406 Mr Wilson, On Convection of Heat. 



On Convection of Heat. By Harold A. Wilson, M.A., D.Sc, 

 Fellow of Trinity College, Cambridge. 



[Read 15 February 1904.] 



The object of the following paper is to determine the distribu- 

 tion of temperature, in certain cases, in media moving between 

 fixed boundaries, the conditions at the boundaries being supposed 

 known. The theory of conduction of heat in bodies at rest 

 relatively to their boundaries has of course been worked out very 

 completely by Fourier, Lord Kelvin and other investigators, but 

 so far as the writer is aware, very little* has been done towards 

 the solution of the corresponding problems in moving media. 



The methods employed in the present paper are two, corre- 

 sponding with the two methods invented by Fourier and Lord 

 Kelvin respectively for the solution of problems on conduction of 

 heat, in bodies at rest. The first method is to find a solution of 

 the differential equation which satisfies the given conditions, while 

 the second method is to regard the sources of heat as made up 

 of point sources and to obtain the temperature at any point as 

 the sum of all the temperatures due respectively to each point 

 source. 



It will be convenient first to obtain the general differential 

 equation, in Cartesian coordinates, which the distribution of tem- 

 perature must satisfy in a medium moving in any manner. Let 

 the thermal conductivity of the medium be denoted by k, its 

 density by p and its specific heat by s. We shall suppose that h is 

 the same in all directions and independent of the temperature and 

 density of the medium. The specific heat s will be taken to be 

 proportional to p so as to include the case where the medium is a 

 gas at constant pressure, but in the problems which will be 

 considered in this paper the variations of temperature will be 

 supposed so small that p may be regarded as a constant. If the 

 medium is a viscous fluid, then, when it flows past fixed 

 boundaries, heat will be developed in it, but in what follows we 

 shall suppose that the amount of heat developed in this way is 

 negligible. 



Let x, y, z be the coordinates of a point P in the medium 

 and let the velocity components of the medium at P be u, v, w 

 respectively. Consider a small rectangular parallelipiped whose 

 sides are Ax, Ay, Az situated with P at its centre. The heat 



* A case of the flow of a liquid through a tuhe has been solved by L. Graetz, 

 Wied. Ann. xvm. p. 78. 



