We have seen in the last lecture that the streamlines associated with the doublet are
 |
If we replace sinθ by y/r, and the minus sign be absorbed in C1 , we get
 |
 | (21.17a) |
Putting 
we get
we get  | (21.17b) |
Equation (21.17b) represents a family of circles with
- radius :
- centre :
- For x = 0, there are two values of y, one of them=0.
- The centres of the circles fall on the y-axis.
- On the circle, where y = 0, x has to be zero for all the values of the constant.
- family of circles formed(due to different values of C1 ) is tangent to x-axis at the origin.
These streamlines are illustrated in Fig. 21.5.
Fig 21.5 Streamlines and Velocity Potential Lines for a Doublet
Due to the initial positions of the source and the sink in the development of the doublet , it is certain that
- the flow will emerge in the negative x direction from the origin
and
- it will converge via the positive x direction of the origin.
Velocity potential lines
 |
In cartresian coordinate the equation becomes
 | (21.18) |
Once again we shall obtain a family of circles
- radius:
- centre:
- The centres will fall on x-axis.
- For y = 0 there are two values of x, one of which is zero.
- When x = 0, y has to be zero for all values of the constant.
- These circles are tangent to y-axis at the origin.
In addition to the determination of the stream function and velocity potential, it is observed that for a doublet
 |
As the centre of the doublet is approached; the radial velocity tends to be infinite.
It shows that the doublet flow has a singularity.
Since the circulation about a singular point of a source or a sink is zero for any strength, it is obvious that the circulation about the singular point in a doublet flow must be zero i.e. doublet flow 
=0
=0 | (21.19) |
Applying Stokes Theorem between the line integral and the area-integral
 | (21.20) |
From Eq. 21.20 the obvious conclusion is 
i.e., doublet flow is an irrotational flow.
i.e., doublet flow is an irrotational flow.