A THEORY OF FLUID FRICTION. 81 
It will be seen that for constant value of the product of the velocity by the 
length of surface, the resistance varies as the velocity. 
The curves of resistance for varying lengths are plotted in Fig. 9, Plate 61, 
for speeds of 800, 600, 400, 300, 200 and 100 feet per minute, as determined from 
the above considerations, assuming F = .03538 and tan « = 10.000 for speed of 
600 feet per minute. Curves of resistance for the same speeds as given by Mr. W. 
Froude for shellacked surface are shown in Fig. 10, Plate 61. 
The theory as above exemplified does not fit in with the numerous experiments 
in which the resistance has been shown to vary as the square of the velocity. It 
seems to be only for rough surfaces, however, that the resistance appears to vary 
as the square of the velocity, and in such cases the effect of the elevation of the sur- 
face of the water would appear to be too important to be neglected. For the same 
length of surface, the resistance at low speeds would be less than when this effect 
is negligible, while its influence diminishes as the speed increases. This may ac- 
count for the higher exponent for rough surfaces. 
The theory may be summarized as follows :— 
1. Frictional resistance per unit of surface varies as the square of the equiva- 
lent rubbing velocity determined by considering the flow to be laminar. 
2. Friction between adjacent lamine of the fluid varies as the rate of change 
of velocity occurring at their boundary. 
3. In the case of a fixed body and a moving fluid, the frictional resistance ex- 
perienced up to a given distance abaft the cutwater must be balanced by the loss of 
head in the wake at the same distance abaft the cutwater. 
4. The width of the wake at a given distance abaft the cutwater and the re- 
duction of velocity in the wake at various distances from the fixed body may be 
determined by the consideration that the loss of head outboard of any lamina is 
equal to the frictional resistance experienced by the lamina before peeing the 
given distance from the cutwater. 
5. The determination of the loss of head in the wake is complicated by the 
condition that the volume of the fluid passing the fixed body is not altered, al- 
though the velocity in the wake is decreased. The consequent elevation of the 
surface of the fluid becomes of considerable importance when the velocity is rela- 
tively small or the reduction in velocity occurs quickly on account of roughness of 
surface. 
6. The energy and action of the eddies may be assumed to be such that the 
loss of head may be determined as though the flow were entirely laminar. 
In conclusion, it may be noted that laminar flow in the open seems hardly in 
accordance with ail the facts as observed, since the friction of wind on water pro- 
duces waves. 
