186 
PROFESSOR O. REYNOLDS ON THE THEORY OF LUBRICATION 
In which the left-hand members are the stresses on plane perpendicular to the first 
suffix in directions parallel to the second, the first three being the normal stresses, the 
last six the tangential stresses. 
The values of these substituted in the equations of motion 
bu v dp xx dp vx dp zx 
dp, v/JL dp yx dp zy 
dy dz 
bt 
to _ V 
Pdt-P Y 
bw 
dx 
„ dp xz dpy z d/pz 
m=p L+ j^ + i^ + ik 
bp (du dv dw 
bt P\dx dy dz 
0 0 
give the complete equations of motion for the interior of a viscous fluid. 
These equations involve terms severally depending on the inertia and the weight of 
the fluid, also the variation of stress in the fluid. 
In the case of lubrication the spaces between the solid surfaces are so small com¬ 
pared with 
f± 
U 
that the motion of the fluid is shown to be free from eddies as already explained 
(Art. II). Also that the forces arising from weight and inertia are altogether small 
compared with the stresses arising from viscosity. 
The equations which result from the substitution from (9) and (10) in the first 3 of 
(11) may therefore be simplified by the omission of the inertia and gravitation terms, 
which are the terms involving p a.s a factor. 
In the case of oil the remaining terms may still further be simplified by omitting 
the terms depending on the compressibility of the fluid. 
Also if, as is the case, /x is nearly constant the terms involving dp may be omitted 
or considered of secondary importance. 
From equations (11) we then have 
dp__ 
~dx~P 
dp 
dy ^ 
dp 
dz 
:p 
(d 2 u d 2 u d?u 
d~v d 2 v d 2 v 
dh? 
/ d-w d?w d~w 
\rfx ; 2 dy 2 dz 3 
„ du dv , dw 
0 =—A -L 
dx ' dy 'dz 
-J 
(12) 
