Orr — Stability or Instahilitij of Motions of a Perfect Liquid. 17 



CHAPTER I. 



Rectilinear Motion in Plane Layees, chiefly the case of a Liquid 



Sheaking Uniformly. 



Art. 1. Lord Bayleigh's Investigations. 



The oscillations which are possible in a stream of liquid, supposed 

 frictionless, flowing between two fixed parallel planes, have been discussed 

 in a series of papers by Lord Rayleigh. It appears desirable to give a brief 

 account of some of his investigations. In one of his earliest papers on the 

 subject,* he supposes that the axis of y is drawn at right angles to these 

 planes, and that the velocity in the steady motion is U in the direction of 

 the axis of x, U being a function of y only. He considers only two- 

 dimensioned disturbances; in these denote the x, y components of velocity 

 by U+u, v; let ^denote the vorticity in the steady motion, i.e. ^dUjdy, and 

 Z, denote the additional vorticity, i. e. ^ {dujdy - dvjdx). Since, in the absence 

 of friction, the vorticity of each element remains constant, we have 



l.^(Z.Z).{U.n)l-JZ.l)..l^^(Z.Z)-,, (1) 



or, if we retain only the first powers of small quantities, 



which may be written in the form 



d ^^d\fdu dv'\ d^U ^ ,^, 



+ ^:^J(:7--:57J + ^':5:t = 0. (3) 



/ d d \ (du dv\ d^ I 



\dt dxj \dy dxj dy 



Introducing the supposition that as functions of x, % and v vary as e***, and 

 using the equation of continuity 



duldx + dvjdy = 0, (4) 



or, as it now becomes, '■■ttO 



iku + dvjdy = 0, (5) 



we obtain, on elimination of u, 



f^ + iklA {d'v/dy' - h^v) - ikvd'U/dy^ = 0. (6) 



* " On the Stability or Instability of certain Fluid Motions," Proc. Lond. Math. Soc. xi., p. 57, 

 1880, Collected Scientific Papers, I., p. 484. 



R. I. A. PKOC, VOL. XXVII., SECT. A. [3] 



