If the velocity along the free streamlines is made U instead of unity, the only effect is 

 to replace Equation [lllp] by w = 2aU In t/{2 + n), and to multiply all velocities by U; q is 

 then replaced in all formulas by q/U. The flow pattern is unaltered. V may be negative, so 

 that the fluid is entering through the slot. 



If p^ is the pressure at infinity in the mass of fluid where §■ = 0, and p, the pressure at 

 the surface of the jet itself, then, from the Bernoulli equation 



1 2 



This equation fixes U when p^ - pr is given. 



In the presence of gravity the velocity is not uniform along the free surface and the 

 problem is more difficult. 



(For notation and method; see Section 34; Reference 1, Article 75, where a is replaced 

 by {n + 2) 6/77; Reference 2, Section 11.53.) 



112. TWO-DIMENSIONAL BORDA'S MOUTHPIECE. 



The transformations in the last section are easily modified so as to allow the plane 

 boundaries to be inclined to each other at any angle, with preservation of the symmetry. 

 The integration is simple if the planes are made parallel, so as to form a parallel-sided 

 mouthpiece enclosing the issuing jet. Let the a;-axis be rotated so as to lie in the plane of 

 symmetry, with the edges of the sides at (0, a), (0, - a), as in Figure 181. 



On the ^-plane, the boundaries AB, /I'S'now coincide and lie on the positive real 

 axis; in the figure they are drawn slightly separated for clarity. If amp ^ = on AB, 

 amp ^ = - 2 77 on ^'6'; for, on the s-plane the direction of the velocity rotates through a 

 clockwise angle of 360 deg in passing from AB through the fluid to A'B'. The t diagram is 

 the same as before, and Equation [llld] holds again. Substituting in it, first In ^ = when 

 f = - 1, then In (1, = -2-ni when t =\, noting that here In (-1) = ttz, In (0) = 0, and determining 

 K and L, Equation [llld] becomes 



In ^= 2 In [^ + {t^ - 1) ] -2 77Z, 



whence, since e "^' = 1, 



As before, 



C^Vt + {t^ -1) ] . [112a] 



w ^ cf) + ixjj ^ c\xit [112b] 



278 



