60 
Proceedings of the Royal Society of Edinburgh. [Sess. 
VI. — The Kinetic Energy of Viscous Flow through a Circular 
Tube. By Professor A. H. Gibson, D.Sc., University College, 
Dundee. 
(MS. received October 11, 1913. Read December 15, 1913.) 
In the stream-line flow of a viscous fluid through a circular pipe of radius 
a, the velocity of flow at any radius x is given by 
where 
v = k(a 2 - x 2 ), 
k=— . . 
4 fx dl 
From this it follows that the kinetic energy of the moving column per 
unit volume is given by 
— I 27rxv 3 dx 
2gJo 
I — . 2t rife* [ a {aPx 
2 g Jo 
- 3 cfix 3 + 3 a 2 x b - x 7 )dx 
w 
2 9 
Jtt Wa 8 . 
( 1 ) 
The mean velocity v of flow through such a tube is given by 
v = J ka 2 , 
so that the apparent kinetic energy, or the product of the mass flow and 
one-half the square of its mean velocity, is 
— • ira 2 [^ka 2 f 
2 Q 
= “ • (2) 
The true kinetic energy is therefore twice that calculated as in (2) 
from the mean velocity. 
In the majority of experiments carried out to determine the co- 
efficient of viscosity of a fluid, the head necessary to maintain a 
measured velocity of flow through a tube of known diameter and length 
