62 Lord Rayleigh on the Question of 



motion there is a finite slip at the walls, and this is inconsistent 

 with even the smallest viscosity. And, further, there are 

 kindred problems relating to the behaviour of a viscous fluid 

 in contact with fixed walls for which it can actually be proved* 

 that certain features of the motion which could not enter into 

 the solutions were the viscosity ignored from the first are 

 nevertheless independent of the magnitude of the viscosity, 

 and therefore not to be eliminated by supposing the viscosity 

 to be infinitely small. Another case that may be instanced is 

 that of a large stream of viscous fluid flowing past a spherical 

 obstacle. As Sir Gr. Stokes has shown, the steady motion is 

 the same whatever be the degree of viscosity ; and yet it is 

 entirely different from the flow of an inviscid fluid in which 

 no rotation can be generated. Considerations such as this 

 raise doubts as to the interpretation of much that has been 

 written on the subject of the motion of inviscid fluids in 

 the neighbourhood of solid obstacles. 



The principal object of the present communication is to 

 test the first of the two latter suggestions. It will appear 

 that, as in the case of motion between parallel plane walls, so 

 also for the case of a tube of circular section, no disturbance 

 of the steady motion is exponentially unstable, provided vis- 

 cosity be altogether ignored. 



Referring the motion to cylindrical coordinates z, r, 0, 

 parallel to which the component velocities are w, u, v, we 

 havef 



du v 2 __ dQ dv uv _ ldQ dw _ dQ 

 "SF "" 7 ~" ~d? d* + "7 ~ ^ d0' W~"dz~' 



d d , d v d d 



Wt = dt+ U dr + rTe +W dz> 



where —Q,=Y+p/p. 



These are the general equations. In order to apply them 

 to the present problem of small disturbances from a steady 

 motion represented by 



u=0, v=0, w=W, 



where W is a function of r only, we will regard the complete 

 motion as expressed by u, v, W + w, and neglect the squares 

 of the small quantities u, v, w, which express the disturbance. 



* "Ob the Circulation of Air in Kundt's Tubes," Phil. Trans. 

 November 1883. 



t Basset's ' Hydrodynamics/ § 470. 



