10 Proceedings of the Eoijnl Irish Academy. 



stratified motion were, as a function of the time, proportional to c'"^, then 

 ii could have no imaginary part." In the paper quoted from he discusses flow 

 in C3iindrical layers, as through a straight circular pipe ; and, as a particular 

 case of a more general result, he concludes that when the distribution of 

 velocity is that which actually exists in the case of a viscous liquid, the 

 steady motion is not unstable. He considers that case also of 13 ow m 

 cylindrical layers in which the particles describe circles about a common 

 axis, and concludes that the motion is stable if the rotation either continually 

 increases or continually decreases in passing outwards from the axis. This 

 condition is satisfied if the law of velocity is that which obtains in a viscous 

 liquid between long concentric cylinders of which one is fixed and tlie other 

 made to rotate. It has been found experimentally by Mallock^ and by 

 Couette- that, under these cii'cumstances, the motion of water is unstable 

 if the velocity be sufficiently great. 



Accordingl}", in the second and third of the three classes of motion 

 referred to, the behaviour of natural liquids, as tested by Eeynolds, Mallock, 

 and Couette, appears to differ from that attributed to perfect liquids by Lord 

 Eayleigh. (I am not aware that any experiments have been made dealing 

 directly with the first class of motions, that in plane layers.) There is thus 

 a difficulty in reconciling theory and experunent. 



Portions of Lord Eayleigh's argument have, however, been criticised 

 adversely by Lord Kelvin^ and by Love.^ 



"When viscosity is taken into account, the mathematical difiiculties 

 involved in a discussion of the question of stability are much greater. Lord 

 Kelvin^ has attacked the question under such conditions. He has considered 

 two problems of motion in plane layers — one that of a liquid undergoing shear 

 at a uniform rate, the other that of a liquid flowing between two fixed 

 parallel planes — and concludes that in each case the motion is stable for 

 sufficiently small disturbances, but that for disturbances exceeding a certain 

 magnitude the motion becomes unstable, and that this limiting magnitude is 

 smaller the smaller the viscosity — a view to which Eeynolds has been led by 

 his experiments. His mode of solving the latter problem applies equally to 

 the former, as he points out : but these solutions have been rejected by Lord 

 Eayleigh. Lord Kelvin has also given another solution of the former 

 problem which Lord Eayleigh regards as satisfactory. 



1 "Experiments on Fluid Viscosity," Phil. Trans., A, t. clxxxrii., p. 41 (1896). 



- Annales de Chimie et (ie Physique [6], 21, p. 433. 



3 Phil. Mag., Sept., 1887, oth Series, t. xxiv. : Brit. Ass. Rep., 1880, p. 492. 



■* Proe. I.ond. Math. Soc, t. xxvii., p. 199. 



'= Phil. Mag., Aug. and Sept., 1887, 5th series, t. xxiv. 



