434 
MESSRS. T. E. THORPE AHD J. W. RODGER ON THE RELATIONS 
experiments of Koch (‘Wied. Ann.,’ 14, 1) on the flow of mercury through glass 
tubes, and those of Couette on the flow of water through paraffin tubes, seem to 
show that even when the tube is not wetted the same state of things prevails.* 
Under the conditions above given, the loss of pressure may thus be wholly 
attril)uted to overcoming the viscosity of the liquid. 
Assume the velocity of the liquid molecules to be the same at points equidistant 
from the axis of the tube, and to be zero at the wall of the tube and greatest at the 
axis, and consider the forces acting upon an elementary cylinder of liquid situated 
l:)etween the two sections of the tube at which pressure is measured, and having for 
its axis the axis of the tube. 
If r be the radius of such an elementary hollow cylinder, dx its length, measured 
in the direction of the axis of the tube, dr its thickness, and P the pressure exerted 
on one end of the cylinder, then the total pressure on this end will be 27rrP<:/r. On 
the other end of the cylinder the pressure will be 27rr [P + {dFjdx) dx^ dr. The 
difference of these two pressures ^irr (dFjdx) dxdr is spent in overcoming viscosity 
or internal friction, inasmuch as the external pressures which are normal to the 
direction of movement must be in equilibrium with the weight of the liquid. 
Within the cylinder, the adjacent liquid is moving more freely and tends to carry 
the cylinder along with it, whereas on the exterior surface of the cylinder the 
adjacent liquid, which is moving more slowly, exerts a retarding effect. The 
difference of these two friction-effects corresponds to the loss of pressure. 
In order to estimate the magnitude of the friction-effects assumptions have now to 
he made. When a liquid is at rest its surface is plane, the force between two con¬ 
tiguous strata of liquid is therefore normal to their surface of separation. It is only 
when tlie liquid moves that this force has a tangential component. It is thus assumed 
that the magnitude of this component is a function of the relative velocity of the 
strata, becoming zero when the relative velocity is zero. For sinall velocities, such as 
those usually attained in capillary tubes, it is further assumed that the tangential 
component is simply proportional to the relative velocity. The tangential component 
is also assumed to be proportional to the area of the surface of contact, and to be 
independent of the curvature of the surface. After making these assumptions it may 
readily Ije shown that if p be the difference in the pressures at two sections of the 
tube distant I from one another, tlien V, the volume of liquid carried through the 
tube per unit time, is given by 
* Tlie question of slipping at a liquid boundary lias recently been raised by Bassett (‘ Roy. Soc. 
Proc.,’ .52, 273, 1893). Trustwortby exjierimental support to the idea that slipping really takes place 
seems, however, to be wanting. Besides the results quoted above, and those summarised by Codette 
(Ann. dc Chimie et de Phys. [6], 21, 490, 1890), the work of Whetham (‘Phil. Trans.,’ Anl. 181, A. 
(1890), p. .559) is conclusive in sliOAving that during linear movement the liquid layer in contact with 
tlie solid wall is stationary, and from the experiments of Couette, the same condition appear.s to hold 
even when tlie movement is tuiliulent. 
