l/J = Q 



Figure 46 — Streamlines in an angle of 

 a = jt/S radians: w = Az^. 



Figure 47 — Streamlines around a right 

 angle: a = 877/2, «' = 3^^^. 



If a = 27T, the flow is around the edge of a semi-infinite plane. In this case 



w = Az^/^, 4, = /lrl/2 cos i, = 4rl/2 gin J, q =Ar ^ [39f,g,h,i] 



Streamlines for this case are shown in Figure 48. 



A uniform flow parallel to the plane may be added, producing streamlines as shown in 

 Figure 49; see Cisotti, Reference 24. 



The mathematical transformation 2'= z", where n is real, is useful in constructing 

 transformations for special purposes. Geometrically, it merely rotates all radii from the origin, 

 except the positive real axis, about the origin as center until on the s -plane they make an 

 angle with the positive real axis n times as great as on the 2-plane. The change is like the 

 opening or shutting of a fan. If n is an integer, the 2-plane is mapped n times onto the 2'-plane; 

 the mapping is backwards if n is negative. If \n\ < 1, the entire 2-plane is mapped onto a 

 sector of angle 27rn radians. If n is not integral, the transformation is many-valued, with 2=0 

 as a branch point. In any case, circles centered at the origin transform into arcs of similar 

 circles. 



The more general transformation 2'= Cz" also stretches all radii from the origin in a 

 ratio equal to |C| and rotates everything through an additional angle equal to amp C. 



(For notation and method; see Section 34; Reference 1, Article 63; Reference 2 Section 

 6.0.) 



80 



