j4. Y. ^o^^^^' 



[ 9 ] 



II. 



THE STABILITY OE INSTABILITY OF THE STEADY MOTIONS 

 OF A PEEFEOT LIQUID AND OF A VISCOUS LIQUID. Part I. : 

 A PERFECT LIQUID. 



By WILLIAM M'F. ORPt, M.A., 



Professor of Mathematics in the Eoyal College of Science for Ireland. 



Read November 12. Ordered for Publication December 17, 1906. Published March 29, 1907. 



Introduction and Summary of Contents. 



It is a well-known experimental fact that when a liqiiid of small viscosity, 

 such as water, flows through a straight circular pipe under applied pressure 

 or under the action of gravity, the steady motion — in which, of course, each 

 particle describes a straight line — may be unstable. The subject has been 

 investigated experimentally by Osborne Reynolds,^ who found that the 

 motion is stable so long as the mean velocity does not exceed a certain 

 limit depending on the radius of the pipe and on the nature of the liquid. 

 This limit, beyond which instability sets in and the motion becomes turbulent, 

 he found to vary directly as the kinematic viscosity, and inversely as the 

 radius of the pipe — results to which he was led also by considerations of the 

 theory of dimensions. 



The question has also been attacked theoretically, chiefly by Lord Rayleigh, 

 Lord Kelvin, and Reynolds himself. Lord Rayleigh^ has ignored the effect 

 of viscosity in the disturbed motion — a simplification which renders such 

 problems much more amenable to mathematical treatment. One series of 

 his papers deals with flow in plain strata between fixed parallel walls ; and he 

 arrives at the conclusion that the motion is not unstable, provided the law of 

 flow is such that the velocity-gradient continually increases or continually 

 decreases (algebraically) from one wall to the other. Quoting his own words," 

 " To be more precise, it was proved that if the deviation from the regularly 



1 "An experimental investigation of the circumstances wliich determine whether the motion of 

 water shall be direct or sinuous, and of the law of resistance in piirallel cliannels," Phil. Tians., 

 t. clxxiv.. Part iii., p. 935 (1883) ; Sc. Papers, t. ii., p. 51. 



^ Detailed references are given in the text. 



^ "On the question of the Stability of the Flow of Liquids," Pliil. Mag., t. xxxiv., p. 6], July, 

 1892 ; Sc. Papers, t. iii., p. 576. 



R.I. A. PROC, VOL. XX^^I., SECT, A. [2] 



