where [xi] is, like f, nearly constant for a given surface, and is
termed the coefficient of friction.
The following are average values of the coefficient of friction for
water, obtained from experiments on large plane surfaces, moved in an
indefinitely large mass of water.
+------------------------------------+--------------+-----------------+
| | Coefficient | Frictional |
| | of Friction, | Resistance in |
| | [xi] | lb. 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 " | .00418 | .00405 |
| Coarser sand surface " | .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 = f[omega]v³ foot-pounds per second \ (3).
= [xi]G[omega]v³/2g " " /
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
[xi] = [alpha] + [beta]/v
in which expression the second term is of greatest importance at very
low velocities, and of comparatively little importance at velocities
over about ½ ft. per second. Values of [xi] 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.
[Illustration: 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 lb. per square foot. Also the resistance varied as the
1.949th power of the velocity.
[Illustration: FIG. 79.]
§ 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 3/16 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:--
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 lb. 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 v^(2.16).
For other surfaces the resistance varied as v^(2.00).
With boards 50 ft. long,
For varnish or tinfoil the resistance varied as v^(1.83).
For sand the resistance varied as v^(2.00).
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
lb. 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
the portion of surface which succeeds the first will be rubbing,
not against stationary water, but against water partially moving in
its own direction, and cannot therefore experience so much resistance
from it."
§ 69. The following table gives a general statement of Froude's
results. In all the experiments in this table, the boards had a fine
cutwater and a fine stern end or run, so that the resistance was
entirely due to the surface. The table gives the resistances per
square foot in pounds, at the standard speed of 600 feet per minute,
and the power of the speed to which the friction is proportional, so
that the resistance at other speeds is easily calculated.
+------------+---------------------------------------------------------------------------+
| | Length of Surface, or Distance from Cutwater, in feet. |
| +------------------+------------------+------------------+------------------+
| | 2 ft. | 8 ft. | 20 ft. | 50 ft. |
| +------+-----+-----+------+-----+-----+------+-----+-----+------+-----+-----+
| | A | B | C | A | B | C | A | B | C | A | B | C |
+------------+------+-----+-----+------+-----+-----+------+-----+-----+------+-----+-----+
| Varnish | 2.00 | .41 |.390 | 1.85 |.325 |.264 | 1.85 |.278 |.240 | 1.83 |.250 |.226 |
| Paraffin | .. | .38 |.370 | 1.94 |.314 |.260 | 1.93 |.271 |.237 | .. | .. | .. |
| Tinfoil | 2.16 | .30 |.295 | 1.99 |.278 |.263 | 1.90 |.262 |.244 | 1.83 |.246 |.232 |
| Calico | 1.93 | .87 |.725 | 1.92 |.626 |.504 | 1.89 |.531 |.447 | 1.87 |.474 |.423 |
| Fine sand | 2.00 | .81 |.690 | 2.00 |.583 |.450 | 2.00 |.480 |.384 | 2.06 |.405 |.337 |
| Medium sand| 2.00 | .90 |.730 | 2.00 |.625 |.488 | 2.00 |.534 |.465 | 2.00 |.488 |.456 |
| Coarse sand| 2.00 |1.10 |.880 | 2.00 |.714 |.520 | 2.00 |.588 |.490 | .. | .. | .. |
+--------- --+------+-----+-----+------+-----+-----+------+-----+-----+------+-----+-----+
Columns A give the power of the speed to which the resistance is
approximately proportional.
Columns B give the mean resistance per square foot of the whole
surface of a board of the lengths stated in the table.
Columns C give the resistance in pounds of a square foot of surface at
the distance sternward from the cutwater stated in the heading.
Although these experiments do not directly deal with surfaces of
greater length than 50 ft., they indicate what would be the
resistances of longer surfaces. For at 50 ft. the decrease of
resistance for an increase of length is so small that it will make no
very great difference in the estimate of the friction whether we
suppose it to continue to diminish at the same rate or not to diminish
at all. For a varnished surface the friction at 10 ft. per second
diminishes from 0.41 to 0.32 lb. per square foot when the length is
increased from 2 to 8 ft., but it only diminishes from 0.278 to 0.250
lb. per square foot for an increase from 20 ft. to 50 ft.
If the decrease of friction sternwards is due to the generation of a
current accompanying the moving plane, there is not at first sight any
reason why the decrease should not be greater than that shown by the
experiments. The current accompanying the board might be assumed to
gain in volume and velocity sternwards, till the velocity was nearly
the same as that of the moving plane and the friction per square foot
nearly zero. That this does not happen appears to be due to the mixing
up of the current with the still water surrounding it. Part of the
water in contact with the board at any point, and receiving energy of
motion from it, passes afterwards to distant regions of still water,
and portions of still water are fed in towards the board to take its
place. In the forward part of the board more kinetic energy is given
to the current than is diffused into surrounding space, and the
current gains in velocity. At a greater distance back there is an
approximate balance between the energy communicated to the water and
that diffused. The velocity of the current accompanying the board
becomes constant or nearly constant, and the friction per square foot
is therefore nearly constant also.
§ 70. _Friction of Rotating Disks._--A rotating disk is virtually a
surface of unlimited extent and it is convenient for experiments on
friction with different surfaces at different speeds. Experiments
carried out by Professor W. C. Unwin (_Proc. Inst. Civ. Eng._ lxxx.)
are useful both as illustrating the laws of fluid friction and as
giving data for calculating the resistance of the disks of turbines
and centrifugal pumps. Disks of 10, 15 and 20 in. diameter fixed on a
vertical shaft were rotated by a belt driven by an engine. They were
enclosed in a cistern of water between parallel top and bottom fixed
surfaces. The cistern was suspended by three fine wires. The friction
of the disk is equal to the tendency of the cistern to rotate, and
this was measured by balancing the cistern by a fine silk cord passing
over a pulley and carrying a scale pan in which weights could be
placed.
If [omega] is an element of area on the disk moving with the velocity
v, the friction on this element is f[omega]v^n, where f and n are
constant for any given kind of surface. Let [alpha] be the angular
velocity of rotation, R the radius of the disk. Consider a ring of the
surface between r and r + dr. Its area is 2[pi]r dr, its velocity
[alpha]r and the friction of this ring is f2[pi]r dr[alpha]^n r^n. The
moment of the friction about the axis of rotation is
2[pi][alpha]^n fr^(n + 2)dr, and the total moment of friction for the
two sides of the disk is
_
/R
M = 4[pi][alpha]^n f | r^(n+2) dr = {4[pi][alpha]^n /(n + 3)}fR^(n+3). .
_/0
If N is the number of revolutions per sec.,