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# Dipole Point Sources

Consider the flow pattern generated by point source of strength located on the symmetry axis at , and a point source of strength (i.e., a point sink) located on the symmetry axis at . It follows, by analogy with the analysis of the previous section, that the stream function and velocity potential at a general point, , lying in the meridian plane, are

 (7.36)

and

 (7.37)

respectively. Here, , , , and are defined in Figure 7.2.

In the limit that the product remains constant, while , we obtain a so-called dipole point source. According to the sine rule of trigonometry,

 (7.38)

However, , so we obtain

 (7.39)

In fact, , which leads to

 (7.40)

Thus, in the limit and , we get

 (7.41)

Hence, according to Equation (7.37),

 (7.42)

Equation (7.36) implies that

 (7.43)

Thus, in the limit and , we obtain

 (7.44)

where use has been made of Equation (7.41), as well as the fact that . Figure 7.3 shows the stream function of a dipole point source located at the origin.

Incidentally, Equations (7.26), (7.28), (7.33), (7.35), (7.42), and (7.44) imply that the terms in the expansions (7.23) and (7.24) involving the constants , , and correspond to a point source at the origin, uniform flow parallel to the -axis, and a dipole point source at the origin, respectively. Of course, the term involving is constant, and, therefore, gives rise to no flow.

Next: Flow Past a Spherical Up: Axisymmetric Incompressible Inviscid Flow Previous: Point Sources
Richard Fitzpatrick 2016-03-31