76 A THEORY OF FLUID FRICTION. 
This force reduces the pressure head on the fluid of the core, and this loss of 
head in a given length of the pipe may be expressed as— 
F.tana.L.2nR 
(as w. th? (2) 
In this equation, w is the weight of a unit volume of the fluid. 
After the flow in a pipe has become steady, the pressure head at any cross- 
section must be constant across the section. The velocity varies, however, and so the 
total head is not uniform across the section. The loss of pressure head in a given 
length will be the same for a core of any size, and it will be seen from equation (1) 
that for this loss of pressure head to be constant, tan a must vary inversely as the 
radius of the core. This is a property of the common parabola, having its vertex on 
the axis of the pipe, and such would seem to be the curve of velocity. 
If the wall of the pipe were very smooth, so that there would be no eddying adja- 
cent to it, this curve of velocity would hold for the full diameter of the pipe. Assum- 
ing that the friction varies as the square of the velocity of the fluid adjacent to the 
wall of the pipe, the friction of the wall of the pipe on the fluid would be 
f. Ve. L .2nR, 
in which f is the coefficient of friction as generally used, and V, is the velocity adja- 
cent to the wall. It will be seen that, when eddying is neglected, this equals the fric- 
tion in the fluid at the wall of the pipe, or— 
f. Ve. L.anR, = F.tan o,.L. 20R,, 
or fio eS IP EA, Oy, (2) 
For a similar curve of velocities across the section, it will be noticed that, if the 
velocity be doubled, the diameter of the pipe must be halved, since tan a varies as 
V 2 ONE : 
RZ and also as V*. This fact has been noticed in experiments. 
Near the wall of the pipe, however, the flow is not uniform, but is full of eddies, 
as is evidently indicated by the fact that a stationary wall, however smooth, cannot 
replace surrounding fluid moving at the speed suited to an extension of the curve 
of velocities. The curve of velocities is modified by this eddying, somewhat as indi- 
cated in dotted lines on Fig. 1, Plate 57. The total head in the region of the eddies 
is probably the same as if there were no eddying, since the pressure head must be 
the same as for the rest of the core of fluid and the energy of the eddy motion is 
likely to compensate for the lack of forward velocity of the fluid. The extent of 
the eddying would probably increase as the roughness of the surface increases, and 
also as the velocity increases. It may be also that the eddies keep on growing as the 
distance from the end of the pipe decreases, but this is hardly likely, as they may be 
