Streamlines, Velocity Potential for a Doublet

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  

(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

(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.