10)
By definition of streamlines and velocity potentials, we have,
vr 

1 
1 

 0 , v 
 0, v  
; vr 
r
r 
r 
r
So,
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NPTEL Course Developer for Fluid Mechanics
Module 04; Lecture 29
v 
Dr. Niranjan Sahoo
IIT-Guwahati
1 
 K


r 
r
r
(11)
It indicates that the tangential velocity varies inversely with distance from the origin with
becomes infinite at origin  r  0 .
A vortex motion may be ‘rotational or irrotational’ depending on the orientation of
fluid element in the flow field. The irrotational vortex will occur when the fluid element
does not rotate about its own axis and is not decided by the path followed by the element.
The irrotational vortex is also called as “free vortex” and is governed by Eq. (10). In case
of rotational vortex (also referred to “forced vortex”) the fluid element is artificially
rotated with certain angular velocity   about its axis. So, the constant K in Eq. (10) is
replaced by  . A “combined vortex” may be defined as a forced vortex with central core
and free vortex behavior outside the core. Mathematically, it is written as,
v   r
r  r0
K
r
r  r0
v 
(12)
where r0 corresponds to the radius of the central core.
Circulation   
A vortex motion is mathematically associated with a term called ‘circulation” which is
defined as the line integral of the tangential component of the velocity taken around a
closed curve in the flow field. It is expressed as,
   V .ds
(13)
C
where the integration is performed around any arbitrary closed curve C , V is the
velocity vector and ds is the differential length along the curve as shown in Fig. 4. For
irrotational flow,
V   so that
V .ds   .ds  d
(14)
Therefore,


C
d  0
(15)
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NPTEL Course Developer for Fluid Mechanics
Module 04; Lecture 29
Dr. Niranjan Sahoo
IIT-Guwahati
Any arbitrary curve C
V
ds
Fig. 4: Circulation around a closed curve C .
In general, the ‘circulation’ is zero for irrortational flow. However, it is not true in case of
‘free vortex’ defined by   K (Eq. 11) where the circulation around a circular path can
be represented by,
2

 Kd  2 K
so that K 
0

2
(16)
Now, the velocity potential and stream function for free vortex can be expressed in terms
of circulation as,



2
and   

ln r
2
(17)
Doublet
The source and sink of equal strength located along same axis can be combined to form
another basic flow known as “doublet” (Fig. 5). The combined stream function for the
pair can be written as,
 
m
1   2 
2
y
(18)
P
r
r

r


x
a
source
a
sink
Fig. 5: Schematic representation of a doublet.
It follows from Eq. (18) that
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NPTEL Course Developer for Fluid Mechanics
Module 04; Lecture 29
Dr. Niranjan Sahoo
IIT-Guwahati
tan 1  tan 2
 2 
tan  
  tan 1  2  
1  tan 1 tan 2
 m 
(19)
From Fig. 5, we get,
tan 1 
r sin 
,
r cos   a
tan  2 
r sin 
r cos   a
(20)
Now, Eq. (19) can be simplified as,
 2
tan  
 m
m
 2ar sin 
 2ar sin  
or   
tan 1  2
 2
2
2 
2
 r a
 r a 
 2ar sin 
For small angles, tan 1  2
2
 r a
(21)
 2ar sin 
, so from Eq. (21)
 2
2
 r a
 
mar sin 
 r 2  a2

(22)

In Eq. (22), if a  0 and m   keeping the product
ma

constant, then
r
1

2
r a
r
2
Hence,
 
where K 
ma

K sin 
r
(23)
is called the strength of doublet. Thus, a doublet is formed as a source
and sink of equal strength approach one another while increasing their strength. The
stream lines are governed by Eq. (23) and the corresponding velocity potential is,

K cos 
r
(24)
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NPTEL Course Developer for Fluid Mechanics
Module 04; Lecture 29
Dr. Niranjan Sahoo
IIT-Guwahati
Example 1
The tangential velocity variation of velocity in a washbasin is given by, V 
c
. Referring
r
to the following figure, dtermine the circulation: (a) around a closed curve formed by two
streamlines with r  R1 and r  R2 and two radius vectors with an angle  between
them; (b) around a closed curve in the form of concentric circle of radius R1
B
R1
C

D
A
R2
Solution:
By definition, circulation is defined by,


C
V .ds
Referring to above figure, there is no velocity along the radial direction i.e.
 BC   DA  0
Circulation around ABCD,
 ABCD   AB   BC  CD   DA
 2 K  0  2 K  0  0
The circulation in the form of concentric circle of radius R1  2 K
This is a free vortex.
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NPTEL Course Developer for Fluid Mechanics
Module 04; Lecture 29
Dr. Niranjan Sahoo
IIT-Guwahati
EXERCISES
1.
Consider the superposition of a source with a uniform flow stream as shown in the
following figure.
y
P
U
r
r

x
b
uniform flow
source
The stagnation point is created at a distance b from the source where velocities for both
the flow are equal in magnitude and opposite in direction. If m is volume flow rate
emanating from the line of the source, U is the uniform velocity, determine;
(a) Location of stagnation point ‘b’
(b) Radial and tangential velocity at a point ‘P’ downstream of the flow as shown in
the figure
2.
A two-dimensional incompressible flow field described by equation V  Cr in
which V is the tangential velocity and C is a constant. Determine circulation: (i) around
a circle with radius R ; (ii) around a closed path formed by the arcs of two circles of radii
R1 and R2 with an angle  between them. Also, find the vorticity of the flow.
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