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

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

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

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

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

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

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

which implies that

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

$$\eta = \frac{w}{2}\,\sinh\left(\pi\,\frac{x}{w}\right)\sin\left(\pi\,\frac{y}{w}\right).$$ (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,

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

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

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

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

which can be integrated to give

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