HEAT FROM SMALL CYLINDERS IN A STREAM OF FLUID. 
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greatly simplified form, reducing in many cases to one or other of the known partial 
differential equations of mathematical physics. The case of two-dimensional flow 
lends itself especially well to such an investigation as well as to a comparison 
with the results of experiment. If a = constant represent the stream lines and 
/3 = constant the equipotentials obtained from the solution of the hydrodynamical 
problem of flow past a cylinder of any form of cross-section (fig. 1), Boussinesq first 
showed that the general equation could be transformed to the linear form 
s 2 6 > , d 2 e 0 00 
dot 2 + d/3 2 ~ ^ 1 0/3 ’ 
(2) 
where 6 is the temperature at any point of the liquid, k its heat conductivity supposed 
to be independent of the temperature and therefore constant throughout the liquid, 
Fig. 1. Boussinesq’s transformation. 
c is the specific heat per unit volume, and V the velocity of the stream at a great 
distance from the cylinder. The constant n is defined by the relation 
2 n — cV/k = SotY/k, .. ... (3) 
where s is the specific heat per unit mass of the fluid and a its density. 
The complete solution of (2) requires a knowledge of the conditions of heat-transfer 
over the interface between solid and liquid, a point which can only be settled by 
referring to the results of experiment. If the surface of the cylinder be the particular 
stream-line a = 0, and the critical equipotentials be the curves /3 = 0 and (3 — /3 0 , the 
heat-flux per unit length of the cylinder is given by 
H = — I k (30/0a) o d/3, .(4) 
where the integral is taken to include both branches of the stream-line cl — 0. 
It will be noticed that Boussinesq’s transformation reduces the problem to the 
simple case of calculating the temperature distribution in a uniform stream flowing 
parallel to the axis of x, (a — 0), when the distribution of temperature or heat-flux is 
prescribed over the interval x = 0 to x = /3 0 . A solution of this simple transformed 
