_Water is practically non-elastic._ A pressure of 30,000 lbs. to the
square inch has been applied and its contraction has been found to
be less than one-twelfth. Experiment appears to show that for each
atmosphere of pressure it is condensed 47-1/2 millionth of its bulk.
The mechanical properties of liquids are determined on the hypothesis
that liquids are incompressible; according to known general principles
this is found to be for all practical purposes true, yet liquids are
more compressible than solids. If water be confined in a perfectly
rigid cylindrical vessel, its compression would equal 1/300000 of its
length for every pound per unit of area of the end pressure.
Water is nearly 100 times as compressible as steel, yet for almost all
practical purposes, liquids may be considered as non-elastic bodies
without involving sensible error.
_The pressure upon the horizontal base of any vessel containing a
fluid, is equal to the weight of a column of the fluid, found by
multiplying the area of the base into the perpendicular height of the
column, whatever be the shape of the vessel._
This follows, since here the distance of the center of gravity of the
base from the surface of the fluid, is the same as the perpendicular
height of the column. With a given base and height, therefore, the
pressure is the same whether the vessel is larger or smaller above,
whether its figure is regular or irregular, whether it rises to the
given height in a broad open funnel, or is carried up in a slender tube.
Hence, _any quantity of water, however small, may be made to balance
any quantity, however great_. This is called the _hydrostatic paradox_.
The experiment is usually performed by means of a water-bellows,
as represented in Fig. 84. When the pipe AD is filled with water,
the pressure upon the surface of the bellows, and consequently the
force with which it raises the weights laid on it, will be equal to
the weight of a cylinder of water, whose base is the surface of the
bellows, and height that of the column AD. Therefore, by making the
tube small, and the bellows large, the power of a given quantity of
water, however small, may be increased indefinitely. The pressure of
the column of water in this case corresponds to the force applied by
the piston in the hydrostatic press.
[Illustration: FIG. 84.]
We have already seen that the pressure on the bottom of a vessel
depends neither on the form of the vessel nor on the quantity of the
liquid, but simply on the height of the liquid above the bottom. But
the pressure thus exerted must not be confounded with the pressure
which the vessel itself exerts on the body which supports it. The
latter is always equal to the combined weight of the liquid and the
vessel in which it is contained, while the former may be either smaller
or greater than this weight, according to the form of the vessel. This
fact is often termed the _hydrostatic paradox_, because at first sight
it appears paradoxical.
CD (Fig. 85) is a vessel composed of two cylindrical parts of unequal
diameters, and filled with water to _a_. From what has been said
before, the bottom of the vessel CD supports the same pressure as if
its diameter were everywhere the same as that of its lower part; and it
would at first sight seem that the scale MN of the balance, in which
the vessel CD is placed, ought to show the same weight as if there
had been placed in it a cylindrical vessel having the same weight of
water, and having the diameter of the part D. But the pressure exerted
on the bottom of the vessel is not all transmitted to the scale MN;
for the _upward_ pressure upon the surface _n o_ of the vessel is
precisely equal to the weight of the _extra_ quantity of water which a
cylindrical vessel would contain, and balances an equal portion of the
_downward_ pressure on _m_. Consequently the pressure on the plate MN
is simply equal to the weight of the vessel CD and of the water which
it contains.
[Illustration: FIG. 85.]
_Pressure exerted anywhere upon a mass of liquid is transmitted
undiminished in all directions, and acts with the same force on all
equal surfaces, and in a direction at right angles to those surfaces._
To get a clearer idea of the truth of this principle, let us conceive
a vessel of any given form in the sides of which are placed various
cylindrical apertures, all of equal size, and closed by movable
pistons. Let us, further, imagine this vessel to be filled with liquid
and unaffected by the action of gravity; the pistons will, obviously,
have no tendency to move. If now a weight of P pounds be placed upon
the piston A (Fig. 86), which has a surface A, it will be pressed
inwards, and the pressure will be transmitted to the internal faces of
each of the pistons B, C, D, and E, which will each be forced outwards
by a pressure P, their surfaces being equal to that of the first
piston. Since each of the pistons undergoes a pressure, P, equal to
that on A, let us suppose two of the pistons united so as to constitute
a surface 2_a_; it will have to support a pressure 2P. Similarly, if
the piston were equal to 3_a_, it would experience a pressure of 3P;
and if its area were 100 or 1,000 times that of _a_, it would sustain a
pressure of 100 or 1,000 times P. In other words, the pressure on any
part of the internal walls of the vessel would be proportional to the
surface.
The principle of the equality of pressure is assumed as a consequence
of the constitution of fluids.
By the following experiment it can be shown that pressure is
transmitted in all directions; a cylinder provided with a piston is
fitted into a hollow sphere (Fig. 87). in which small cylindrical jets
are placed perpendicular to the sides. The sphere and the cylinder
being both filled with water, when the piston is moved the liquid
spouts forth from all the orifices, and not merely from that which is
opposite to the piston.
[Illustration: FIG. 86.]
[Illustration: FIG. 87.]
The reason why a satisfactory quantitative experimental demonstration
of the principle of the equality of pressure cannot be given is that
the influence of the weight of the liquid and of the friction of the
pistons cannot be altogether eliminated.
NOTE.—_The influence of the weight (or gravity) of water and its
fractional resistance in practical use_ is so great upon all the
processes of numbers and of the application of the natural laws
governing the operation of fluids, as stated under the heading of
Hydraulic Data, that separate pages will hereafter be found devoted
to a more extended explanation of this subject of gravity and
friction of water.
Yet an approximate verification may be effected by the experiment
represented in Fig. 88. Two cylinders of different diameters are
joined by a tube and filled with water. On the surface of the liquid
are two pistons, P and p, which hermetically close the cylinders, but
move without friction. Let the area of the large piston, P, be, for
instance, thirty times that of the smaller one, p. That being assumed,
let a weight, say of two pounds, be placed upon the small piston; this
pressure will be transmitted to the water and to the large piston, and
as this pressure amounts to two pounds on each portion of its surface
equal to that of the small piston, the large piston must be exposed
to an upward pressure thirty times as much, or of sixty pounds. If
now, this weight be placed upon the large piston, both will remain in
equilibrium; but if the weight is greater or less, this is no longer
the case.
[Illustration: FIG. 88.]
It is important to observe that in speaking of the transmission of
pressure to the sides of the containing vessel, these pressures must
always be supposed to be perpendicular to the sides.
_Equilibrium or state of rest of superposed liquids._ In order that
there should be equilibrium when several heterogeneous liquids are
superposed in the same vessel, each of them must satisfy the conditions
necessary for a single liquid, and further there will be a stable
state of rest only when the liquids are arranged in the order of their
decreasing densities from the bottom upwards.
The last condition is experimentally demonstrated by means of the
_phial of four elements_. This consists of a long narrow bottle
containing mercury, water, colored red, saturated with carbonate of
potash, alcohol, and petroleum. When the phial is shaken the liquids
mix, but when it is allowed to rest they separate; the mercury sinks
to the bottom, then comes the water, then the alcohol, and then the
petroleum. This is the order of the decreasing densities of the bodies.
The water is saturated with carbonate of potash to prevent its mixing
with the alcohol.
This separation of the liquids is due to the same cause as that which
enables solid bodies to float on the surface of a liquid of greater
density than their own. It is also on this account that fresh water, at
the mouths of rivers, floats for a long time on the denser salt water
of the sea; and it is for the same reason that cream, which is lighter
than milk, rises to the surface.
_The pressure upon any particle of a fluid of uniform density is
proportioned to its depth below the surface._
[Illustration: FIG. 89.]
_Example 1._ Let the column of fluid ABCD Fig. (1) be perpendicular
to the horizon. Take any points, _x_ and _y_, at different depths,
and conceive the column to be divided into a number of equal spaces
by horizontal planes. Then, since the density of the fluid is uniform
throughout, the pressure upon _x_ and _y_, respectively, must be in
proportion to the number of equal spaces above them, and consequently
in proportion to their depths.
_Example 2._ Let the column be of the same perpendicular height as
before, but inclined as is Fig. (2); then its quantity, and of course
its weight, is _increased_ in the same ratio as its length exceeds its
height; but since the column is partly supported by the plane, like any
other heavy body, the force of gravity acting upon it is _diminished_
on this account in the same ratio as its length exceeds its height;
therefore as much as the pressure on the base would be augmented by
the increased length of the column, just so much it is lessened by the
action of the inclined plane; and the pressure on any part of C_c_
will be, as before, proportioned to its perpendicular depth; and the
pressure of the inclined column AC_ac_ will be the same as that of the
perpendicular column ABCD.
_Fluids rise to the same level in the opposite arms of a recurved
tube._
[Illustration: FIG. 90.]
Let ABC, (Fig. 90) be a recurved tube: if water be poured into one arm
of the tube, it will rise to the same height in the other arm. For, the
pressure acting upon the lowest part at B, in opposite directions, is
proportioned to its depth below the surface of the fluid. Therefore,
these depths must be equal, that is, the height of the two columns
must be equal, in order that the fluid at B may be at rest; and unless
this part is at rest, the other parts of the column cannot be at rest.
Moreover, since the equilibrium depends on nothing else than the
_heights of_ the respective columns, therefore, the opposite columns
may differ to any degree in quantity, shape, or inclination to the
horizon. Thus, if vessels and tubes very diverse in shape and capacity,
as in Fig. p. 84 be connected with a reservoir, and water be poured
into any one of them, it will rise to the same level in them all.
The reason of this fact will be further understood from the application
of the principle of _equal momenta_, for it will be seen that the
velocity of the columns, when in motion, will be as much greater in the
smaller than in the larger columns, as the quantity of matter is less;
and hence the opposite momenta will be constantly equal.
Hence, water conveyed in aqueducts or running in natural channels, will
rise just as high as its source. Between the place where the water
of an aqueduct is delivered and the spring, the ground may rise into
hills and descend into valleys, and the pipes which convey the water
may follow all the undulations of the country, and the water will run
freely, provided no pipe is laid higher than the spring.
_Pressure of water due to its weight._ The pressure on any particle of
water is proportioned to its depth below the surface. The pressure of
still water in pounds per square inch against the sides of any pipe,
channel, or vessel of any shape whatever, is due solely to the “head”
or height of the level surface of the water above the point at which
the pressure is considered and is equal to ·43302 lbs. per square foot,
every foot of head or 62·355 lbs. per square foot for every foot of
head at 62° F.
The pressure per square inch is equal in all directions downwards,
upwards or sideways and is independent of the shape or size of the
containing vessel; for example, the pressure on a plug forced inward on
a square inch of the surface of water is suddenly communicated to every
square inch of the vessel’s surface, however great and to every inch of
the surface of any body immersed in it.
[Illustration: Various shapes of vessel.]
It is this principle which operates with such astonishing effect in
hydrostatic presses, of which familiar examples are found in the
hydraulic pumps, by the use of which boilers are tested. By the mere
weight of a man’s body when leaning on the extremity of a lever, a
pressure may be produced of upwards of 20 tons; it is the simplest and
most easily applicable of all contrivances for increasing human power,
and it is only limited by want of materials of sufficient strength to
utilize it.