504 



XL HYDRODYNAMICS. 



If the motion be non- vortical, the left-hand members vanish, 

 and we immediately obtain the integral 



33) 



F + P -f 4 <? = const, 



for the expression on the left has been assumed independent of t, 

 and by the equations is shown to be independent of x, y, z. 



It is to be noticed that if we multiply equations 32) respectively 

 by u, v, w y or by |, 77, and add, the left-hand member vanishes 



identically. But the operator u^ h^o h w ~ denotes differentiation 



in the direction of the line of the vector- velocity q, or stream-line 



r\ r\ r\ 



(see p. 333), and | ~ l~ ^2 ^ ^7T differentiation i n the direction 

 of the line of the vector co, or vortex -line. Consequently even though 

 there is vortical motion, along a stream -line or a vortex -line the 



sum F -f P -f Y *f * s cons tant in steady motion, though its value 

 changes as we go from one line to another. 



If the fluid is incompressible P = , if there are no applied 

 forces F=0, and equation 33) becomes 



34) 



P 1 



=~ = const. 



so that where the velocity is small the pressure 

 is great and vice versa. By constricting a tube 

 the velocity is made large and the pressure 

 accordingly is smaller than at other parts of the 

 tube. This is the principle of jet exhaust pumps, 

 like that of Bunsen (Fig. 160), the air being 

 sucked in at the narrow portion of the jet. The 

 same principle is made use of in the Venturi 

 water-meter. The main being reduced in diameter 

 at a certain portion and the difference of pressure 

 at that point and in the main being measured, 

 the velocity is computed. If the pressure at two 

 cross -sections S 1 and S 2 are p and p 2 we have 



35) 



Fig. 160. 



or 



36) 



But by the equation of continuity, the velocity being solenoidal, 



37) 



