Chapter I: The Data of Hydraulics1

§ 1. _Properties of Fluids._--The fluids to which the laws of practical hydraulics relate are substances the parts of which possess very great mobility, or which offer a very small resistance to distortion independently of inertia. Under the general heading Hydromechanics a fluid is defined to be a substance which yields continually to the slightest tangential stress, and hence in a fluid at rest there can be no tangential stress. But, further, in fluids such as water, air, steam, &c., to which the present division of the article relates, the tangential stresses that are called into action between contiguous portions during distortion or change of figure are always small compared with the weight, inertia, pressure, &c., which produce the visible motions it is the object of hydraulics to estimate. On the other hand, while a fluid passes easily from one form to another, it opposes considerable resistance to change of volume.

It is easily deduced from the absence or smallness of the tangential stress that contiguous portions of fluid act on each other with a pressure which is exactly or very nearly normal to the interface which separates them. The stress must be a pressure, not a tension, or the parts would separate. Further, at any point in a fluid the pressure in all directions must be the same; or, in other words, the pressure on any small element of surface is independent of the orientation of the surface.

§ 2. Fluids are divided into liquids, or incompressible fluids, and gases, or compressible fluids. Very great changes of pressure change the volume of liquids only by a small amount, and if the pressure on them is reduced to zero they do not sensibly dilate. In gases or compressible fluids the volume alters sensibly for small changes of pressure, and if the pressure is indefinitely diminished they dilate without limit.

In ordinary hydraulics, liquids are treated as absolutely incompressible. In dealing with gases the changes of volume which accompany changes of pressure must be taken into account.

§ 3. Viscous fluids are those in which change of form under a continued stress proceeds gradually and increases indefinitely. A very viscous fluid opposes great resistance to change of form in a short time, and yet may be deformed considerably by a small stress acting for a long period. A block of pitch is more easily splintered than indented by a hammer, but under the action of the mere weight of its parts acting for a long enough time it flattens out and flows like a liquid.

All actual fluids are viscous. They oppose a resistance to the relative motion of their parts. This resistance diminishes with the velocity of the relative motion, and becomes zero in a fluid the parts of which are relatively at rest. When the relative motion of different parts of a fluid is small, the viscosity may be neglected without introducing important errors. On the other hand, where there is considerable relative motion, the viscosity may be expected to have an influence too great to be neglected.

_Measurement of Viscosity. Coefficient of Viscosity._--Suppose the
plane ab, fig. 1 of area [omega], to move with the velocity V
relatively to the surface cd and parallel to it. Let the space between
be filled with liquid. The layers of liquid in contact with ab and cd
adhere to them. The intermediate layers all offering an equal
resistance to shearing or distortion, the rectangle of fluid abcd will
take the form of the parallelogram a´b´cd. Further, the resistance to
the motion of ab may be expressed in the form

R = [kappa][omega]V, (1)

where [kappa] is a coefficient the nature of which remains to be
determined.

If we suppose the liquid between ab and cd divided into layers as
shown in fig. 2, it will be clear that the stress R acts, at each
dividing face, forwards in the direction of motion if we consider the
upper layer, backwards if we consider the lower layer. Now suppose the
original thickness of the layer T increased to nT; if the bounding
plane in its new position has the velocity nV, the shearing at each
dividing face will be exactly the same as before, and the resistance
must therefore be the same. Hence,

R = [kappa]´[omega](nV). (2)

But equations (1) and (2) may both be expressed in one equation if
[kappa] and [kappa]´ are replaced by a constant varying inversely as
the thickness of the layer. Putting [kappa] = [mu]/T, [kappa]´ =
[mu]/nT,

R = [mu][omega]V/T;

or, for an indefinitely thin layer,

R = [mu][omega]dV/dt, (3)

an expression first proposed by L. M. H. Navier. The coefficient [mu]
is termed the coefficient of viscosity.

According to J. Clerk Maxwell, the value of [mu] for air at [theta]°
Fahr. in pounds, when the velocities are expressed in feet per second,
is

[mu] = 0.0000000256 (461° + [theta]);

that is, the coefficient of viscosity is proportional to the absolute
temperature and independent of the pressure.

The value of [mu] for water at 77° Fahr. is, according to H. von
Helmholtz and G. Piotrowski,

[mu] = 0.0000188,

the units being the same as before. For water [mu] decreases rapidly
with increase of temperature.

§ 4. When a fluid flows in a very regular manner, as for instance when It flows in a capillary tube, the velocities vary gradually at any moment from one point of the fluid to a neighbouring point. The layer adjacent to the sides of the tube adheres to it and is at rest. The layers more interior than this slide on each other. But the resistance developed by these regular movements is very small. If in large pipes and open channels there were a similar regularity of movement, the neighbouring filaments would acquire, especially near the sides, very great relative velocities. V. J. Boussinesq has shown that the central filament in a semicircular canal of 1 metre radius, and inclined at a slope of only 0.0001, would have a velocity of 187 metres per second,[2] the layer next the boundary remaining at rest. But before such a difference of velocity can arise, the motion of the fluid becomes much more complicated. Volumes of fluid are detached continually from the boundaries, and, revolving, form eddies traversing the fluid in all directions, and sliding with finite relative velocities against those surrounding them. These slidings develop resistances incomparably greater than the viscous resistance due to movements varying continuously from point to point. The movements which produce the phenomena commonly ascribed to fluid friction must be regarded as rapidly or even suddenly varying from one point to another. The internal resistances to the motion of the fluid do not depend merely on the general velocities of translation at different points of the fluid (or what Boussinesq terms the mean local velocities), but rather on the intensity at each point of the eddying agitation. The problems of hydraulics are therefore much more complicated than problems in which a regular motion of the fluid is assumed, hindered by the viscosity of the fluid.

RELATION OF PRESSURE, DENSITY, AND TEMPERATURE OF LIQUIDS

§ 5. _Units of Volume._--In practical calculations the cubic foot and
gallon are largely used, and in metric countries the litre and cubic
metre (= 1000 litres). The imperial gallon is now exclusively used in
England, but the United States have retained the old English wine
gallon.

1 cub. ft. = 6.236 imp. gallons = 7.481 U.S. gallons.
1 imp. gallon = 0.1605 cub. ft. = 1.200 U.S. gallons.
1 U.S. gallon = 0.1337 cub. ft. = 0.8333 imp. gallon.
1 litre = 0.2201 imp. gallon = 0.2641 U.S. gallon.

_Density of Water._--Water at 53° F. and ordinary pressure contains
62.4 lb. per cub. ft., or 10 lb. per imperial gallon at 62° F. The
litre contains one kilogram of water at 4° C. or 1000 kilograms per
cubic metre. River and spring water is not sensibly denser than pure
water. But average sea water weighs 64 lb. per cub. ft. at 53° F. The
weight of water per cubic unit will be denoted by G. Ice free from air
weighs 57.28 lb. per cub. ft. (Leduc).

§ 6. _Compressibility of Liquids._--The most accurate experiments show
that liquids are sensibly compressed by very great pressures, and that
up to a pressure of 65 atmospheres, or about 1000 lb. per sq. in., the
compression is proportional to the pressure. The chief results of
experiment are given in the following table. Let V1 be the volume of a
liquid in cubic feet under a pressure p1 lb. per sq. ft., and V2 its
volume under a pressure p2. Then the cubical compression is (V2 -
V1)/V1, and the ratio of the increase of pressure p2 - p1 to the
cubical compression is sensibly constant. That is, k = (p2 - p1)V1/(V2
- V1) is constant. This constant is termed the elasticity of volume.
With the notation of the differential calculus,

/ / dV \ dp
k = dp / ( - -- ) = - V --.
/ \ V / dV

_Elasticity of Volume of Liquids._

+-----------+------------+-----------+------------+------------+
| | Canton. | Oersted. | Colladon | Regnault. |
| | | | and Sturm. | |
+-----------+------------+-----------+------------+------------+
| Water | 45,990,000 | 45,900,000| 42,660,000 | 44,000,000 |
| Sea water | 52,900,000 | .. | | .. |
| Mercury |705,300,000 | .. |626,100,000 |604,500,000 |
| Oil | 44,090,000 | .. | | .. |
| Alcohol | 32,060,000 | .. | 23,100,000 | .. |
+-----------+------------+-----------+------------+------------+

According to the experiments of Grassi, the compressibility of water
diminishes as the temperature increases, while that of ether, alcohol
and chloroform is increased.

§ 7. _Change of Volume and Density of Water with Change of
Temperature._--Although the change of volume of water with change of
temperature is so small that it may generally be neglected in ordinary
hydraulic calculations, yet it should be noted that there is a change
of volume which should be allowed for in very exact calculations. The
values of [rho] in the following short table, which gives data enough
for hydraulic purposes, are taken from Professor Everett's _System of
Units_.

_Density of Water at Different Temperatures._

+-------------+----------+----------+
| | | G |
| Temperature.| [rho] |Weight of |
+-----+-------+Density of|1 cub. ft.|
|Cent.| Fahr. | Water. | in lb. |
+-----+-------+----------+----------+
| 0 | 32.0 | .999884 | 62.417 |
| 1 | 33.8 | .999941 | 62.420 |
| 2 | 35.6 | .999982 | 62.423 |
| 3 | 37.4 | 1.000004 | 62.424 |
| 4 | 39.2 | 1.000013 | 62.425 |
| 5 | 41.0 | 1.000003 | 62.424 |
| 6 | 42.8 | .999983 | 62.423 |
| 7 | 44.6 | .999946 | 62.421 |
| 8 | 46.4 | .999899 | 62.418 |
| 9 | 48.2 | .999837 | 62.414 |
| 10 | 50.0 | .999760 | 62.409 |
| 11 | 51.8 | .999668 | 62.403 |
| 12 | 53.6 | .999562 | 62.397 |
| 13 | 55.4 | .999443 | 62.389 |
| 14 | 57.2 | .999312 | 62.381 |
| 15 | 59.0 | .999173 | 62.373 |
| 16 | 60.8 | .999015 | 62.363 |
| 17 | 62.6 | .998854 | 62.353 |
| 18 | 64.4 | .998667 | 62.341 |
| 19 | 66.2 | .998473 | 62.329 |
| 20 | 68.0 | .998272 | 62.316 |
| 22 | 71.6 | .997839 | 62.289 |
| 24 | 75.2 | .997380 | 62.261 |
| 26 | 78.8 | .996879 | 62.229 |
| 28 | 82.4 | .996344 | 62.196 |
| 30 | 86 | .995778 | 62.161 |
| 35 | 95 | .99469 | 62.093 |
| 40 | 104 | .99236 | 61.947 |
| 45 | 113 | .99038 | 61.823 |
| 50 | 122 | .98821 | 61.688 |
| 55 | 131 | .98583 | 61.540 |
| 60 | 140 | .98339 | 61.387 |
| 65 | 149 | .98075 | 61.222 |
| 70 | 158 | .97795 | 61.048 |
| 75 | 167 | .97499 | 60.863 |
| 80 | 176 | .97195 | 60.674 |
| 85 | 185 | .96880 | 60.477 |
| 90 | 194 | .96557 | 60.275 |
|100 | 212 | .95866 | 59.844 |
+-----+-------+----------+----------+

The weight per cubic foot has been calculated from the values of
[rho], on the assumption that 1 cub. ft. of water at 39.2° Fahr. is
62.425 lb. For ordinary calculations in hydraulics, the density of
water (which will in future be designated by the symbol G) will be
taken at 62.4 lb. per cub. ft., which is its density at 53° Fahr. It
may be noted also that ice at 32° Fahr. contains 57.3 lb. per cub. ft.
The values of [rho] are the densities in grammes per cubic centimetre.

§ 8. _Pressure Column. Free Surface Level._--Suppose a small vertical
pipe introduced into a liquid at any point P (fig. 3). Then the liquid
will rise in the pipe to a level OO, such that the pressure due to the
column in the pipe exactly balances the pressure on its mouth. If the
fluid is in motion the mouth of the pipe must be supposed accurately
parallel to the direction of motion, or the impact of the liquid at
the mouth of the pipe will have an influence on the height of the
column. If this condition is complied with, the height h of the
column is a measure of the pressure at the point P. Let [omega] be the
area of section of the pipe, h the height of the pressure column, p
the intensity of pressure at P; then

p[omega] = Gh[omega] lb.,

p/G = h;

that is, h is the height due to the pressure at p. The level OO will
be termed the free surface level corresponding to the pressure at P.

RELATION OF PRESSURE, TEMPERATURE, AND DENSITY OF GASES

§ 9. _Relation of Pressure, Volume, Temperature and Density in
Compressible Fluids._--Certain problems on the flow of air and steam
are so similar to those relating to the flow of water that they are
conveniently treated together. It is necessary, therefore, to state as
briefly as possible the properties of compressible fluids so far as
knowledge of them is requisite in the solution of these problems. Air
may be taken as a type of these fluids, and the numerical data here
given will relate to air.

_Relation of Pressure and Volume at Constant Temperature._--At
constant temperature the product of the pressure p and volume V of a
given quantity of air is a constant (Boyle's law).

Let p0 be mean atmospheric pressure (2116.8 lb. per sq. ft.), V0 the
volume of 1 lb. of air at 32° Fahr. under the pressure p0. Then

p0V0 = 26214. (1)

If G0 is the weight per cubic foot of air in the same conditions,

G0 = 1/V0 = 2116.8/26214 = .08075. (2)

For any other pressure p, at which the volume of 1 lb. is V and the
weight per cubic foot is G, the temperature being 32° Fahr.,

pV = p/G = 26214; or G = p/26214. (3)

_Change of Pressure or Volume by Change of Temperature._--Let p0, V0,
G0, as before be the pressure, the volume of a pound in cubic feet,
and the weight of a cubic foot in pounds, at 32° Fahr. Let p, V, G be
the same quantities at a temperature t (measured strictly by the air
thermometer, the degrees of which differ a little from those of a
mercurial thermometer). Then, by experiment,

pV = p0V0(460.6 + t)/(460.6 + 32) = p0V0[tau]/[tau]0, (4)

where [tau], [tau]0 are the temperatures t and 32° reckoned from the
absolute zero, which is -460.6° Fahr.;

p/G = p0[tau]/G0[tau]0; (4a)

G = p[tau]0G0/p0[tau]. (5)

If p0 = 2116.8, G0 = .08075, [tau]0 = 460.6 + 32 = 492.6, then

p/G = 53.2[tau]. (5a)

Or quite generally p/G = R[tau] for all gases, if R is a constant
varying inversely as the density of the gas at 32° F. For steam R =
85.5.