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MESSRS. T. E. STANTON AND J. R. PANNELL ON SIMILARITY OF 
under certain conditions, of the similarity in motions of fluids of widely differing 
viscosities and densities which has been predicted, and further by extending tbe 
observations through a range in the velocity of flow which has not hitherto been 
attempted to investigate the limits of accuracy of the generally accepted formulae 
used in calculations of surface friction. 
Previous Expeiionental Investigations. 
Apart from the researches on similarity of motion of fluids, which have been in 
progress in tbe Aeronautical Department of the National Physical Laboratory during 
the four last years, the only previous experimental investigation on the subject, as far 
as the authors are aware, has been that of Osborne Reynolds, to which a brief 
reference may be made. 
By tbe introduction of colouring matter into water flowing through glass tubes 
Reynolds showed that the motion was stream-line or lamellar in character at low 
values of the velocity of flow, and eddying or sinuous at high velocities, and that the 
change from lamellar motion to eddying motion took place suddenly at a definite 
value of the velocity (called the critical velocity), the value of which was inversely 
proportional to the diameter of the tube and directly proportional to the kinematical 
viscosity of the water. 
Expressing this in symbols, if 
d = diameter of the pipe, 
v c = the critical velocity,* 
n = the coefficient of viscosity of the water, 
P = the density of the water, 
v = kinematical viscosity of the water (=p./p), 
dp/dx = rate of fall of pressure along the pipe. 
Reynolds’s discovery was that for geometrically similar tubes 
v c d/v was constant. 
Further, on making a series of observations of the values of dp/dx over as large a range 
in the velocity of flow as possible in similar pipes of different diameters, Reynolds 
found that for all conditions of flow, stream line or eddying, when the values of vd/v 
were identical the corresponding values of pd 3 /p. 2 . dp/dx were identical. It appeared, 
therefore, that the general law of resistance could be expressed by the equation 
pd? dp _ Jvd\ 
p~ dx \ v ) 
( 1 ) 
* Throughout the present paper the symbol v is used to denote the mean velocity of flow through the 
pipe. Where reference is made to the value of the velocity at the axis of the pipe this is denoted 
by Anax- 
