A THEORY OF FLUID FRICTION. 77 
considered as rollers between the fast-moving core of fluid and the fixed wall of the 
pipe, and their size would be practically constant, for uniform flow of the core, so 
long as the roughness of the surface of the pipe is unchanged. 
If the value of /’, in equation (2), were experimentally determined for various 
fluids and through a range of temperature, then the loss of head in a given length 
of pipe of any diameter and for any velocity of flow might be determined from the 
value of the coefficient of friction f obtained by experiment on any size of pipe hav- 
ing the same roughness of surface. 
If a portion of the length of the pipe is roughened, the volume of flow will be un- 
changed, but the curve of velocities across the section will be altered. The velocity 
at the axis of the pipe will be increased, and the rubbing velocity will be decreased. 
This is illustrated in Fig. 2, Plate 57. 
FLOW IN OPEN CHANNELS. 
Consider a smooth, open channel or trough of constant section and of uniform 
slope in which water is flowing steadily. Let y be the depth of the water and wv the 
velocity. Suppose a portion of the length of the channel to be roughened and let Y 
be the depth of the water and V the velocity in this roughened length after steady 
motion is attained, the volume of flow being unchanged. In determining where the 
change in velocity occurs, we find four conditions depending on the relative value 
of y and v. 
If any reduction in velocity causes a decrease in the total head, the change in 
velocity will occur on the rough surface, since the friction will be greater until the 
velocity has been reduced, and it is this increase in friction which causes the loss of 
head. The change back to the greater velocity will occur on the smooth surface, 
since there is a gain in total head, and the friction will be less on the smooth surface 
until the original velocity is acquired again (Fig. 3, Plate 58). 
If any reduction in velocity causes an increase in the total head, the change in 
velocity will occur on the smooth surface, where the friction is less than normal. 
The change back to the greater velocity will occur on the rough surface, where the 
friction is greater than normal and absorbs the excess of head (Fig. 4, Plate 58). 
If the reduction in velocity causes first a decrease and then an increase in total 
head, the final change being a decrease, it would appear that the whole decrease in 
velocity will occur on the rough surface. In order that there may be no increase in 
head at any stage of the change, it is probable that the surface water is retarded 
sooner than the bottom water, and the velocity head is greater in proportion to the 
depth than it would be if the velocity wer uniform from top to bottom. The change 
back to the greater velocity will occur partly on the rough surface, and partly on the 
smooth, since the increased friction on the rough surface due to the increased veloc- 
ity will cause a loss of head, and then the decrease in friction on the smooth surface 
will cause the head to increase again (Fig 5, Plate 59). 
If the reduction in velocity causes first a decrease and then an increase in total 
