...contd...Analysis of Potential Flows through Complex Variables

Let us combine a source and sink now. Refer to Figure 21.4. The complex potential is given by

(22.24)

This follows

with

and

We know,

or,

Therefore,

(22.25)

We also find

and

That results in

(22.26)

Under the limiting condition of   (the flow becomes a doublet)

We also know,

The strength of the doublets given by

We get family of velocity potential from  constant lines

or,

or,

(22.27)

This is the equation for constant lines (also see Figure 21.5)

Now from equation (22.25), for the limiting case of , one can write

[as under the limiting case]

or,

or,

The streamlines associated with the doublet are

constant lines, which can be expressed as

or,

or,

or,

(22.28)

This is the equation for constant lines  (see Figure 21.5)

Therefore for the doublet

and,

It can also be written as

(22.30)

If the represents a doublet.

Under the limiting conditions,

(22.31)

Thus, by using the elementary complex potential for source, sink, doublet, uniform flow, vortex flow etc more complicated fields can be constructed via the method of superposition. Especially, external flow past objects of various shapes can be simulated. However, for the first course, we shall follow simpler approach and construct various complex flows without using the route of complex potentials. We shall take up such exercises in the following lectures.