In this case the streamline for i/f = follows the a;-axis and the curve S. That S passes 

 through the stagnation points at {-Xq, 0) can be verified by first replacing coth {Vy/A) by 

 A/Vy from the first terms of the hyperbolic series [33i]. 



The formulas may represent none ire ulatory flow past a cylinder represented by the un- 

 divided curve S, or flow past two cylinders of a certain shape with circulation - 2nA about 

 them. 



The most interesting case is that in which the vortices, when assumed to move with 

 the fluid, actually stand still. This is realized when A/2c = f/ or A/V = 2c, so that the ve- 

 locity at either vortex due to the other just cancels the stream velocity U. Streamlines for 

 this case are shown in Figure 86. The large oval curve is S; its semidiameters are 2.09c 

 and 1.73c, approximately. 



(For notation and method; see Section 34; Reference 1, Article 155; Reference 2, 

 Section 13.30.) 



56. OTHER COMBINATIONS INVOLVING LINE SOURCES OR DIPOLES 



The following cases may be mentioned. 



(1) Sources or dipoles only. Streamlines due to three equal and symmetrically placed 

 line sources, with the fluid at rest at infinity, are shown in Figure 27 of Durand's Aerody- 

 namic Theory^, and for two sources and a sink in Figure 28. For a source and a dipole at 

 the same point, see Reference 36. 



(2) Source near a cylinder whose contour is elliptic or of certain other types: Morris^^ 

 and Wrinch37; parabola-like: Sharpe^^; a circular arc: Caldonazzo^^ and Sestini^O; gee 

 Figure 87 for streamlines in one case with the source at the center of the circular arc. 



(3) Source inside a rectangular cylinder: Jaffe^" and Miiller.'*^ 



Figure 87 - Line source on axis of a 



circular-arc shell. (Copied from 



Reference 39.) 



129 



