Figure 29 — Illustrating the transformation f{z) = 9 + /i// for two families of orthogonal curves 

 c{> = constant and 1// = constant. 



type through an angle of 90 degree in the counterclockwise direction. Furthermore, the mag- 

 nitude of the velocity, which has in both types the same value q = (u^+w^)^/^ = {u'^ + v'^Y^'^, 

 can be written, in view of Equation [22e]; 





dw 

 7^ 



[25h] 



When (/) is the velocity potential, it is convenient to think of w or + i i/y as a complex 

 potential. Its derivative is related to the velocity by the equation 



dw 

 dz 



[251] 



Equations [25i] and [25li] furnish usually the most convenient means of finding the velocity 

 from Equations [22b] and [25b, c]. The points at which dw/dz - are the stagnation points or 

 or points of zero velocity. 



At a singular point where dw/dz becomes infinite, q would be infinite. In applications 

 of the theory, such points must be excluded. By the insertion of a boundary they may be caused 

 to lie in a region to which the fluid does not penetrate. 



So long as dw/dz is single valued, no harm results if w itself is many-valued. In that 

 case a many-valued potential or stream function is obtained, or both. 



But if, also, dw/dz is many-valued, so would be the velocity, in virtue of the relation 

 expressed in Equation [25i]. A many-valued velocity, however, is physically impossible. In 

 such cases the s-plane must be cut or divided by a curve, representing a physical boundary, . 



48 



