1. The frictional resistance is independent of the pressure between
the fluid and the solid against which it flows. This may be verified
by a simple direct experiment. C. H. Coulomb, for instance, oscillated
a disk under water, first with atmospheric pressure acting on the
water surface, afterwards with the atmospheric pressure removed. No
difference in the rate of decrease of the oscillations was observed.
The chief proof that the friction is independent of the pressure is
that no difference of resistance has been observed in water mains and
in other cases, where water flows over solid surfaces under widely
different pressures.
2. The frictional resistance of large surfaces is proportional to the
area of the surface.
3. At low velocities of not more than 1 in. per second for water, the
frictional resistance increases directly as the relative velocity of
the fluid and the surface against which it flows. At velocities of ½
ft. per second and greater velocities, the frictional resistance is
more nearly proportional to the square of the relative velocity.
In many treatises on hydraulics it is stated that the frictional
resistance is independent of the nature of the solid surface. The
explanation of this was supposed to be that a film of fluid remained
attached to the solid surface, the resistance being generated between
this fluid layer and layers more distant from the surface. At
extremely low velocities the solid surface does not seem to have much
influence on the friction. In Coulomb's experiments a metal surface
covered with tallow, and oscillated in water, had exactly the same
resistance as a clean metal surface, and when sand was scattered over
the tallow the resistance was only very slightly increased. The
earlier calculations of the resistance of water at higher velocities
in iron and wood pipes and earthen channels seemed to give a similar
result. These, however, were erroneous, and it is now well understood
that differences of roughness of the solid surface very greatly
influence the friction, at such velocities as are common in
engineering practice. H. P. G. Darcy's experiments, for instance,
showed that in old and incrusted water mains the resistance was twice
or sometimes thrice as great as in new and clean mains.
§ 66. _Ordinary Expressions for Fluid Friction at Velocities not
Extremely Small._--Let f be the frictional resistance estimated in
pounds per square foot of surface at a velocity of 1 ft. per second;
[omega] the area of the surface in square feet; and v its velocity in
feet per second relatively to the water in which it is immersed. Then,
in accordance with the laws stated above, the total resistance of the
surface is