THE LAW OF POISEUILLE 15
be somewhat distorted, in which case there must be a drop in
pressure pa.
6. Heat is produced as the fluid passes through the tube and
therefore the temperature may be different at different points of
the tube and since the temperature greatly affects the viscosity
of most substances, this may affect the amount of work done in
the passage through the tube. If the fluid is incompressible it
will have the same mean velocity through each cross-section of
the capillary, and the pressure must fall in a linear manner at least
so long as the flow is linear. If on the other hand the substance
is compressible, the velocity must increase as the fluid passes
through the tube, because of the expansion which results from
the decrease of pressure. With the expansion there is a lowering
of the temperature. Let the resultant effect of these changes in
the temperature upon the effective pressure be pT. It may be
either positive or negative.
At the exit of the capillary the fluid has no effective pressure
but it still possesses all of its kinetic energy which causes the
fluid to go for a considerable distance out into the reservoir R,
dragging some of the fluid in R with it and producing eddies, so
that the kinetic energy is finally dissipated in overcoming viscous
resistance outside of the capillary, and not in adding to the effec-
tive pressure, as Applebey (1910) has supposed.
The sum of these possible losses of effective pressure is then
p = Pk + PS + Pv + Pe + ps + PT (2)
We shall consider first the case where p = pv, supposing that the
fluid is incompressible, as is nearly the case in liquids.
Let the radius of the capillary be R and the radius of a hollow
cylinder coaxial with the capillary be r. It is evident from the
symmetrical arrangement that at every point in such a cylinder,
the velocity must be identical. Let this velocity be v. The
dv
rate of deformation must be -r~ and the tangential force due to
(Jv
the viscous resistance, acting from right to left, will be ?? ~r (cf.
Eq. (1)). Over the whole surface of the cylinder whose length
is Z, this force must amount to