combinedmoment of momentum of these jets increases by 



pa^U^(a^/2)-pa^U^ {a^/2) = (a^ _ a^) pU^./2, 



in a cloci<wise direction. The reaction on the wall must be an equal counterclockwise moment 

 of force about M. Hence the force F, must act at a point displaced from M to the left, or 

 toward the stagnation point C, through a distance e such that e Fj = (a^ - a, )pU /2 or, from 

 Equation [116d] and Equations [116a, b], 



a 

 = — cot a 



[116e] 



Further details can be discovered by resorting to the method of complex variables. 

 Only a few results will be cited here. 



The distance h from M to the stagnation point C is 



a a \ a I ^ \ 



h - — cot a + — (cos a ) In (2 sin a ) + In cot -2-+ asina 



2 77 L 2 I 2 / 



. [116f] 



At perpendicular incidence or a = 77/2, the equations for the free streamlines, with the 

 origin taken at the stagnation point C or M and with the a;-axis drawn along the wall, are 



, / 1 1 \ 



- a\ — + — In cot — I , V = 

 \ 2 77 2 / 



1 1 

 — + — In cot 



il-T) 



[116g,h] 



where 6 is the angle, taken positive, between the direction of the tangent to the streamline and 

 the wall. On the median plane, where a; = 0, in terms of the velocity q, i( U > 0, 



y = 



V + q , U 



In - 2 tan — + n 



U-q q 



[116i] 



Along the wall the absolute value of x is given in terms of q by the same expression that 

 represents y along the median plane. Each half of the jet has a plane of symmetry through 

 C inclined at 45 deg to the wall. 



Flow nets for a = n/2 and « = SttA are reproduced from Reference 183 in Figures 190 

 and 191. In the figures v stands for f , and cji and \jj are reversed in sign in accord with the 

 older convention. The numerical values refer to the case a = 1, v^ - U = 1. The broken curves 

 are curves of constant velocity, the value of the ratio w = q/V being indicated for each. 



(See Reference 2, Section 11.41; Reference 50 and 183.) 



292 



