﻿Convective Cooling in Fluids. 923 



u\ v\ and iv' being the accelerations of the fluid parallel to 

 the axes. 



If k be the thermal conductivity of the fluid and c its heat 

 capacity per unit volume, then t', the rate of change of tem- 

 perature for a given particle with respect to time, is given by 



t'=*V 2 t (4) 



c v J 



Also, the derivative t', like the derivatives u\ v', iv' of the 

 velocities, is obtained by finding the increase in r when 

 A!, y. z, increase by udt, vdt, and wdt ; in this way we have 

 the quadruple equation 



(«', ,«, *, o = » 5( "'!'"'' T) + v ***»>*) 



d(u, r, w, t) 



+« — Y z • • • • ( 5 ) 



To the five differential equations (3) and (4) it is 

 necessary to add tho following seven boundary conditions, 

 in the first of which, /, m, n, denote the three direction- 

 cosines of the normal drawn from the interior of the fluid to 

 any element of surface of the body. 



At the surface of the solid u=v = w==Q and r = 0, 



At infinite distance (P 9 u, v, w, r) = 0. / 



In words, at the surface of the solid the fluid takes the 

 temperature 6 of the solid, and the velocity is zero. 



Following Boussinesq, let us endeavour to replace the 

 independent variables cc, y, z and the functions t, v, v, w, P 

 by others, £, 77, f, T, U, V, W, II, respectively proportional to 

 each of them, but whose ratios are chosen in a manner to 

 eliminate the parameters 0,ga, k/c,p,v. 



Let us consider the following substitutions : 



y /0gac\ l < 3 f0gac\ 13 . (9gac\ l l 3 



Kir) + Kir) y. Hir) z > 



01 



U =W) L ' v= \^r) y ' w= \r^) w ' • (7) 



pp) 11 



It is readily found that the substitutions eliminate the 

 parameters satisfactorily if cv/k is a constant and equal to 

 unity. For liquids cv/k may h;ive very large values 

 (glycerine 8000, etc., see later), but for gases, as indicated 

 by the Kinetic Theory, it is constant and approximately 



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