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.
