166 
PROFESSOR 0. REYNOLDS ON THE THEORY OF LUBRICATION 
diameter up to a velocity of 6 metres per second, which corresponds to a value of 
u/a— 20,000. 
On the other hand it is found by Darcy* and others in large tubes that the 
VL 
resistance varies as the square of the velocity for values of -, as low as 1. Thus in a 
tube of '6 mm. we have jx constant for all rates of distortion below 20,000, while in a 
tube of 500 mm. diameter /x is a function of the distortion for all values greater 
than 1. 
It is, therefore, clear that if p, is a function of the distortion it must also be a func¬ 
tion of the dimensions of the channels, and in that case /x cannot be considered as a 
property of the fluid only. 
The change in the law of resistance from the simple ratio has, however, been shown 
by the author to be due to a change in the character of the motion of the fluid from 
that of direct parallel motion to that of sinuous or eddying motion, t 
In the latter case, although the mean motion at any point taken over a sufficient 
time is parallel to the pipe, it is made up of a succession of motions crossing the pipe 
in different directions. 
The question as to whether, in the case of sinuous motion, /x is to be considered as 
a function of the velocity or not, depends on whether we regard f as expressing the 
instantaneous shearing stress at a point, or the mean over a sufficient time. Whether 
we regard the symbols in the equations of motion as expressing the instantaneous 
motion or the mean taken over a sufficient time. 
If the latter, then /x must be held to include, in addition to the mean stress, the 
momentum per second parallel to u carried by the cross streams in the negative 
direction across the surface over which f is measured. 
If, however, we regard the motion at each instant, then we must restrict our 
definition of viscosity by making f the instantaneous value of the intensity of 
resistance at a point. 
This is a quantity which we have, and can have no means of measuring except 
under circumstances which secure that f is constant for all points over a given surface, 
and for all instants over a given time. 
It thus appears that there are two essentially distinct viscosities in fluids. The one 
a mechanical viscosity arising from the molar motion of the fluid, the other a physical 
property of the fluid. It is worth wffiile to point out that, although the conditions 
under which the first of these, the mechanical viscosity, can exist depend primarily on 
the physical viscosity, the actual magnitudes of these viscosities are independent, or 
are only connected in a secondary manner. This is shown by a very striking but 
little noticed fact. When the motion of the fluid is such that the resistance is as the 
* Recherches Expl. Paris, 1852. 
I “ An Experimental Investigation of the circumstances which determine whether the Motion of 
Water shall be Direct or Sinuous.” Phil. Trans., vol. 174 (1883), p. 935. 
