As 2 -* - c, either X-»ooorA-»-oo, and in either case one exponential becomes negligibly 

 small, while in the other - i fi may be ignored. It thus appears that dz'/dz -♦ oo if 71 < 1, but 

 dz'/dz -♦ if n > 1. 



The transformation has several uses; see Sections 80 and 89. 



(See Reference: v. Karman and Trefftz'^ and Milller^^.) 



89. CIRCULAR-ARC CYLINDER, BOSS OR GROOVE 



By means of the preceding transformation the flow can be found past any cylinder whose 

 contour consists of two circular arcs. Only the symmetrical case will be treated here; compare 

 Figure 142a and b. 



Figure 142 — Examples of a symmetrical circular-arc cylinder (a) or (b), 

 or a circular-arc groove (c) in a plane wall. 



To be streamlines, the arcs must transform into part of the real axis of it. Let the 

 edges of the cylinder be at (i c, 0) on the s-plane, so that the arcs have a common chord of 

 length 2c, and let each arc make a numerical angle y = m n with their common chord produced. 

 Thus < OT < 1, and the internal angle at each edge of the cylinder is 2 (1 - m) n; the radius 

 of each arc \s R - c/sin y. Then the transformation Equation [88a] flattens both arcs onto the 

 same segment of the real axis of 2' provided n - 1/m; the region outside of the cylinder thus 

 goes into the whole 2 '-plane, and at infinity z' ^ z. 



213 



