N. Y, AoAOEMr 



[ 69 ] ■ 



III. 



THE STABILITY OK INSTABILITY OF THE STEADY MOTIONS 

 OF A PEEFECT LIQUID AND OF A VISCOUS LIQUID. Part II. : 

 A VISCOUS LIQUID. 



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



Professor of Mathematics in the Eoyal College of Sc^'ence for Ireland. 



Eead June 24. Ordered for Publication June 26. Published October 28, 1907. 



Inteoduction and Summary of Contents. 



In Part I.* reference was made to a well-known difficulty in reconciling 

 theory and experiment in the case of the steady motion of liquids. The 

 flow through pipes and between concentric cylinders, one of which is 

 rotated, had been found experimentally to be unstable if the velocity is 

 great enough ; while, on the other hand, Lord Eayleigh had shown that, in 

 these cases, if the effect of viscosity be neglected in the disturbed motion, 

 the fundamental free disturbances are strictly periodic, the values of the 

 " free periods " being real. An explanation of the difficulty was given by 

 showing that it is necessary to push Lord Eayleigh's investigations a step 

 farther by resolving a disturbance into its constituent fundamental ones 

 by quasi-¥ oiiriev analysis, and that, when this is done for disturbances of 

 initially simple type in some of the most important and simplest cases 

 of flow, it is found that the disturbance will, for suitable values of the 

 constants, increase very much, so that the motion is practically unstable. 



The present investigation attempts to discover how far this conclusion 

 must be modified when viscosity is taken account of. 



It may be stated at once that I have not succeeded in throwing much 

 additional light on this matter ; but a good deal of the work had been done 

 before I discovered that the slight extension of Lord Eayleigh's analysis which 

 is contained in Part I. would explain the difficulty, at least qualitatively ;t and 

 I therefore decided to carry the investigation as far as I could : I may 

 moreover plead that I found some portions of the analysis interesting on 

 their own account. 



*Proc. R.I. A., vol. xxvii., Section A, No. 2. 



t I consider that a proof of instability for a perfect liquid is a proof of instability also for a 

 viscous liquid if the viscosity be small enough. 



K. I. A. PROC, VOL. XXVII., SECT. A. [10] 



