116. JETON A WALL. 



Let a two-dimensional jet of width a fall upon an infinite plane wall, approaching with 

 uniform velocity V at an angle a. with the face of the wall, as in Figure 189. Part of the fluid 

 in the jet will flow away toward one side, part toward the other. 



Figure 189 — Two-dimensional jet 

 striking a wall. 



In the two departing jets, of widths a^ and a^, the velocity will ultimately become 

 uniform and equal to that in the incident jet, since, as in Section 41, it is uniform along the 

 free streamlines. Hence, the incompressibility of the fluid requires that aj + ^2 = °" 

 Furthermore, the component of momentum parallel^to the wall must be conserved, since no 

 force acts on the fluid in this direction. In unit time a mass paV of fluid, carrying momentum 

 paV^ , is lost from the incident jet and reappears as masses pa^V and pa^l) in the departing 

 jets; hence paV'^ cos a = pa^U^ - pa^U^, and a cos a = Oj - a^. From these two equations 



I, = — (1 + cos a ), o„ = (1 - cos a ), = tan -^ . [116a, b,c] 



1 2 "22 a, 2 



The decrease in the component of momentum perpendicular to the wall, on the other 

 hand, equals the total force on the wall, so that 



Fj = paU^ sin a [116d] 



where F. is the force per unit of length of the wall in a direction perpendicular to the planes 

 of flow. The effective line of action of F^ can be found from the conservation of moment of 

 momentum. About the axis M along which the median plane of the incident stream cuts the 

 wall, the incident stream has a zero moment of momentum because of symmetry, but the median 

 planes of the departing jets lie at distances a^/2 and a^/2 from M. Hence, in unit time the 



291 



