Conformal Maps

(6.53) | ||

(6.54) |

In other words, the points and in the complex -plane map to the points and in the complex -plane, respectively. If , then

(6.55) | ||

(6.56) |

where . Hence,

(6.57) |

Thus, it follows that

(6.58) |

and

(6.59) |

We can think of and as infinitesimal vectors connecting neighboring points in the complex -plane to the point . Likewise, and are infinitesimal vectors connecting the corresponding points in the complex -plane. It is clear, from the previous two equations, that, in the vicinity of , the mapping from the complex -plane to the complex -plane is such that the lengths of and expand or contract by the same factor, and the angle subtended between these two vectors remains the same. (See Figure 6.7.) This type of mapping is termed

Suppose that is a well-behaved function of the complex variable . It follows that . Hence, the functions and can be interpreted as the velocity potential and stream function, respectively, of some two-dimensional, incompressible, irrotational flow pattern, where and are Cartesian coordinates. However, if , where is well-behaved, then , where is also well-behaved. It follows that . In other words, the functions and can be interpreted as the velocity potential and stream function, respectively, of some new, two-dimensional, incompressible, irrotational flow pattern, where and are Cartesian coordinates. In other words, we can use a conformal map to convert a given two-dimensional, incompressible, irrotational flow pattern into another, quite different, pattern. Incidentally, a conformal map converts a line source into a line source of the same strength, and a vortex filament into a vortex filament of the same intensity. (See Exercise 12.)

As an example, consider the conformal map

Writing , it is easily demonstrated that and . Hence, the positive -axis ( ) maps to the line , the negative -axis ( ) maps to the line , and the region ( ) maps to the region . Moreover, the points , map to the points , . (See Figure 6.8.) As we saw in Section 6.6, in the region , the velocity potential

(6.61) |

corresponds to the flow pattern generated by a vortex filament of intensity , located at the point , , in the presence of a rigid plane at . Hence,

(6.62) |

corresponds to the flow pattern generated by a vortex filament of intensity , located at the origin, in the presence of two rigid planes at . This follows because the line is mapped to the lines , and the point , is mapped to the origin. Moreover, if the line is a streamline in the -plane then the lines are also streamlines in the -plane. Thus, these lines could all correspond to rigid boundaries. The stream function associated with the previous complex velocity potential,

(6.63) |

is shown in Figure 6.9.

As a second example, consider the map

(6.64) |

This maps the positive -axis to the positive -axis, the negative -axis to the positive -axis, the region to the region , , and the point , to the point , . As we saw in Section 6.6, in the region , the velocity potential

(6.65) |

corresponds to the flow pattern generated by a line source of strength , located at the point , , in the presence of a rigid plane at . Thus, the complex velocity potential

(6.66) |

corresponds to the flow pattern generated by a line source of strength , located at the point , , in the presence of two orthogonal rigid planes at and . The stream function associated with the previous complex potential,

(6.67) |

is shown in Figure 6.10.

As a final example, consider the map

where is real and positive. Writing , we find that

(6.69) | ||

(6.70) |

Thus, the map converts the circle in the -plane, where , into the ellipse

(6.71) |

in the -plane, where

(6.72) | ||

(6.73) |

Note that the center of the ellipse lies at the origin, and its major and minor axes run parallel to the - and the -axes, respectively. As we saw in Section 6.4, in the -plane, the complex velocity potential

(6.74) |

represents uniform flow of unperturbed speed , running parallel to the -axis, around a circular cylinder of radius , centered on the origin. Thus, assuming that , in the -plane, the potential represents uniform flow of unperturbed speed , running parallel to the -axis [which follows because at large the map (6.68) reduces to , and so the flow at large distances from the origin is the same in the complex - and -planes], around an elliptical cylinder of major radius , aligned along the -axis, and minor radius , aligned along the -axis. Note that and . The corresponding stream function in the -plane is

(6.75) |

where

(6.76) | ||

(6.77) | ||

(6.78) |

Figure 6.11 shows the streamlines of the flow pattern calculated for .