Convection of /Feat and Similitude. G93 



The equations required for this paper may he deduced from 

 his. However, they may also be obtained from the principle 

 of similitude. Rayleigh * has pointed this out for the case 

 of forced convection. It is instructive to work out the 

 case for natural convection from this alternative point of 



view 



Let the fundamental units he those of mass (M), length (L), 

 time (T), and temperature (®) . The derived units are given 

 in brackets below. For an inviscid fluid, Boussinesq's 

 equations show that the mean heat-loss "A" (MT" :; ) per 

 unit area of a body per unit time depends upon 



k, (M C ~ 3 L0 -1 ) thermal conductivity of the fluid. 



c. (MT -2 !* -1 ©" 1 ) capacity for heat per unit volume of 



the fluid. 

 0, (0) temperature excess of the body. 



<(, (B _1 ) coefficient of density reduction of the 



fluid per degree lise of temperature. 

 //, (LT~ 2 ) acceleration due to gravity. 



/, (L) linear dimensions of the body. 



Sinc^ we deal with the capacity for heat per unit volume 

 of the fluid, reflection shows why the density (p) of the fluid 

 does not enter. The convection currents, essentially gravity 

 currents, depend upon the density of the heated fluid relative 

 to that of the cool fluid, and not upon the absolute densitv. 



Boussinesq regards the effect of the thermal expansion of 

 the fluid as negligible except in so far as it alters the weight 

 of unit volume of the fluid — that is, except in so far as "a" 

 and "g " occur as a product. Consequently, let 



* = r,T(#; (1) 



then we readily derive : 



By mass 1 = u + v, 



By length = u — v+x + z, 



By time - 3 = - 3w - 2r - 2x, 



By temperature = —u — v + w — x : 



whence 



u = 1-2.6-, w = 1 + .r, : = 3.t- -1, r'= 2x. . (2) 



* Rayleigh, ' Nature,' xcv. p. 66 (1915). 



