AF^^- pUUh^-h^)[— -l] =- ph^U^l —— ^ [108r] 



in terms of the exit velocity U^ = h^ ^ /^2' 



If h. - hr, is held constant while h. -> ~, the case of the last section is reproduced, and 

 in this case A F, -♦ 0. If, on the other hand, h^/h^ is very small, A /^, = P ^9 ^2'^^' ^ppro'^im^tely. 

 This is the familiar suction force due to deficit of pressure on the walls of a vessel in the 

 neighborhood of a relatively small orifice. 



By putting together two flows of this type, one reversed as if by reflection in the direc- 

 tion of y, the flow can be represented through a plane-sided orifice of width ^h^, located in 

 the plane end of a two-dimensional semi-infinite tank whose sides are 2Aj apart. 



The velocity may be reversed at all points without affecting q or the geometrical flow 

 net. 



(For notation and method; see Section 34; Reference 2, Section 10.7.) 



109. CHANNELS OF VARIOUS FORMS. 



Channels with sides variously composed of straight lines, or in part curved, are de- 

 scribed by Love and Miyadzu;^^^ for the introduction of a gate see Reference 177. Channels 

 with curved walls are described by Sakurai.^^^ 



Branching channels are treated in Reference 2, Section 10.8; see also articles by 

 Agostinelli,*^^ Cisotti,^^^ and Boverio.^^° 



FREE STREAMLINES 



110. NATURE OF FREE STREAMLINES. 



Where a free surface occurs, as at the top of a mass of liquid or on the boundary of a 

 region of cavitation, the usual requirement is uniformity of pressure. Problems involving this 

 boundary condition are often difficult to solve. 



If the motion is steady, however, and if gravity is absent, constancy of pressure is 

 equivalent to constancy of q, the magnitude of the velocity, as is evident from the Bernoulli 

 Equation [lOd]. Furthermore, in steady motion the free surface is composed of streamlines. 

 Thus in steady motion the boundary condition along a free streamline is that q is constant. 

 This boundary condition is readily handled. 



Alternatively, the space adjacent to the moving fluid, instead of being empty or filled 

 with gas of negligible density, may be assumed to be filled with fluid of the same kind but at 

 rest. The steady motion of the remaining fluid is not thereby affected, provided viscosity is 

 entirely absent; it suffices to assume that the pressure in the stationary fluid or wake is the 

 same as the constant pressure along the boundary surface between the two. At this boundary 

 the velocity is then discontinuous, and the motion is there rotational; a sheet of vortices may 



270 



