| Coefficient of Friction, ξ | Frictional Resistance in ℔ per sq. ft. f | |
| New well-painted iron plate | .00489 | .00473 |
| Painted and planed plank (Beaufoy) | .00350 | .00339 |
| Surface of iron ships (Rankine) | .00362 | .00351 |
| Varnished surface (Froude) | .00258 | .00250 |
| Fine sand surface (Froude) | .00418 | .00405 |
| Coarser sand surface (Froude) | .00503 | .00488 |
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
in which expression the second term is of greatest importance at very low velocities, and of comparatively little importance at velocities over about 12 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.
 |
| Fig. 78. |
Fig. 78 shows Coulomb’s apparatus. LK supports the wire and disk: ag is the brass wire, the torsion of which causes the oscillations; DS is a graduated disk serving to measure the angles through which the apparatus oscillates. To this the friction disk is rigidly attached hanging in a vessel of water. The friction disks were from 4.7 to 7.7 in. diameter, and they generally made one oscillation in from 20 to 30 seconds, through angles varying from 360° to 6°. When the velocity of the circumference of the disk was less than 6 in. per second, the resistance was sensibly proportional to the velocity.
Beaufoy’s Experiments.—Towards the end of the 18th century Colonel Mark Beaufoy (1764–1827) made an immense mass of experiments on the resistance of bodies moved through water (Nautical and Hydraulic Experiments, London, 1834). Of these the only ones directly bearing on surface friction were some made in 1796 and 1798. Smooth painted planks were drawn through water and the resistance measured. For two planks differing in area by 46 sq. ft., at a velocity of 10 ft. per second, the difference of resistance, measured on the difference of area, was 0.339 ℔ per square foot. Also the resistance varied as the 1.949th power of the velocity.
§ 68. Froude’s Experiments.—The most important direct experiments on fluid friction at ordinary velocities are those made by William Froude (1810–1879) at Torquay. The method adopted in these experiments was to tow a board in a still water canal, the velocity and the resistance being registered by very ingenious recording arrangements. The general arrangement of the apparatus is shown in fig. 79. AA is the board the resistance of which is to be determined. B is a cutwater giving a fine entrance to the plane surfaces of the board. CC is a bar to which the board AA is attached, and which is suspended by a parallel motion from a carriage running on rails above the still water canal. G is a link by which the resistance of the board is transmitted to a spiral spring H. A bar I rigidly connects the other end of the spring to the carriage. The dotted lines K, L indicate the position of a couple of levers by which the extension of the spring is caused to move a pen M, which records the extension on a greatly increased scale, by a line drawn on the paper cylinder N. This cylinder revolves at a speed proportionate to that of the carriage, its motion being obtained from the axle of the carriage wheels. A second pen O, receiving jerks at every second and a quarter from a clock P, records time on the paper cylinder. The scale for the line of resistance is ascertained by stretching the spiral spring by known weights. The boards used for the experiment were 316 in. thick, 19 in. deep, and from 1 to 50 ft. in length, cutwater included. A lead keel counteracted the buoyancy of the board. The boards were covered with various substances, such as paint, varnish, Hay’s composition, tinfoil, &c., so as to try the effect of different degrees of roughness of surface. The results obtained by Froude may be summarized as follows:—
 |
| Fig. 79. |
1. The friction per square foot of surface varies very greatly for different surfaces, being generally greater as the sensible roughness of the surface is greater. Thus, when the surface of the board was covered as mentioned below, the resistance for boards 50 ft. long, at 10 ft. per second, was—
| Tinfoil or varnish | 0.25 | ℔ per sq. ft. |
| Calico | 0.47 | 〃 〃 |
| Fine sand | 0.405 | 〃 〃 |
| Coarser sand | 0.488 | 〃 〃 |
2. The power of the velocity to which the friction is proportional varies for different surfaces. Thus, with short boards 2 ft. long,
| For tinfoil the resistance varied as v2.16. |
With boards 50 ft. long,
| For varnish or tinfoil the resistance varied as v1.83. |
3. The average resistance per square foot of surface was much greater for short than for long boards; or, what is the same thing, the resistance per square foot at the forward part of the board was greater than the friction per square foot of portions more sternward. Thus,
| Mean Resistance in ℔ per sq. ft. | |||
| Varnished surface | 2 | ft. long | 0.41 |
| 50 | 〃 | 0.25 | |
| Fine sand surface | 2 | 〃 | 0.81 |
| 50 | 〃 | 0.405 | |
This remarkable result is explained thus by Froude: “The portion of surface that goes first in the line of motion, in experiencing resistance from the water, must in turn communicate motion to the water, in the direction in which it is itself travelling. Consequently
