The Stability of Fluid Motion. 434 
experiments with smooth glass tubes ; a much lower number, of the order of 
400, has been found from metal tubes. The only available direct results for 
flow in an open stream appear to be those given by Hopf,* who found R to 
be of the order of 300. These results agree in character with the theoretical 
calculations, which is all that could be expected. 
It is of interest to note that this appears to contradict a statement by 
Reynolds} in one of his earlier papers. He classes separately circumstances 
condusive to steady motion and those conducive to unsteady motion: among 
the former a free surface, and in the latter solid bounding walls. However, 
this opinion seems to be based on visual observation of eddies caused by the 
wind beneath the oiled surface of water. “At a sufficient distance from 
the windward edge of an oil-calmed surface there are always eddies beneath 
the surface, even when tlie wind is light. . . . Without oil I was unable 
to perceive any indication of eddies.” 
This introduces a different property of a boundary surface, namely, that of 
initiating disturbances. The mathematical statement ignores this property 
and specifies only control of the velucity functions: the disturbances are 
supposed to be initiated by some extraneous agency, and it is tacitly assumed 
that all types of disturbance are equally probable. It may be, for instance, 
that the theoretical results for flow through pipes should be compared with 
experiments on rough pipes rather than those with perfectly smooth walls. 
However, we may conclude that a solid boundary is conducive to stability in 
so far as it ensures that there is no slipping of the fluid in contact with it. 
6. In determining the minimum value of R from the differential equation (7), 
there are only two factors: the distribution of steady velocity, U, and the 
boundary conditions for the disturbance. The comparison in the previous 
section, between an open stream and flow between fixed walls, involved 
changes in both these factors. We may separate the effect of the boundary 
conditions by assuming the same value of U as in (10), but expressing the 
property of the supposed moving plane in contact with the upper surface by 
u=0,v = 0, instead of by v= 0, px = 0. To anticipate the argument of 
the next sections, we should expect a value of R intermediate between 96 
and 117. 
We have the same equation (11) for y, together with py = 0, dy/de = 0 
at a=0,and a=p. It follows that only the solutions wa and we are 
involved, and we have 
ra dwy3/da—rwWs3 dipo]/ da => 0, (18) 
* L. Hopf, ‘ Ann. der Phys.,’ vol. 32, p. 777 (1910). 
+ O. Reynolds, ‘Scientific Papers,’ vol. 2, pp. 57, 59. See also A. H. Gibson, ‘Phil. 
Mag.,’ vol. 25, p. 81 (1918). 
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