The doublet solution is generated by a superposition of a source and a sink. It is such a basic element in the superposition of complex flows that here, it is considered a fourth-elementary flow.
Consider a source-sink pair of equal strength placed a distance h apart along the x-axis, as shown in Fig. 4.11. For convenience, assume that the left singularity (i. e., the source) is located at the origin of coordinates.
At a Point P, which is arbitrarily located, ¥, = ^0, and y2 = -^02 by
applying Eq. 4.31. Superposing (adding) these two stream functions results in a new stream function that also satisfies the Laplace’s Equation and represents a new, more complicated flow field. Thus,
¥ = ¥i + ¥2= 2П(01-02) = 2П(-Д0)
because 01=л-Д0 and 02 =n-(n-0,-A0).