Complex Velocity Potential

For instance, suppose that

where is real. It follows that

(6.24) | ||

(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 directed along the -axis. Furthermore, as is easily demonstrated, the complex velocity potential associated with uniform flow of speed whose direction subtends a (counter-clockwise) angle with the -axis is .

Suppose that

where is real. Because (Riley 1974), it follows that

(6.27) | ||

(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 , located at the origin. (See Figure 5.3.) As a simple generalization of this result, the complex potential of a line source, of strength , located at the point , , is , where . 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

where is real. It follows that

(6.30) | ||

(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 located at the origin. (See Figure 5.5.) As a simple generalization of this result, the complex potential of a vortex filament, of intensity , located at the point , , is , where . According to Equation (6.29), the complex velocity potential of a vortex filament is singular at the location of the filament.

Suppose, finally, that

where , , and , are real. It follows that

(6.33) | ||

(6.34) |

Thus, according to the analysis of Section 5.8, the complex velocity potential (6.32) corresponds to uniform flow of unperturbed speed , running parallel to the -axis, around an impenetrable circular cylinder of radius , centered on the origin. (See Figures 5.6, 5.7, and 5.8.) Here, is the circulation of the flow about the cylinder. It can be seen that on the surface of the cylinder ( ), 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.