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Schwarz-Christoffel Theorem

The Schwarz-Christoffel theorem is an important mathematical result that allows a polygonal boundary in the $ z$ -plane to be mapped conformally onto the real axis, $ \eta=0$ , in the $ \zeta $ -plane. It is conventional to map the region inside the polygon in the $ z$ -plane onto the upper half, $ \eta>0$ , of the $ \zeta $ -plane. If the interior angles of the polygon are $ \alpha $ , $ \beta $ , $ \gamma$ , $ \cdots$ then the appropriate map is

$\displaystyle \frac{d\zeta}{dz} = K\,(\zeta-a)^{1-\alpha/\pi}\,(\zeta-b)^{1-\beta/\pi}\,(\zeta-c)^{1-\gamma/\pi}\cdots,$ (6.79)

where $ K$ is a constant, and $ a$ , $ b$ , $ c$ , $ \cdots$ are the (real) values of $ \zeta $ that correspond to the vertices of the polygon (Milne-Thompson 2011).

It is often convenient to take the point in the $ \zeta $ -plane that corresponds to one of the vertices of the polygon--say, that given by $ \zeta=a$ --to be at infinity. In this case, the factor $ (\zeta-a)$ in Equation (6.79) becomes effectively constant, and can be absorbed into a new constant of proportionality, $ K'$ .

Figure: Conformal transformation of a semi-infinite strip in the $ z$ -plane into the upper half of the $ \zeta $ -plane.
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\centerline{\epsffile{Chapter06/schwarz1.eps}}
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Consider, for example, a semi-infinite strip in the $ z$ -plane, for which $ \alpha=0$ , $ \beta=\pi/2$ , and $ \gamma=\pi/2$ . This is mapped onto the upper half of the $ \zeta $ -plane, with the zero-angle vertex corresponding to a point at infinity in the $ \zeta $ -plane, by means of the transformation

$\displaystyle \frac{d\zeta}{dz}= K'\,(\zeta-b)^{1/2}\,(\zeta-c)^{1/2},$ (6.80)

where $ K'$ is real and positive. We can integrate the previous expression to give

$\displaystyle \zeta = \frac{1}{2}\,(b+c)+\frac{1}{2}\,(b-c)\,\cosh[K'\,(z-z_0)].$ (6.81)

Here, the points $ \zeta=b$ and $ \zeta=c$ in the $ \zeta $ -plane correspond to the vertices $ z=z_0$ and $ z=z_0+{\rm i}\,\pi/K'$ , respectively, in the $ z$ -plane. (See Figure 6.12.)

Suppose that $ b=0$ , $ c=-w$ , $ z_0=0$ , and $ \pi/K'=w$ . In this case, the transformation (6.81) becomes

$\displaystyle \zeta = \frac{w}{2}\left[\cosh\left(\pi\,\frac{z}{w}\right)-1\right],$ (6.82)

which implies that

$\displaystyle \xi$ $\displaystyle =\frac{w}{2}\left[\cosh\left(\pi\,\frac{x}{w}\right)\cos\left(\pi\,\frac{y}{w}\right)-1\right],$ (6.83)
$\displaystyle \eta$ $\displaystyle = \frac{w}{2}\,\sinh\left(\pi\,\frac{x}{w}\right)\sin\left(\pi\,\frac{y}{w}\right).$ (6.84)

The transformation (6.82) maps the semi-infinite strip in the $ z$ -plane that is bounded by the lines $ y=0$ , $ x=0$ , and $ y=w$ into the upper half of the $ \zeta $ -plane. The transformation also maps the origin of the $ \zeta $ -plane to the origin of the $ z$ -plane, and the $ \xi$ -axis to the lines $ y=w$ , $ x=0$ , and $ y=0$ . Now, in the $ \zeta $ -plane,

$\displaystyle F= -\frac{Q}{\pi}\,\ln \zeta$ (6.85)

is the complex potential associated with a line source of strength $ 2\,Q$ , located at the origin. By symmetry, the $ \xi$ -axis corresponds to a streamline, and can, therefore, be replaced by a rigid boundary. It follows that, in the $ z$ -plane, the same potential is that due to a line source, of strength $ Q$ , located in the lower left-hand corner of a semi-infinite strip bounded by rigid planes at $ y=0$ , $ x=0$ , and $ y=w$ . (Incidentally, the source strength in the $ z$ -plane is $ Q$ , rather than $ 2\,Q$ , because, by symmetry, half the output from the source in the $ \zeta $ -plane goes below the $ \xi$ -axis and, therefore, does not map to the semi-infinite strip in the $ z$ -plane.) The corresponding stream function is

$\displaystyle \psi = -\frac{Q}{\pi}\,{\rm arg}(\zeta) =- \frac{Q}{\pi}\,\tan^{-1}\left(\frac{\eta}{\xi}\right),$ (6.86)

and is illustrated in Figure 6.13.

Figure: Stream function due to a line source located in the left-hand corner of a semi-infinite strip bounded by rigid planes at $ y=0$ , $ x=0$ , and $ y=w$ .
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An infinite strip in the $ z$ -plane is a polygon with two zero-angle vertices, both at infinity. The required transformation follows from Equation (6.79) by setting $ \alpha=\beta=0$ , $ (\zeta-a)/a\rightarrow -1$ , and $ K\,a\rightarrow K'$ . Thus, we obtain

$\displaystyle \frac{d\zeta}{dz} = K'\,(\zeta-b),$ (6.87)

which can be integrated to give

$\displaystyle \zeta = b + {\rm e}^{\,K'\,(z-z_0)}.$ (6.88)

This transformation maps the point $ \zeta=b+1$ to the point $ z=z_0$ , the $ \xi$ -axis to the two lines $ y={\rm Im}(z_0)$ and $ y={\rm Im}(z_0)+\pi/K'$ , and the upper half of the $ \zeta $ -plane to the region between the lines $ y={\rm Im}(z_0)$ and $ y={\rm Im}(z_0)+\pi/K'$ , as illustrated in Figure 6.14. It is clear that the transformation (6.60), studied in the preceding section, is just a special case of the transformation (6.88).

Figure: Conformal transformation of an infinite strip in the $ z$ -plane into the upper half of the $ \zeta $ -plane.
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\centerline{\epsffile{Chapter06/schwarz2.eps}}
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next up previous
Next: Free Streamline Theory Up: Two-Dimensional Potential Flow Previous: Conformal Maps
Richard Fitzpatrick 2016-03-31