We can now form different flow patterns by superimposing the velocity potential and stream functions of the elementary flows stated above.     

In order to develop a doublet, imagine a source and a sink of equal strength K at equal distance s from the origin along x-axis as shown in Fig. 21.4.

Fig 21.4      Superposition of a Source and a Sink

From any point p(x, y) in the field, r₁ and r₂ are drawn to the source and the sink. The polar coordinates of this point (r, θ) have been shown.

The potential functions of the two flows may be superimposed to describe the potential for the combined flow at P as

(21.11)

Similarly, 

	(21.12)
where

Expanding θ₁ and θ₂ in terms of coordinates of p and s

(21.13)

(21.14)

Using  

we find

or,

Hence the stream function and the velocity potential function are formed by combining Eqs (21.12) and (21.13), as well as Eqs(21.11) and (21.14) respectively

Hence     -------    Stream Function	(21.15)
----- Potential Function	(21.16)

Doublet is a special case when a source as well as a sink are brought together in such a way that

•     and at the same time the

• strength   is increased to an infinite value.

These are assumed to be accomplished in a manner which makes the product of s and (in limiting case) a finite value c

This gives us