556 XL HYDRODYNAMICS. 



heat. The example just treated is an example of this, for we find 

 at once 



2 f) QJ 



and the vorticity is propagated like the velocity. 



As a final example, let us consider a case of laminar motion in 

 which u, as a function of y, has a discontinuity, this having an 

 important application to the theory of thin plane jets and flames, 

 including sensitive flames. 1 ) We will suppose that at the time t = 

 for y < u has a certain constant value, and that for y > it has 

 a different constant value. It is easy to see that this is equivalent 

 to supposing that there is no vorticity except in an infinitely thin 

 lamina at y = 0. For we have 



s s 



/-* /-* 



263) /^ = --l / ^dy = ~(u -u,) 



/ i > i/ 2 / c i/ 2 ^ * ' 



t/ t/ y 



e c 



where u is the velocity on one side, u 2 that on the other of the 

 layer of thickness 2s. Now if the thickness decrease without limit, 

 while g increases without limit, the integral may still be finite, as 

 we shall suppose. 



We have then to find two solutions u and f of equation 258), 

 so related that g = -^- Let us put s = '* > and try to find a 

 particular solution that is a function of s alone. We have 



du du ds _ 1 du y 



dt ~~ ds dt 2 ds }/i* 



du du ds du I 



d^ == ~ds^~'ds'y^ ) 



d*u _ 1 d*u ds _ d*u 

 dy 9 yt ds* dy t ds* 



so that our equation becomes the ordinary differential equation, 



1) Rayleigh, On the Stability, or Instability, of certain Fluid Motions 

 Proc. London Math. Soc., xi., pp. 57 70, 1880. Scientific Papers, Vol.1, p. 474. 



