Let's think of a source and sink pair with strengths $ \Lambda $ and $ -\Lambda $, respectively. Now let's say they are a distance l apart and we were to move them closer to one another. This would lead to a singularity called a doublet, which is frequently used in incompressible flow theory.
Let there be a source and sink as described above. For ant point P in the flow, the stream function would be
[ 1 ]
$$ \psi = \frac{\Lambda}{2 \pi} (\theta_1 - \theta_2) = -\frac{\Lambda}{2 \pi} \Delta \theta $$
where $ \Delta \theta = \theta_2 - \theta_1 $ as seen in Figure 1.
If we let l be the distance between the source and sink, as l approaches zero and the magnitudes of the source and sink strengths increase such that $ l \Delta $ remains constant, we obtain a special flow pattern called a doublet. The strength of a doublet is denoted as $ \kappa $ and is defined as $ \kappa \equiv l \Delta $. Hence, the stream function for a doublet is
[ 2 ]
$$ \psi = \lim_{\begin{matrix} l \rightarrow 0 \\ \kappa = l \Delta = const \end{matrix}} (-\frac{\Lambda}{2 \pi} d\theta) $$
Fig. 1 - Image of a Source-Sink pair and its limiting case for a Doublet
Using the notation in Figure 1, as $ \Delta \theta \rightarrow d\theta \rightarrow 0 $, we can say that
[ 3 ]
$$ a = l \sin \theta $$ $$ b = r - l \cos \theta $$ $$ d\theta = \frac{a}{b} = \frac{l \sin \theta}{r - l \cos \theta} $$
Hence leaving us with
[ 4 ]
$$ \psi = \lim_{\begin{matrix} l \rightarrow 0 \\ \kappa = const \end{matrix}} (-\frac{\kappa}{2 \pi} \frac{l \sin \theta}{r - l \cos \theta}) $$
This leaves a stream function of
[ 5 ]
$$ \psi = -\frac{\kappa}{2 \pi} \frac{\sin \theta}{r} $$
for doublets. Now, if we wanted to get velocity potential for a doublet, that would be given by
[ 6 ]
$$ \psi = \frac{\kappa}{2 \pi} \frac{\cos \theta}{r} $$
Now, let's take a look at a doublet.
Fig. 2 - Image of a doublet with strength kappa.
We will talk about doublets more in later tutorials and how they can be superimposed with uniform flows!