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Next: Complex Velocity Up: Two-Dimensional Potential Flow Previous: Cauchy-Riemann Relations


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

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

where $ V_0$ is real. It follows that

$\displaystyle \phi (r,\theta)$ $\displaystyle =-V_0\,r\,\cos\theta,$ (6.24)
$\displaystyle \psi(r,\theta)$ $\displaystyle =-V_0\,r\,\sin\theta.$ (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

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

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

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

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

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

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

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

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

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


next up previous
Next: Complex Velocity Up: Two-Dimensional Potential Flow Previous: Cauchy-Riemann Relations
Richard Fitzpatrick 2016-03-31