Stability of Viscous Fluid Motion. 613 



while u, v themselves satisfy the " equation of continuity " 



r+P-0 ( 3 ) 



ax ay 



In other applications of (1), e. g. to the diffusion of heat 

 or dissolved matter in a moving fluid, f is a new dependent 

 variable, not subject to (2), and representing temperature or 

 salinity. We may then regard the motion as known while 

 f remains to be determined. In any case ^D% 2 /Dt = v£S7 2 & 

 If the fluid move within fixed boundaries, or extend to 

 infinity under suitable conditions, and we integrate over the 

 area included, 



so that 



Id 



2dt 



tf?d>vdy = v\iz\7 2 Zdxdy 



-i&-$i@?A ®>* (4) 



by Green's theorem. The boundary integral disappears, if 

 either f or dfldn there vanishes, and then the integral on 

 the left necessarily diminishes as time progresses*. The 

 same conclusion follows if f and dfldn have all along the 

 boundary contrary signs. Under these conditions £ tends to 

 zero over the whole of the area concerned. The case where 

 at the boundary f is required to have a constant finite value 

 Z is virtually included, since if we write Z + f for f, Z dis- 

 appears from (1), and f everywhere tends to the value Z. 



In the hydrodynamical problem of the simple shearing 

 motion, f is a constant, say Z, u is a linear function of y, 

 say U, and u=0. If in the disturbed motion the vorticity 

 be Z+f, and the components of velocity be U + " and v, 

 ■equation (1) becomes 



|+cp+.)2+r|«y% • ■ • m 



in which f, u, and v relate to the disturbance. If the dis- 

 turbance be treated as infinitesimal, the terms of the second 

 order are to be omitted and we get simply 



5 + uf =*V 5 ?. (G) 



at ax 



* Compare Orr, I. c. p. 115. 



