Thus the vector velocity (w', v') is rotated positively or counterclockwise through 90 degree 

 relative to {u,v). The second type of flow also constitutes the first type as furnished by the 

 modified transformation, 



w. (z) - (ji' + i \Ji' = - i w (2) 



The flow net, or the pattern of equipotential curves and streamlines, is geometrically 

 the same for both types of flow. The magnitude of the velocity is also the same in the two 

 types, namely, 



q^{u^ +v^)'^=^{u'^ +v''^f^ = 1^1 [34e] 



\d2 I 



Furthermore, since dw/dz = dw/dx = d(ty/dx + i d\Jj/dx, 



dw dw dw. 



— u + iv=' — , -u'+iv'=-i — = — ^ [34f,g] 



dz dz dz 



Usually u and v are most easily found from Equation [34f] by separating dw/dz into its 

 real and imaginary parts; in order to do this, it may be necessary to rationalize a denominator 

 by multiplying by its complex conjugate. Frequently, values of u and v obtained in this manner 

 will be given without writing down dw/dz. In some cases, however, use of Equations [34a, b] 

 is more convenient. 



Stagnation points occur in both types where dw/dz = and hence 7 = 0. At such points 

 the transformation may fail to be conformal, and equipotential curves and streamlines may meet 

 at other angles than 90 degree. 



Singular points for the transformation occur tvherever dw/dz -» 00. Since at such points 

 ^ -> 00, they must be excluded from the body of the fluid by inserting suitable boundaries. It is 

 convenient, however, to allow a singularity to fall on a boundary; in a physical case, it can 

 then always be imagined to be removed from the region of the fluid by slightly altering the 

 shape of the boundary. 



When polar coordinates r, d are employed, the component of the velocity in the radial 

 direction is denoted by q^, that in the transverse direction of increasing 6 by q^; these 

 components are calculated as 





dd 



Many-valued functions are to be understood as defined so that they vary continuously 

 with z, or with x and y, in all variations that are possible without crossing any boundaries 

 that may be present. If it is appropriate in a given case to choose a single set of values for 

 such a function, this is to be done in- such manner that the function takes on its ordinary values 

 at points on the positive a;-axis, or the positive real axis for s. 



The symbol v^will be used only for the positive square root of a positive real number. 



71 



