AMPLIFICATION OF THE LAW OF POISEUILLE 49

4° to nearly 50°. Plotting the logarithmic homologues they
obtained a family of curves exactly similar to those in Fig. 14,
so that it is unnecessary to reproduce them. The points of
intersections between the curves for linear and for turbulent
flow lie on a perfectly straight line as is true in Fig. 14. This
proves that the critical velocity is directly proportional to the
viscosity. Indeed plotting the critical velocities read from their
curves against the temperatures, one obtains a curve which is
almost identical with that obtained by calculation from the
viscosities according to the assumed law.

Compressible Fluids. — As a compressible fluid flows through
a capillary under pressure, expansion takes place as the pressure
is relieved. The expansion may give rise to several effects which
must be taken into consideration. (1) The velocity increases as
the fluid passes along the tube. (2) There must be a component
of the flow which is toward the axis of the tube. (3) The expan-
sion may cause a change of temperature. This may affect the
flow in two ways (a) by changing the volume and consequently
the velocity and (V) by changing the viscosity of the medium and
consequently the resistance to. the flow. (4) As the density
changes, the viscosity may also change, unless the viscosity is
independent of the density. (5) We must also consider whether
the kinetic energy correction is changed when the velocity
increases as the fluid passes along the tube.

Por incompressible fluids, we have seen that the viscosity
measurement may be made without reference to the absolute
pressure. But with compressible fluids this is not the case,
because the rate of expansion depends upon the absolute pres-
sures, in the two reservoirs at the level of the capillary, Pi and P%.
We will first suppose that Boyle's law holds, the flow taking place
isothermally. For this case, as we shall see, page 243, the vis-
cosity is independent of the density. Let Z7, P, and p represent
the mean velocity, absolute pressure, and density at any cross-
section of the tube. Since at any instant the quantity Q of the
fluid passing every cross-section is constant, we have from Eq. (4)

dp irgR4* P

"

But -7^— n is constant and therefore -57- = -^-j — ~
loij Jr dl I