534 Lord Rayleigh on Convection Currents in 



Also, since the fluid is now treated as incompressible, 



du dv dw __ « . 



dx dy dz ° • • V / 



The equation for the conduction of heat is, 



D0 _ faH ®l dW \ rec\ 



Bt ~ K \dx* + dy* + d?)> • • • • W 



in which k is the divisibility for temperature. These are 

 the equations employed by Boussinesq. 



In the particular problems to which we proceed the fluid 

 is supposed to be bounded by two infinite fixed planes at 

 2 = and 2=f, where also the temperatures are maintained 

 constant. In the equilibrium condition u, v, w vanish and 

 6 being a function of z only is subject to d' 2 $ldz 2 = 0, or 

 d0 /dz = ft, where ft is sl constant representing the tempera- 

 ture gradient. If the equilibrium is stable, ft is positive; 

 and if unstable with the higher temperature below, ft is 

 negative. It will be convenient, however, to reckon 6 as 

 the departure from the equilibrium temperature ©. The 

 only change required in equations (4) is to write m for P, 

 where 



m = F-pry$®dz (7) 



In equation (6) DO/Dt is to be replaced by D6/~Dt + wft. 



The question with which we are principally concerned is 

 the effect of a small departure from the condition of equi- 

 librium, whether stable or unstable. For this purpose it 

 suffices to suppose u, v, w, and to be small. When we 

 neglect the squares of the small quantities, D/D/ identities 

 itself with d/dt and we get 



du 1 dm dv _ 1 dm dw _ 1 dm ^ .~. 



dt = " pdx> fo~~~ply' dt~~~ pdz +r) ' ' ' W 



ti+P w = K {da? + df + d?)> ' * ' (9) 



which with (5) and the initial and boundary conditions 

 suffice for the solution of the problem. The boundary 

 conditions are that w = 0, # = 0, when 2 = or f. 



We now assume in the usual manner that the small 

 quantities are proportional to 



e ilx e imy e nt^ (] Q) 



