136 Proceedings of the Royal Society of Edinburgh. [Sess. 
XV. — On the Criterion for Stable Flow of a Fluid in a Uniform 
Channel. By H. Levy, M.A., D.Sc., Assistant-Professor of Mathe- 
matics, Imperial College of Science, South Kensington. 
(MS. received May 27, 1921. Read June 20, 1921.) 
§ 1. The conditions determining the stable flow of a viscous fluid in a 
uniform channel and in a uniform circular pipe were investigated experi- 
mentally for water by Osborne Reynolds in a searching series of papers.* 
For a channel he concluded that so long as the non-dimensional group 
U Ijvt where 
U = mean velocity in the channel, 
l = breadth of the channel, 
v — kinematic viscosity of the fluid, 
was maintained below a certain value, any slight disturbance imposed on 
the steady flow tended to die out, and the steady streaming persisted, but 
in the neighbourhood of and above this critical value a condition of 
turbulence and eddying set in immediately the steady stream-line motion 
was disturbed. 
Many theoretical investigations of this problem, having for their object 
the mathematical formulation of the conditions determining this critical 
state, have been attempted, but no satisfactory analysis has yet been forth- 
coming. The mathematical difficulties of a direct attack are so formidable 
that simplifications are invariably introduced ; but these, while they may 
render the mathematical development amenable to treatment, nevertheless 
involve physical assumptions whose interpretation is frequently obscure. 
This is amply borne out by the fact that theoretical estimates of the 
critical value of U Ijv vary over exceedingly wide ranges. In the present 
paper it is proposed to approach this question from a new standpoint. 
§ 2. On general grounds of dimensions it is clear that any problem in 
the flow of a viscous fluid with given boundary will centre round the 
particular value assigned to XJl/v, and the author has indicated how this 
fact may be utilised to approach a solution of any such problem in general.f 
A full experimental investigation of the question in relation to the critical 
flow of air, water, and oil in pipes has been conducted by Stanton and 
* Scientific Papers , vol. ii, p. 51 et seq. ; Phil. Trans., 1883. 
+ Phil. Mag., xli, April 1921, “ On a Method of Analysis suitable for the Differential 
Equations of Mathematical Physics.” 
