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# Schwarz-Christoffel Theorem

The Schwarz-Christoffel theorem is an important mathematical result that allows a polygonal boundary in the -plane to be mapped conformally onto the real axis, , in the -plane. It is conventional to map the region inside the polygon in the -plane onto the upper half, , of the -plane. If the interior angles of the polygon are , , , then the appropriate map is

 (6.79)

where is a constant, and , , , are the (real) values of that correspond to the vertices of the polygon (Milne-Thompson 2011).

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

Consider, for example, a semi-infinite strip in the -plane, for which , , and . This is mapped onto the upper half of the -plane, with the zero-angle vertex corresponding to a point at infinity in the -plane, by means of the transformation

 (6.80)

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

 (6.81)

Here, the points and in the -plane correspond to the vertices and , respectively, in the -plane. (See Figure 6.12.)

Suppose that , , , and . In this case, the transformation (6.81) becomes

 (6.82)

which implies that

 (6.83) (6.84)

The transformation (6.82) maps the semi-infinite strip in the -plane that is bounded by the lines , , and into the upper half of the -plane. The transformation also maps the origin of the -plane to the origin of the -plane, and the -axis to the lines , , and . Now, in the -plane,

 (6.85)

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

 (6.86)

and is illustrated in Figure 6.13.

An infinite strip in the -plane is a polygon with two zero-angle vertices, both at infinity. The required transformation follows from Equation (6.79) by setting , , and . Thus, we obtain

 (6.87)

which can be integrated to give

 (6.88)

This transformation maps the point to the point , the -axis to the two lines and , and the upper half of the -plane to the region between the lines and , 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).

Next: Free Streamline Theory Up: Two-Dimensional Potential Flow Previous: Conformal Maps
Richard Fitzpatrick 2016-01-22