440 On the Flow of Water in a Straight Pipe. 



motion. Our equations of disturbed motion serving to inves- 

 tigate the question of stability are the same hydrodynamical 

 equations with certain terms neglected. These equations, as 

 Lord Kelvin has proved, show that the undisturbed motion is 

 stable. But if we introduce something that is not contained 

 in the hydrodynamical equations, as Mr, Basset has done, we 

 find the sought instability. 



In other words, our hydrodynamical equations, which we 

 know to be strictly true only for small relative velocities, are 

 now shown to be, in the case of water, of very limited im- 

 portance. They are not able to express the eddying motion. 



In the motion which they are able to express, any surface 

 drawn within the liquid is supposed to be strained in a con- 

 tinuous manner. In the eddying motion these surfaces 

 are continually breaking and again reforming, an opinion 

 which seems not to be new to hydraulicians *, 



The opinion that it is with a real breaking that we have to 

 do is strongly supported by a striking fact observed by Prof. 

 Reynolds. The eddies appear only at a certain distance from 

 the entrance of the pipe. This distance is diminishing when 

 the velocity increases, but diminishes asymptotically. Now 

 it is a known fact that the breaking of bodies, solid, plastic, 

 or plastico-viscous, depends not only on the amount of the 

 strain, but also on the velocity of straining. Even hard 

 bodies sustain a great strain when the straining is slow 

 enough ; on the other hand, flexible bodies, when very rapidly 

 strained, break down. 



On the other hand, when the liquid enters the pipe tumul- 

 tuously, but with small mean velocity, the viscosity begins to 

 act at advantage, the breaking ceases, and the eddies die out. 

 Reynolds has shown that this reversal from tumultuous to 

 quiet motion occurs at a critical mean velocity, which ceteris 

 paribus is about 6*3 times smaller than the other mean velocity 

 which renders the quie' 1 motion impossible. 



All other features of the phenomenon — the dependence of 

 critical mean velocity, i. e. of the critical rate of straining, on 

 the viscosity and on the size of the tube — are clearly in best 

 accord with the hypothesis of breaking for a certain critical 

 rate of straining. 



The existence of two critical velocities — a greater which 

 makes the quiet motion impossible, and a smaller which 

 makes the tumultuous motion impossible — is very interesting, 

 and shows similitude to many other physical phenomena. 



* See Boussinesq, " Essai sur la theorie des eaux courantes," Mem, 

 Sav. Mr. vol. xxiii. p. 5. 



