;J42 Profs. H. T. Barnes and E. G. Coker. [Nov. 4, 



from a study of the work of Eeynolds that the change from stream- 

 line to eddy motion may take place within a wide range of velocities. 

 We may observe the critical velocity in two ways : either by observing 

 the velocity at which the stream-lines break up into eddies, or by 

 obtaining the velocity at which the eddies from initially disturbed 

 water do not become smoothed out into stream-lines in a long uniform 

 pipe. The first change may be at any velocity within certain limits 

 depending on the initial steadiness of the inflowing water, while in 

 the second, the change can take place at only one velocity. It 

 therefore depends on whether we start with initially quiet water or 

 disturbed water what value will be attained. Below what we will call 

 the lower limit of critical velocity or stream-line flow, the stream-lines 

 will form the stable flow whatever may be the condition of the water 

 before entering the pipe. If it be in a disturbed state, a short length 

 of pipe is required before the eddies are smoothed out, but stream- 

 lines finally appear further on and subsequently persist as the stable 

 flow. The greater the initial disturbance in the water, the greater 

 will be the length of pipe probably required before the eddies disappear. 

 The flow at which the eddies persist throughout the entire pipe, 

 however long, indicates that the change has taken place from one 

 kind of flow to the other. This point of change is the true critical 

 velocity, and although the production of eddies at the mouth of the 

 pipe may vary, it will be independent of them. If the initial 

 disturbance, however, is not sufficient to prevent the water from 

 starting in stream-line motion, the critical velocity is raised. The 

 change in flow in this case takes place by the birth of eddies in the 

 pipe itself, and the point of change is in some way related to or limited 

 by the degree of steadiness in the water. In regard to this point, 

 Keynolds says, on p. 955 of his original memoir : " The fact that the 

 steady motion breaks down suddenly shows that the fluid is in a state 

 of instability for disturbances of the magnitude which cause it to break 

 down. But the fact that in some conditions it will break down for a 

 large disturbance, while it is stable for a small disturbance, shows that 

 there is a certain residual stability so long as the disturbances do not 

 exceed a given amount. 



" The only idea that I had formed before commencing the experi- 

 ments was that at some critical velocity the motion must become 

 unstable, so that any disturbance from perfectly steady motion would 

 result in eddies. 



" I had not been able to form any idea as to any particular form of 

 disturbance being necessary. But experience having shown the 

 impossibility of obtaining absolutely steady motion, I had not doubted 

 but that appearance of eddies would be almost simultaneous with the 

 condition of instability. I had not, therefore, considered the dis- 

 turbances except to try and diminish them as much as possible. I 



