Dipole Point Sources

Consider the flow pattern generated by point source of strength $Q$ located on the symmetry axis at $z=+d/2$ , and a point source of strength $-Q$ (i.e., a point sink) located on the symmetry axis at $z=-d/2$ . It follows, by analogy with the analysis of the previous section, that the stream function and velocity potential at a general point, $P$ , lying in the meridian plane, are

$$\psi = \frac{Q}{4\pi}\left(\cos\theta_2-\cos\theta_1\right),$$ (7.36)

and

$$\phi = \frac{Q}{4\pi}\left(\frac{1}{r_2}-\frac{1}{r_1}\right),$$ (7.37)

respectively. Here, $r_1$ , $r_2$ , $\theta _1$ , and $\theta _2$ are defined in Figure 7.2.

Figure 7.2: A dipole source.

In the limit that the product $D=Q\,d$ remains constant, while $d\rightarrow 0$ , we obtain a so-called dipole point source. According to the sine rule of trigonometry,

$$\frac{r_1}{\sin\theta_2} = \frac{r_2}{\sin\theta_1} =\frac{d}{\sin(\theta_2-\theta_1)}.$$ (7.38)

However, $\sin(\theta_2-\theta_1)= 2\,\sin[(\theta_2-\theta_1)/2]\,\cos[(\theta_2-\theta_1)/2]$ , so we obtain

$$r_1-r_2 = \frac{d\,(\sin\theta_2-\sin\theta_1)}{2\,\sin[(\theta_2-\theta_1)/2]\,\cos[(\theta_2-\theta_1)/2]}.$$ (7.39)

In fact, $\sin\theta_2-\sin\theta_1 = 2\,\cos[(\theta_2+\theta_1)/2]\,\sin[(\theta_2-\theta_1)/2]$ , which leads to

$$r_1 - r_2= \frac{d\,\cos[(\theta_2+\theta_1)/2]}{\cos[(\theta_2-\theta_1)/2]}.$$ (7.40)

Thus, in the limit $\theta_2\rightarrow\theta_1\rightarrow \theta$ and $r_2\rightarrow r_1\rightarrow r$ , we get

$$r_1-r_2\rightarrow d\,\cos\theta.$$ (7.41)

Hence, according to Equation (7.37),

$$\phi = \frac{Q}{4\pi}\left(\frac{r_1-r_2}{r_1\,r_2}\right)\rightarrow \frac{D}{4\pi}\,\frac{\cos\theta}{r^{\,2}}.$$ (7.42)

Equation (7.36) implies that

$$\psi = \frac{Q}{4\pi}\left(\frac{z-d/2}{r_2}-\frac{z+d/2}{r_1}\ri... ...frac{1}{r_1}\right)-\frac{d}{2}\left(\frac{1}{r_2}+\frac{1}{r_1}\right)\right].$$ (7.43)

Thus, in the limit $\theta_2\rightarrow\theta_1\rightarrow \theta$ and $r_2\rightarrow r_1\rightarrow r$ , we obtain

$$\psi\rightarrow \frac{Q}{4\pi}\left(\frac{d\,\cos^2\theta}{r}-\fr... ...\theta}{r} =-\frac{D}{4\pi}\,\frac{\varpi^{\,2}}{(\varpi^{\,2}+z^{\,2})^{3/2}},$$ (7.44)

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

Figure 7.3: Contours of the stream function of a dipole 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 $\beta_0$ , $\alpha_1$ , and $\beta_1$ correspond to a point source at the origin, uniform flow parallel to the $z$ -axis, and a dipole point source at the origin, respectively. Of course, the term involving $\alpha _0$ is constant, and, therefore, gives rise to no flow.