the Stability of the Flow of Fluids. 61 



that (at an earlier date *) I attempted an investigation of the 

 stability of stratified flow in two dimensions, fully expecting 

 to find it unstable. The result, however, was to show that in 

 the absence of viscosity the stratified flow between two parallel 

 walls was not unstable, provided that the law of flow were 

 such that the curve representing the velocities in the various 

 strata was of one curvature throughout, a condition satisfied 

 in the case in question. To be more precise, it was proved 

 that if the deviation from the regularly stratified motion were, 

 as a function of the time, proportional to e int , then n could 

 have no imaginary part. 



On the other hand, if the condition as to the curvature of 

 the velocity curve be violated, n may acquire an imaginary 

 part, and the resulting disturbance of the steady motion is 

 exponentially unstable, as was shown by several examples in 

 the paper referred to, and in a later one f in which the subject 

 was further pursued. 



We are thus confronted with a difficulty. For if the inves- 

 tigation in question can be applied to a fluid of infinitely 

 small viscosity, how are we to explain the observed instability 

 which occurs with moderate viscosities? It seems very 

 unlikely that the first effect of increasing viscosity should be 

 to introduce an instability not previously existent, while, as 

 observation shows, a large viscosity makes for stability. 



Several suggestions towards an explanation of the discre- 

 pancy present themselves. In the first place, irregularities in 

 the walls, not included in the theoretical investigation, may 

 play an essential part. Again, according to the view of Lord 

 Kelvin, the theoretical stability for infinitely small disturb- 

 ances at all viscosities may not extend beyond very narrow 

 limits ; so that in practice and under finite disturbances the 

 motion would be unstable, unless the viscosity exceeded a 

 certain value. Two other suggestions which occurred to me 

 at the time of writing my first paper as perhaps pointing to 

 an explanation may now be mentioned. It is possible that 

 there may be an essential difference between the motion in 

 two dimensions to which the calculations related, and that in 

 a tube of circular section on which observations are made. 

 And, secondly, it is possible that, after all, the investigation 

 in which viscosity is altogether ignored is inapplicable to the 

 limiting case of a viscous fluid when the viscosity is supposed 

 infinitely small. There is more to be said in favour of this 

 view than would at first be supposed. In the calculated 



* Proc. Math. Soc. February 12, 1880. 

 t Ibid. November 1887. 



