436 ON CONVECTION CURRENTS IN A [412
unimportant and even the variation with temperature may be disregarded except in so far as it modifies the operation of gravity. If we write p = p0 + Bp, we have
ffP = ffPo (l + Sp/Po) = ffP° - 9P<>a0>
where 8 is the temperature reckoned from the point where p =• p0 and a is the coefficient of expansion. We may now identify p in (1) with p0, and our equations become
^ _ __ I ^ ^__1^? ^ - _ 1 ^ ft (A\
Dt~ p dx} Dt~ p dy' Dt ~ Pdz+y> ...... ()
where p is a constant, 7 is written for ga, and P for p + gpz. Also, since the fluid is now treated as incompressible,
du dv dw ,
j- + -7- -f -j~ = 0 ............................... (5)
da) dy dz ^
The equation for the conduction of heat is
Dt~
in which re is the diffusibility 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 z — 0 and z=%, where also the temperatures are maintained constant. In the equilibrium condition u, v, w vanish and 9 being a function of z only is subject to d^djdz2= 0, or ddjdz = ^, where /3 is a constant representing the temperature gradient. If the equilibrium is stable, /3 is positive ; and if unstable with the higher temperature below, /3 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 •BJ- for P, where
W =P_pry f ®dz ............................... (7)
In equation (6) D6/Dt is to be replaced by D0/Dt + w/3.
The question with which we are principally concerned is the effect of a • small departure from the condition of equilibrium, whether stable or unstable. For this purpose it suffices to suppose u, v, w, and Q to be small. When we neglect the squares of the small quantities, D/Dt identifies itself with djdt and we get
du 1 dor dv 1 diff dw 1 d™ed force,