Oer — Stability or Instability of Motions of a Perfect Liquid. 43 



CHAPTEE II. 



The Case of Flow through a Pipe whose section is a circle or 

 TWO concentric circles. 



Art, 15. Lord Bayleigli's Investigation. 



Lord Eayleigh has discussed* also the question of the stabihty of 

 steady flow through a pipe of circular section, or an annular pipe whose 

 section is two concentric circles, and has concluded that [when the undis- 

 turbed motion is that appropriate to a viscous fluid] no disturbance of the 

 steady motion is exponentially unstable, provided viscosity be altogether 

 ignored. It seems desirable to quote the substance of his discussion at least 

 for a disturbance symmetrical about the axis. Eeferring the motion to 

 cylindrical coordinates z, r, B, parallel to which the component velocities 

 are w, u, 0, we have 



Lu dO Bid dO _,_ ,,^ ,,, ,, , 



r=— = -;— , — — = -— , D LDt = a at + ua dr + vMldz, 

 Vt dr JJt dz 



where - Q = V + p/p, and V is the potential of the impressed forces. In 

 applying these general equations to the present problem of small disturbances 

 from a steady motion represented by tt = 0, vj = W, where ^ is a function 

 of r only, the complete motion is regarded as expressed by u, W + lo, and the 

 squares of the small quantities u, w are neglected. 



Thus :— duldt + Wduldz = dQIdr, (1) 



dwjdt + ud W/dr + Wdwjdz = dQ/dz, (2) 



which, with the equation of continuity, 



d {rit)/dr + rdvj/dz = 0, (3) 

 determine the motion. 



The next step is to introduce the supposition that, as functions of t, z, the 

 variables u, iv, Q are proportional to e«(«^+^'=). 



This gives i(n + k W) u = dQ/dr, (4) 



ltd W/dr + i(n + kW)w = ihQ, (5) 



d (ru)ldr + ikrw = 0. (6) 



Eliminating w, Q, there is obtained the equation 



If the undisturbed motion be that of a viscous fluid, W is of the form 



*" On the Question of the Stability of the Flow of Fluids," Phil. Mag. xxxiv., 1892, p. 59, 

 Scientific Papers, III., p. 578. 



[6*] 



