A THEORY OF FLUID FRICTION. 
By W1Lt1AM GaTEwoop, Esg., MEMBER. 
[Read at the twenty-fourth general meeting of the Society of Naval Architects and Marine Engineers, held in 
New York, November 16 and 17, 1916.] 
The subject of fluid friction has received a great deal of attention, and em- 
pirical formulz are in general use in connection with the flow of water, air and 
other fluids in closed pipes, and with the flow of water in open channels. We have 
empirical formulz also for determining the frictional resistance of vessels, derived 
from an analysis of experiments on flat boards. Until very recently, however, very 
little progress has been made in checking the accuracy of the assumptions made in ex- 
tending the results of these experiments to higher speeds, greater lengths, and ship- 
shape forms. 
It seems that the reason why our knowledge of fe ictional resistance is so uncer- 
tain is due to the fact that there is no generally accepted theory of fluid friction. 
This paper is an attempt to correlate the laws of fluid motion which seem to have 
a bearing on the subject, and to formulate a theory. 
FLOW IN PIPES. 
Experiment has shown that the velocity of a fluid flowing through a pipe is not 
uniform across the section. When the flow is steady, the velocity is greatest at the 
axis of the pipe, diminishes slowly as the distance from the axis increases, and then 
seems to diminish rapidly in the immediate neighborhood of the wall of the pipe. 
When there is a variation in velocity across the section of a fluid in steady flow, 
it seems reasonable to consider that the resistance offered by the slower moving 
fluid to the fluid which is moving faster varies directly as the rate of change in 
velocity. 
On this hypothesis, the frictional force exerted on a central core of the fluid in 
a pipe by the surrounding fluid may be expressed as— 
fF tana. L.2nkR. 
In this expression, F is a coefficient of friction which will vary with the nature and 
temperature of the fluid. It may be called a viscosity coefficient. Tan a is used to 
denote the rate of variation of velocity at the circumference of the core of fluid, and 
may be expressed as so many feet per second per foot of width. L is the length of 
the pipe and F is the radius of the core. If foot, second, pound units are used, the 
frictional force given by the above formula will be the force in pounds. 
