﻿922 Mr, A. H. Davis on Natural 



Part I. — Theoretical. 



Convective cooling is taken to refer to the total heat 

 transfer from a hot body by the medium of a fluid 

 moving pnst the surface. Such cooling is said to be 

 " natural " or " free " when the fluid is still, except for the 

 streams set up by the heat from the hot body itself, and is 

 said to be " forced " when the body is immersed in a fluid 

 stream, usually considered to be moving with such velocity 

 that the currents set up l>y the hot body itself are negligible. 

 The present paper is limited to natural convection. 



In 1820, Fourier * stated the equation of heat conduction 

 in a moving fluid, and in 1881, Lorenz f, upon certain 

 assumptions, gave a formula for heat loss by natural convection 

 for the special case of a vertical plane surface immersed in 

 an infinite viscous fluid. In 1901, Boussinesq J, dealing with 

 inviscid fluids, gave a general solution of the problem of 

 natural convection from heated solids in infinite fluid media. 



The following investigation follows Boussinesq closelv, but 

 introduces the modifications necessary in extending the 

 inquiry to viscous fluids. 



Adopting the same mathematical symbols as those already 

 used, let us consider the natural convective cooling of a hot 

 body immersed in an infinite viscous medium and maintained at 

 a certain temperature, 6 degrees in excess of that of the liquid 

 at infinite distance, to which all temperatures are referred. 



Let p and v be respectively the density of the fluid and its 

 kinematical viscosity. For an element of the fluid at the 

 point x y z, let r, u, v, w, P be the temperature excess 

 (assumed steady, i. e. independent of time '£'}, the three 

 components of its velocity, and the non-hydrostatic part of 

 its pressure. For elements of the fluid at infinite distance 

 these quantities are all zero. 



Let us assume that the dilatation of the fluid by heat is 

 negligible except in so far as the weight of unit volume is 

 altered, so that it occurs in the equations only when multi- 

 plied by i g\ Let gpa be the reduction in weight produced 

 in unit volume by unit rise of temperature, gpar thus being 

 the total reduction effected. The axis of z being vertical, the 

 hydrodynamical equations of continuity and of motion are 



~du . "bv Bw '■ IBP , _ 2 



3.i? 3# Bs pox 



-!? = -t>' + vV't>, ~ = -gaT-w' + vV 2 iv, 



/>(ty , P os J 



* Fourier, Memoires de V Academic, xii. p. 507 (1820). 



t Lorenz, Ann. der Phijsik, xiii. p. 582 (1881). 



X Boussinesq, Camples Rendus, cxxxii. p. 1382 (1901). 



