374 
PROF. LOUIS YESSOT KING ON THE CONVECTION OF 
PART I. 
Mathematical Theory of the Convection of Heat from a Cylinder of any 
Form of Cross-section in a Stream of Fluid 
Section 1. Introduction. 
The general problem of the convection of heat from bodies immersed in moving 
media has recently received considerable attention both from the theoretical and 
experimental point of view. The equation of the conduction of heat in a moving 
fluid was stated by Fourier as long ago as 1820,( 1 ) and a few years later was 
expressed by Poisson ( 2 ) and Ostrogradsky ( 3 ) in the familiar form 
c 
De> 
D t 
d_( 00 \ 
dxY 0 x) 
ayv aj 
(i) 
where 6 is the temperature of the fluid at any point (x, y, z), c the heat capacity of 
the fluid per unit volume, k its thermal conductivity, and D/D t the “ mobile 
operator” D/Dt = d/dt+ud/dx + v d/dy+w d/dz of the hydrodynamical equations. 
In 1901 the problem was taken up by Boussinesq, ( 4 ) whose memoir on the subject 
in 1905 contains a great number of successful calculations of heat losses from bodies 
of various shapes immersed in a stream of fluid. 
A full account of the theoretical development of the subject is given by Russell,( 5 ) 
and extensive references are given to papers and memoirs relating to the convection 
of heat. 
Section 2. Boussinesq’s Transformation. 
Under certain assumptions Boussinesq, by an extremely elegant transformation, 
was able to reduce (l) to a differential equation capable of solution. Assuming a 
frictionless, incompressible fluid, the flow of liquid past an obstacle maps out the field 
in the neighbourhood of an obstacle by stream lines and equipotential surfaces which 
in some cases may constitute a set of orthogonal co-ordinates. If in such cases the 
general equation of heat conduction (l) be expressed in these co-ordinates, it takes a 
if Fourier, ‘ M6moires de l’Academie,’ t. 12, p. 507, 1820. 
( 2 ) Poisson, ‘ Theorie Mathematique de la Chaleur,’ 18-35. 
( 3 ) Ostrogradsky, ‘St. Pet. Ac. Sc. Bll.,’ t. 1, p. 25, 1836. 
( 4 ) Boussinesq, ‘Comptes Rendus,’ vol. 133, p. 257; also ‘Journal de Matkematiques,’ vol. 1, 
pp. 285-332, 1905. 
( 5 ) Russell, ‘Phil. Mag.,’ vol. 20, pp. 591-610, October, 1910. 
