31 33.] VENA CONTRACTA. 27 



vessel. The mass of fluid which in unit time passes the vena 

 contracta is pqS , and the momentum which this carries away with 

 it is pq*S . Hence, substituting the value of q from (14), we have 



gpzS=*gpzS ........................ (16), 



or, since z, z are nearly equal 



S :S=l :2, 



approximately. Since, however, the pressure on the wall is, near 

 the orifice, sensibly less than the statical pressure P + gpz, the 

 total horizontal force acting on the fluid somewhat exceeds the 

 value (15). The left-hand side of (16) is therefore too small, and 

 the ratio : S is really greater than J. 



The above theory is taken from a paper by Mr G. 0. Hanlon, 

 in the 3rd volume of the Proceedings of the London Mathematical 

 Society, and from a note appended thereto by Professor Maxwell. 



In one particular case, viz. where a short cylindrical tube, pro 

 jecting inwards, is attached to the orifice, the assumption on which 

 (16) was obtained is sensibly exact; and the value J- of the co 

 efficient of contraction then agrees with experiment. Compare 

 Art. 97. 



33. Example 2. A gas flows through a small orifice from a 

 receiver, in which the pressure is p l and the density p iy into an 

 open space where the pressure is p 3 *. 



We assume that the motion has become steady. In the re 

 ceiver, at a distance from the orifice, we have p=p l} q = Q, sensibly. 

 This determines the value of C in equation (11). Neglecting the 

 external forces, we find for the velocity of efflux 



where /o 2 is the density of the issuing gas at the vena contracta. If 

 c be the velocity of sound in the gas of the receiver, we have 



(Chapter vm.) c 2 = ^ ; and therefore, taking account of (4), Art. 7, 



* See Joule and Thomson, On the Thermal Effects of Fluids in Motion, Proc. 

 R. S. May, 1856. 



Also Rankine, Applied Mechanics, Arts. 637, 637 A. 



