Separating dx and dy and integrating along the left-hand streamline, 



a 



X = - (tt + 2 cos Q), 



[lllw] 



2+77 



1 1 



In tan ( — 77 + — ) - sin 

 4 2 



[lllx] 



Here < < 0. The right-hand streamline is then the mirror image in the y-axis of this 



2 = 

 one. 



The issuing jet eventually becomes straight; its limiting width is twice the value of 



\x\ when Q = - 77/2 or 2 77 0-/(2 + 77). The ratio of contraction, relative to the width of the orifice 



or 2a, is thus 



= 0.611. 



2 + 77 



Since the velocity in the ultimate jet is uniform, the volume of fluid that issues per second, 

 per unit length, is 2 77 0-/(2 + 77). 



Thus on the s-plane the streamlines do not converge at infinity, as is indicated by the 

 distinct labeling C, J, C" in Figure 179, but on the ^-, In ^-, and ^planes they converge to a 

 finite point, as at ^ = - z or i! = 0. 



At = 0, or the edge of the orifice, dy/dx = sin^ 0/cos ^ ^ = 0, but dQ/dx « 1/sin ^ c 

 Thus, although there is no discontinuity in the slope of the streamline, its curvature at the 

 edge is infinite. 



The free streamlines are plotted to scale in Figure 179, and an enlarged plot of one 

 half of the symmetrical diagram is shown in Figure 180. 



Figure 180 - Efflux as in Figure 179: 

 one side of the issuing jet in 

 more detail. (Copied from 

 Reference 1.) 



Line of Symmetry 



277 



