Clearly Isj -► ~ only as ;! -» 0, or as (^ -» - i; then, also, u-* 0, v -* -1. Thus, in particular, 

 the velocity at infinity is -1 in the approaching stream. Mathematically, the point /or ^=-^ 

 corresponds to all directions of recession to infinity on the 2-plane, such as those labeled /, 

 /', /"in Figure 183. In the same way, C, C, and Con the ^-plane all correspond to the 

 single point C on the s-plane. At these points there is a failure of one or more of the conformal 

 transformations. 



By proceeding as in Section 111 it is found that on A' A 



n / g(3 + q^) 1 . _j 2q 

 ' + — sin 



4 1 (1 . q^f 2 



1 + ?^ 



[113g] 



whereas along the median streamline I" C, or the positive y-axis, 



21 I q{S-q^) 1 . 2q 



^^ + 4 I (1-qY 2 



— sinh' 



l-q^ 



[113h] 



Along the right-hand free streamline, proceeding as in Section 111, where Equations 

 [lllu,v] still hold but w is now given by Equation [113c] and c by Equation [113d], 



21 . sin d 

 dx + idy = - t^dw - (cos d + i sin 6) dO, 



77 + 4 



cos" 6 



21 



77 + 4 



I sec 6 + — I , V = sec d tan - \n tan I ~ + ~ I 



\ 4/77 + 4 \24/ 



, [113i] 



after integrating and choosing the constants of integration to make x = 1/2, y = at /I, where 

 0=0. The other free streamline is symmetrical with this one. 



As -♦ - 77/2, which is the lower limit for d, x ^ o" and y -» - <». Thus the free stream- 

 lines eventually approach parallelism to the direction of incidence. They also become parallel 

 to each other, but so slowly that the separation between them continues to increase without 

 limit. A larger plot of one free streamline is shown in Figure 184, and a few streamlines near 

 the lamina are shown in Figure 185. 



The pressure on the free surface must be the same as the pressure in the fluid at 

 infinity, since the velocity in both locations is unity. On the upper surface of the lamina it 

 is higher. The total net force on the lamina in the direction of the stream per unit of its 

 length is, from the Bernoulli equation. 



1 

 * 2 



1/2 



-1/2 



(1 - q^) dx. 



[113j] 



283 



