/^r^ _^ 



\dw/ ^^gW ^ 



+ e^ (cos i/f + i sin i/y) 



\dw\ 1 T^ ^ 1 



= — = -, ff = (1 + e^t?" + 2 e9 cos v!')^ 



U2I G 



u = (1 + e^ cos 1//), tJ = - 



6-2 ff2 



1 A. ■ 



— e?* sin ip. 



[62d,e] 



[62f,g] 



Since 1-2 e^+ e2<?!> = (1 - 6*^)2 ^ 0, it follows that 2 e^ ^ 1 + e^'^; and the equality 

 sign holds only if ^ = 0. Hence (? = and dw/dz -* <» only if ^ = and cos ip = - 1. Thus 

 singular points occur at a; = - 1 and y = - n, - Stt, i Stt 



The streamline for = is the a;-axis, on which a; = + e'^. Along this streamline, 

 while is negative and numerically large, (p = x, approximately. As <p increases, x increases, 

 and at <?i = 0, a; = 1; as (;i ^ + 00, a; -► + ~. Again, U ip = - n, y = ^ 77 and x = cfo-e^. Flere x in- 

 creases to a maximum of -1 at = 0, and then returns toward - 00 as <j!) -» + 00. The two 

 straight lines on which y = - n and ^ ^ -1 may be regarded as streamlines bent back on them- 

 selves. The intermediate streamlines, (or -n < ip < n, lie between these straight lines; for 

 large negative <p they are almost parallel, but for large positive cp they fan out and cover the 

 entire 2-plane. Half of the flow net, which is symmetrical about the a;-axis, is shown in Fig- 

 ure 95. For \ip\>Tr, curves are obtained which overlap some of those already obtained; since 

 this results in multiple-valued velocities such values of ip cannot be used. 



If plane, semi-infinite boundaries are inserted along the two straight streamlines, a 

 motion is represented in which the fluid is flowing into and through a spout or mouthpiece 

 bounded by two parallel walls 2 n apart. 



Figure 95 — Half of the symmetrical flow 

 net for fluid entering a straight two- 

 dimensional spout extending toward 

 the left to infinity. (Copied from 

 Reference 253.) 



141 



