[ 329 ] 



XXXIII. On the Stability of the Simple Shearing Motion of a 

 Viscous Incompressible Fluid. By Lord Kayleigh, Q.M., 

 F.R.S* 



A PRECISE formulation of the problem for free in- 

 finitesimal disturbances was made by Orr (1907) f. 

 It is supposed that £ (the vorticity) and v (the velocity 

 perpendicular to the walls) are proportional to e xnt e ikx , where 

 n~p + iq. If y 2 v = S, we have 



g-{*-J+';GM-W}B, • ■ • CD 



and d 2 v/dy 2 -Pv = S, (2) 



with the boundary conditions that v = 0, dr/dy = at the 

 walls where yh constant. Here v is the kinematic viscosity, 

 and /3 is proportional to the initial constant vorticity. Orr 



easily shows that the period-equation takes the form 



J S : «* dy . jS s e-* dy- JSj e'* dy . JS 2 e**dy=Q, (3) 



where Si, S 2 are any two independent solutions of (1) and 

 the integrations are extended over the interval between the 

 walls. An equivalent equation was given a little later (1908) 

 independently by Sommerfeld, 



Stability requires that for no value of k shall any of the 

 y's determined by (3) be negative. In his discussion Orr 

 arrives at the conclusion that this condition is satisfied. 

 Another of Orr's results may be mentioned. Be -hows tin! 

 ]> + h[3y necessarily changes sign in the interval between 

 the walls J. 



In the paper quotod reference was made also to the work 

 of v. Mises and Eopf, and it was suggested fcbai the problem 

 might be simplified if it could he shown that q—vk a cannot 

 vanish. If so, it will follow that q is always positive and 

 indeed greater than v/r, inasmuch as this is certainly the 

 case when /3 = 0§. The assumption that q = v/r, by which 

 the real part of the { } in (1) disappears, is indeed a con- 

 siderable simplification, but my hope that it would lead to 

 an easy solution of the stability problem has been dis- 

 appointed. Nevertheless, a certain amount of progress has 



* Communicated by the Author. 



+ Proc. Roy. Irish Acad. vol. xxvii. 



X Phil. Mag. vol. xxviii. p. 018 (1914). 



5 Phil. Mag. vol. xxxiv. p. 69 (L s 92) ; Scientific Papers, vol. iii. p. 583. 



