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LXII. Further Remarks on the Stability of Viscous Fluid 

 Motion. By Lord Kayleigh, O.M., F.E.S.* 



AT an early date my attention was called to the problem 

 of the stability of fluid motion in connexion with the 

 acoustical phenomena of sensitive jets, which may be ignited 

 or unignited. In the former case they are usually referred 

 to as sensitive flames. These are naturally the more con- 

 spicuous experimentally, but the theoretical conditions are 

 simpler when the jets are unignited, or at any rate not 

 ignited until the question of stability has been decided. 



The instability of a surface of separation in a non-viscous 

 liquid, i. e. of a surface where the velocity is discontinuous, 

 had already been remarked by Helmholtz, and in 1879 I 

 applied a method, due to Kelvin, to investigate the character 

 of the instability more precisely. But nothing very practical 

 can be arrived at so long as the original steady motion is 

 treated as discontinuous, for in consequence of viscosity such 

 a discontinuity in a real fluid must instantly disappear. A 

 nearer approach to actuality is to suppose that while the 

 velocity in a laminated steady motion is continuous, the 

 rotation or vorticity changes suddenly in passing from one 

 layer of finite thickness to another. Several problems of 

 this sort have been treated in various papers f. The most- 

 general conclusion may be thus stated. The steady motion 

 of a non-viscous liquid in two dimensions between fixed 

 parallel plane walls is stable provided that the velocity U, 

 everywhere parallel to the walls and a function of y only, is 

 such that d 2 XJ/dy 2 is of one sign throughout, y being the 

 coordinate measured perpendicularly to the walls. It is 

 here assumed that the disturbance is in two dimensions and 

 infinitesimal. It involves a slipping at the walls, but this 

 presents no inconsistency so long as the fluid is regarded as 

 absolutely non-viscous. 



The steady motions for which stability in a non-viscous 

 fluid may be inferred include those assumed by a viscous 

 fluid in two important cases, (i.) the simple shearing motion 

 between two planes for which d 2 U/r/y 2 = 0, and (ii.) the flow 

 (under suitable forces) between two fixed plane walls for 

 which d 2 XJ/dy 2 is a finite constant. And the question pre- 

 sented itself whether the effect of viscosity upon the dis- 

 turbance could be to introduce instability. An affirmative 



* Communicated by the Author. 



t Proc. Loud. Math. Soc. vol. x. p. 4 (1879) ; xi. p. 57 (1880) ; xix. 

 p. 67 (1887) ; xxvii. p. 5 (1895) ; Phil. Mag. vol. xxxiv. p. 59 (1892) ; 

 xxvi. p. 1001 (1913). Scientific Papers, Arts. 58, 66, 144, 216, 194. 



Phil. Maq. S. G. Vol. 28. No. 166. Oct. 1914. 2 R 



