26-28] EFFLUX OF GASES. 29 



found, and is therefore approximately independent* of the external pressure 

 p l so long as this falls below '527/> . The physical reason of this is (as pointed 

 out by Reynolds) that, so long as the velocity at any point exceeds the velocity 

 of sound under the conditions which obtain there, no change of pressure can 

 be propagated backwards beyond this point so as to affect the motion further 

 up the stream. 



These conclusions appear to be in good agreement with experimental results. 



Under similar circumstances as to pressure, the velocities of efflux of 

 different gases are (so far as y can be assumed to have the same value for 

 each) proportional to the corresponding velocities of sound. Hence (as we 

 shall see in Chap, x.) the velocity of efflux will vary inversely, and the rate 

 of discharge of mass will vary directly, as the square root of the density f. 



Rotating Liquid. 



27. Let us next take the case of a mass of liquid rotating, 

 under the action of gravity only, with constant and uniform angular 

 velocity co about the axis of z, supposed drawn vertically upwards. 



By hypothesis, 



u = or?/, v = cox, w = 0, 



x = o, r=o, z=-g. 



The equation of continuity is satisfied identically, and the dynamical 

 equations of Art. 6 become 



1 dp I dp I dp 



-o> 2 # = -- -f. -o) 2 i/ = --- f- t = --- - q....(\\ 



pdx' pdy pdz 



These have the common integral 



f + y'>)-gz + const ............. (2). 



The free surface, p = const., is therefore a paraboloid of revolution 

 about the axis of z, having its concavity upwards, and its latus 

 rectum = 



. dv du 



Since -^ --- =- = 2o), 



dx dy 



a velocity-potential does not exist. A motion of this kind could 

 not therefore be generated in a ' perfect ' fluid, i.e. in one unable 

 to sustain tangential stress. 



28. Instead of supposing the angular velocity o> to be uni- 

 form, let us suppose it to be a function of the distance r from the 



* The magnitude of the ratio p^p^ will of course have some influence on the 

 arrangement of the streams, and consequently on the value of S. 

 t Cf. Graham, Phil. Trans., 1846. 



