MOTION IN RELATION TO THE SURFACE FRICTION OF FLUIDS. 203 
To measure tlle velocity of the current, one of two methods was used according to 
convenience. By one method the total quantity of fluid passing through the pipe in 
a given time was either weighed directly, or passed through a water-meter or a gas¬ 
holder, which had been designed for the purpose of the experiments and carefully 
calibrated. By the other method the velocity at the axis of the pipe was estimated 
by measuring the difference of pressure between that in a small Pitot tube facing the 
current and placed in the axis of the pipe and that in a small hole in the wall of the 
pipe. As is now known* * * § this pressure difference is accurately jrpv 2 , and from this 
relation the speed at the axis was determined. The mean flow was then calculated 
from the known ratio of the mean speed to the speed at the axis ; an investigation 
into which forms part of the present paper. 
To determine the amount of the surface friction two small holes were made in the 
walls of the experimental portion of the pipe, one at each extremity, at a known 
distance apart, and connected to a tilting manometer. The length of this portion 
varied in different pipes from 20 to 50 diameters, according to the resistance to be 
measured. In this way the fall of pressure along a given length of the pipe was 
determined, and from the known diameter of the pipe the surface friction per unit area 
was calculated. The form of tilting manometer used for the estimation of both the 
surface friction and the axial velocity, is that devised by Dr. A. P. Chattock and 
has been previously described, f For the purpose of the present paper it is sufficient 
to state that in this manometer a pressure difference of the order of 0'003 mm. of 
water can easily be detected, which is well within the limits of sensitivity required 
in these experiments. As the fall of pressure in these pipes varied from 0'5 to 
150,000 mm. of water, other manometers were required for the higher pressures, and 
for this purpose water or mercury U-tubes were used for the intermediate pressures, 
and Bourdon pressure gauges for the highest pressures. 
The Relation between the Mean Velocity of Flow and the Velocity at the Axis in 
Tubes of Circular Cross Section. 
In the case of the “stream-line” motion of a viscous fluid through a tube, to which 
the ordinary equations of motions apply, it appears both from theory and direct 
experiment that the ratio of the mean velocity over the section to the velocity at the 
axis is 0‘5. For the case of eddying flow the value of this ratio has been investigated 
by Darcy! f° r water and by Threlfall§ for air. It might be inferred from the 
conditions of similarity of flow discussed above, that the ratio in question would be a 
function of the ratio vd/v, and in a paper on the Mechanical Viscosity of Fluids read 
* ‘ Report of Advisory Committee for Aeronautics,’ 1912-13. 
f ‘ Proc. Inst. Civil Engineers,’ December, 1903, and ‘ Engineering,’ September, 1913. 
J ‘ Compfces Rendus de l’Academie des Sciences,’ vol. 38. 
§ ‘ Proc. Inst. Mechanical Engineers,’ 1904, p. 280. 
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