576 VISCOSITY. [CHAP, xi 



312. The theoretical explanation of the instability of lineal- 

 flow, under the conditions stated, and of the manner in which 

 eddies are maintained against viscosity, is at present obscure. We 

 may refer, however, to one or two attempts which have been .made 

 to elucidate the question. 



Lord Rayleigh, in several papers*, has set himself to examine 

 the stability of various arrangements of vortices, such as might be 

 produced by viscosity. The fact that, in the disturbed motion, 

 viscosity is ignored does not seriously affect the physical value 

 of the results except perhaps in cases where these would imply 

 slipping at a rigid boundary. 



As the method is simple, we may briefly notice the two-dimensional form 

 of the problem. 



Let us suppose that in a slight disturbance of the steady laminar motion 



where U is a function of y only, we have 



u=U 

 The equation of continuity is 



+ = .................................... (ii). 



dx ay 



The dynamical equations reduce, by Art. 143, to the condition of constant 

 angular velocity Z)/Zte = 0, or 



dU 



Hence, neglecting terms of the second order in u', v' t 



I 



( 



d TT d\(dv' du' , 



dt + U Tx te-~ V = 



Contemplating now a disturbance which is periodic in respect to x, we 

 assume that u' t v' vary as e ** a: + ?<r ^ Hence, from (ii) and (v), 



and 



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

 Soc., t. xi., p. 57 (1880), and t. xix., p. 67 (1887) ; " On the Question of the Stability 

 of the Flow of Fluids," Phil. Mag., July 1892. 



