The distance through which the frictional resistance is overcome is v ft. per second. The work expended in fluid friction is therefore given by the equation—
Work expended = fωv3 foot-pounds per second
= ξGωv3/2g ” ”
(3).
The coefficient of friction and the friction per square foot of surface can be indirectly obtained from observations of the discharge of pipes and canals. In obtaining them, however, some assumptions as to the motion of the water must be made, and it will be better therefore to discuss these values in connexion with the cases to which they are related.
Many attempts have been made to express the coefficient of friction in a form applicable to low as well as high velocities. The older hydraulic writers considered the resistance termed fluid friction to be made up of two parts,—a part due directly to the distortion of the mass of water and proportional to the velocity of the water relatively to the solid surface, and another part due to kinetic energy imparted to the water striking the roughnesses of the solid surface and proportional to the square of the velocity. Hence they proposed to take
ξ = α + β/v
in which expression the second term is of greatest importance at very low velocities, and of comparatively little importance at velocities over about 1⁄2 ft. per second. Values of ξ expressed in this and similar forms will be given in connexion with pipes and canals.
All these expressions must at present be regarded as merely empirical expressions serving practical purposes.
The frictional resistance will be seen to vary through wider limits than these expressions allow, and to depend on circumstances of which they do not take account.
§ 67. Coulomb’s Experiments.—The first direct experiments on fluid friction were made by Coulomb, who employed a circular disk suspended by a thin brass wire and oscillated in its own plane. His experiments were chiefly made at very low velocities. When the disk is rotated to any given angle, it oscillates under the action of its inertia and the torsion of the wire. The oscillations diminish gradually in consequence of the work done in overcoming the friction of the disk. The diminution furnishes a means of determining the friction.