248 
CALCULUS 
16. Two-Dimensional Flow. Consider a solid cylinder of arbi 
trary cross-section, S, cut off by two planes perpendicular to its ele 
ments. Let the ends be insulated, and let the remaining surface be 
maintained at prescribed temperatures which shall not change with 
the time. In particular, the surface temperature shall be a con 
tinuous function, and it shall be constant along each element of the 
cylinder. 
The limiting flow, i.e. the steady flow which corresponds to the 
surface conditions, will be one in which the lines of flow all lie in 
planes parallel to the bases ; and the lines of flow in one of these 
planes project on the lines of flow in any other plane. 
Thus the flow is completely described by the flow in one of these 
planes. Let the (x, y)-axes be chosen in this plane. Then u does 
not depend on z ; hence d^u/dz 1 = 0, and Laplace’s Equation reduces 
to 
Equation (8) of § 12 reduces to 
(2) 
provided that the altitude of the cylinder is unity. 
A necessary and sufficient condition for a steady flow, when K is 
constant, is that 
(3) 
r 
for every sub-region, the integral being extended over the complete 
boundary in the positive sense. Equation (3) follows from Laplace’s 
Equation, and conversely ; § 4. 
Flow of Electricity. A two-dimensional 
flow can be realized as follows. Consider a 
piece of tin foil. Let the edge be connected 
with a thick piece of copper, and let one pole 
of a battery be connected with the copper ; 
the other, with an interior point of the tin 
foil. Then a flow of electricity in the tin 
foil will be established, and since the resis 
tance of the copper is negligible, while that 
Fig. 60