592 Dr. A. Russell on the Convection of Heat 



found some approximate solutions which deserve to be more 

 widely known. The author has therefore thought that it 

 would be useful to give the proofs in full of the more practical 

 of Boussinesq^s formulae, laying stress on their limitations, 

 and pointing out some of their applications. The author also 

 discusses the important problem of the heating, due to stream- 

 line convection, of a liquid flowing through a cylindrical 

 tube, and gives a table by means of which approximate 

 solutions can be found without much difficulty. 



2. Historical. 



The differential equations the solutions of which would give 

 the flow of heat through a fluid were first given by Fourier*. 

 The fundamental equation was put into a more manageable 

 form by Poisson f, but neither he nor Fourier gave any 

 solution of it. 



Poisson writes the equation as follows : — 



where is the temperature o£ the fluid at the point (#, y, z), 

 c the capacity for heat per unit volume, k the conductivity, 

 and D6/Dt the rate at which the temperature of a particle of 

 fluid passing through the point (#, y, z) is increasing in the 

 direction of the motion of the fluid at the point. When 

 written in this form it is interesting to notice how similar this 

 equation is to the equation of the flow of heat through a solid 

 body. We may also write 



m he ^6 g^ he 



ln=& + u ^ +v ^ +w ^ • • • ^ 



where u, v, and w are the component velocities of the 

 current at the point (as, y, z) parallel to the . three axes 

 respectively. 



In addition to equation (1) we have the ordinary hydro- 

 dynamical equations J, namely, the equation of continuity 

 and the three equations of Euler. 



A. Oberbeck § discusses the general equations, and gives a 

 solution for a special case. In a valuable paper || L. Lorenz 



* Memoir es de lAcademie, t. xii. p. 507 (1820), or (Euvres de Fourier 

 (Darboux's Edition), t. ii. p. 275. 



t Theorie Mathematique de la Chaleur, chapter iv. (1835). 

 j Lamb's 'Hydrodynamics,' chap. i. 

 S Ann. der Physik,\\i. p. 271 (1879). 

 ij Ann. der Physik, xiii.p. 582 (1881). 



