310 
PROFESSOR HELE-SHAW AND MR. ALFRED HAY 
From these tables the curves shown in figs. 2 and 3 have been drawn, and these 
curves were actually used to determine the relation between different depths of well 
and corresponding permeabilities. 
The authors are indebted to Mr. J. C. W. Humfrey, RSc., Victoria University 
Scholar in Engineering, for valuable assistance in carrying out the above experiments. 
(/>) After the results embodied in the curves, figs. 2 and 3, had been obtained, we 
thought it would be interesting to see in how far they were in agreement with the 
deductions from theory. Let 
t — thickness of liquid film. 
.<? =: tangential stress (per unit area) due to viscosity at a distance 
x from the middle layer. 
v = velocity of flow at distance x from middle layer. 
r) = coefficient of viscosity. 
Then 
s = yj dv/dx. 
At a point distant x = Sx from the middle layer the tangential stress will he 
, dv d~v « 
s — Sx - 
dor 
Hence if w r e consider a layer of unit length, unit width and thickness Sx, the drag 
to which this elementary layer is subject on account of viscosity is given by 
dh- 
Now, since we suppose that the flow is steady, the elementary layer considered is 
not undergoing any acceleration. Hence the backward drag due to viscous resistance 
must be balanced by the forward push due to the difference of pressure over the two 
ends of the elementary layer. 
If dp/dy stand for the gradient of pressure, then 
d? v . dp ~ 
71 w Sx = ~ 
or yd~v/dx 2 = — dp/dy = — f say, 
where f is the fall of pressure per unit length of the liquid layer. 
Integrating this equation once, we get 
r) dv/dx = — fx -f C, 
where C is a constant. Since when x — 0, dv/dx = 0 (the velocity in the middle 
being at its maximum), C = 0. Integrating a second time, we get 
yv = — -|/x 3 + C'. 
