Orr — Stability or Instability of Motions of a Perfect Liquid. 61 



CHAPTEE III. 



Motion in Cylindrical Steata eotating round a Common Axis. 

 Art. 22. Lord Bayleigh^s reference to this case. 



Lord Eayleigh has remarked* that when the fluid is bounded by fixed 

 concentric cylindrical walls, and the stream-lines are circles in planes 

 perpendicular to the axis, the motion is stable, provided that in the steady 

 motion the rotation continually increases or decreases from one boundary to 

 the other. 



As with the preceding cases of steady motion, he evidently refers to the 

 fundamental disturbances solely (and even then, I think, the argument he 

 indicates is inapplicable to those in three dimensions) ; but, as has been 

 shown in the preceding chapters, stability for fundamental disturbances is 

 quite compatible with instability for those of a more general character. 



Art. 23. Tivo-dimensioned Disturbances 'when steady fiovj is tha.t of Viscous 

 Liqidd ; the Fundamental Types ; Resolution of one initially 

 a.rhitrary. 



I proceed to discuss this problem also in some detail. Eeferring to 

 two dimensions alone, we may conveniently use the current function ;//, in 

 terms of which the velocities in the disturbed relative to the steady motion 

 are, radially u = dip/rcW, and circumferentially v = - c^/dr. If F denote 

 the velocity in the steady motion, the vorticity is 



du Id 

 m'-rdr'^'^''-'^'^' 



^,i#^i^_ll(,r)- (1) 



^^2 r^^. ^ r-^. ae- rdr^ ^' ^ ^ 



and the differential equation governing the motion may be conveniently 

 obtained by expressing that this remains constant for any given element of 

 fluid, i.e. that 



d ^^^ ^ d d \ (d^ 1 cl-il 1 d-1 I d ^ ^\ ^ 



* " On the Stabilitj- or Instability of certain Fluid Motions" : Proc. Lond. Math. Soc. xi., 1880 ; 

 Collected Papeis I., pp. 474-487, concluding paragraph. 



