Complex Velocity Potential

Equations (6.17)-(6.18) are identical to Equations (5.21)-(5.22). This suggests that the real and imaginary parts of a well-behaved function of the complex variable can be interpreted as the velocity potential and stream function, respectively, of some two-dimensional, irrotational, incompressible flow pattern.

For instance, suppose that

$$F(z)=-V_0\,z,$$ (6.23)

where $V_0$ is real. It follows that

$$\phi (r,\theta) =-V_0\,r\,\cos\theta,$$ (6.24)

$$\psi(r,\theta) =-V_0\,r\,\sin\theta.$$ (6.25)

It can be seen, by comparison with the analysis of Section 5.4, that the complex velocity potential (6.23) corresponds to uniform flow of speed $V_0$ directed along the $x$ -axis. Furthermore, as is easily demonstrated, the complex velocity potential associated with uniform flow of speed $V_0$ whose direction subtends a (counter-clockwise) angle $\theta_0$ with the $x$ -axis is $F(z)=-V_0\,z\,{\rm e}^{-{\rm i}\,\theta_0}$ .

Suppose that

$$F(z)=- \frac{Q}{2\pi}\,\ln z,$$ (6.26)

where $Q$ is real. Because $\ln z = \ln r + {\rm i}\,\theta$ (Riley 1974), it follows that

$$\phi (r,\theta) =-\frac{Q}{2\pi}\,\ln r,$$ (6.27)

$$\psi(r,\theta) =-\frac{Q}{2\pi}\,\theta.$$ (6.28)

Thus, according to the analysis of Section 5.5, the complex velocity potential (6.26) corresponds to the flow pattern of a line source, of strength $Q$ , located at the origin. (See Figure 5.3.) As a simple generalization of this result, the complex potential of a line source, of strength $Q$ , located at the point $(x_0$ , $y_0)$ , is $F(z)=-(Q/2\pi)\,\ln(z-z_0)$ , where $z_0=x_0+{\rm i}\,y_0$ . It can be seen, from Equation (6.26), that the complex velocity potential of a line source is singular at the location of the source.

Suppose that

$$F(z) = {\rm i}\,\frac{\mit\Gamma}{2\pi}\,\ln z,$$ (6.29)

where ${\mit\Gamma}$ is real. It follows that

$$\phi (r,\theta) =-\frac{\mit\Gamma}{2\pi}\,\theta,$$ (6.30)

$$\psi(r,\theta) =\frac{\mit\Gamma}{2\pi}\,\ln r.$$ (6.31)

Thus, according to the analysis of Section 5.6, the complex velocity potential (6.29) corresponds to the flow pattern of a vortex filament of intensity ${\mit\Gamma}$ located at the origin. (See Figure 5.5.) As a simple generalization of this result, the complex potential of a vortex filament, of intensity ${\mit\Gamma}$ , located at the point $(x_0$ , $y_0)$ , is $F(z)={\rm i}\,({\mit\Gamma}/2\pi)\,\ln(z-z_0)$ , where $z_0=x_0+{\rm i}\,y_0$ . According to Equation (6.29), the complex velocity potential of a vortex filament is singular at the location of the filament.

Suppose, finally, that

$$F(z)= -V_0\left(z+\frac{a^{\,2}}{z}\right) + {\rm i}\,\frac{\mit\Gamma}{2\pi}\,\ln\left(\frac{z}{a}\right),$$ (6.32)

where $V_0$ , $a$ , and ${\mit\Gamma}$ , are real. It follows that

$$\phi (r,\theta) =-V_0\left(r+\frac{a^{\,2}}{r}\right)\cos\theta-\frac{\mit\Gamma}{2\pi}\,\theta,$$ (6.33)

$$\psi(r,\theta) =-V_0\left(r-\frac{a^{\,2}}{r}\right)\sin\theta + \frac{\mit\Gamma}{2\pi}\,\ln\left(\frac{r}{a}\right).$$ (6.34)

Thus, according to the analysis of Section 5.8, the complex velocity potential (6.32) corresponds to uniform flow of unperturbed speed $V_0$ , running parallel to the $x$ -axis, around an impenetrable circular cylinder of radius $a$ , centered on the origin. (See Figures 5.6, 5.7, and 5.8.) Here, ${\mit\Gamma}$ is the circulation of the flow about the cylinder. It can be seen that $\psi=0$ on the surface of the cylinder ($r=a$ ), which ensures that the normal velocity is zero on this surface (because the surface corresponds to a streamline), as must be the case if the cylinder is impenetrable.