Hydrostatics is a science which grew originally out of a number of
isolated practical problems; but it satisfies the requirement of perfect
accuracy in its application to phenomena, the largest and smallest, of
the behaviour of a fluid. At the same time, it delights the pure
theorist by the simplicity of the logic with which the fundamental
theorems may be established, and by the elegance of its mathematical
operations, insomuch that hydrostatics may be considered as the
Euclidean pure geometry of mechanical science.
1. _The Different States of a Substance or Matter._--All substance in
nature falls into one of the two classes, solid and fluid; a solid
substance, the land, for instance, as contrasted with a fluid, like
water, being a substance which does not flow of itself.
A _fluid_, as the name implies, is a substance which flows, or is
capable of flowing; water and air are the two fluids distributed most
universally over the surface of the earth.
Fluids again are divided into two classes, termed a liquid and a gas, of
which water and air are the chief examples.
A _liquid_ is a fluid which is incompressible or practically so, i.e. it
does not change in volume sensibly with change of pressure.
A _gas_ is a compressible fluid, and the change in volume is
considerable with moderate variation of pressure.
Liquids, again, can be poured from one open vessel into another, and can
be kept in an uncovered vessel, but a gas tends to diffuse itself
indefinitely and must be preserved in a closed reservoir.
The distinguishing characteristics of the three kinds of substance or
states of matter, the solid, liquid and gas, are summarized thus in O.
Lodge's _Mechanics_:--
A solid has both size and shape.
A liquid has size but not shape.
A gas has neither size nor shape.
2. _The Change of State of Matter._--By a change of temperature and
pressure combined, a substance can in general be made to pass from one
state into another; thus by gradually increasing the temperature a solid
piece of ice can be melted into the liquid state of water, and the water
again can be boiled off into the gaseous state as steam. Again, by
raising the temperature, a metal in the solid state can be melted and
liquefied, and poured into a mould to assume any form desired, which is
retained when the metal cools and solidifies again; the gaseous state of
a metal is revealed by the spectroscope. Conversely, a combination of
increased pressure and lowering of temperature will, if carried far
enough, reduce a gas to a liquid, and afterwards to the solid state; and
nearly every gaseous substance has now undergone this operation.
A certain critical temperature is observed in a gas, above which the
liquefaction is impossible; so that the gaseous state has two
subdivisions into (i.) a true gas, which cannot be liquefied, because
its temperature is above the critical temperature, (ii.) a vapour, where
the temperature is below the critical, and which can ultimately be
liquefied by further lowering of temperature or increase of pressure.
3. _Plasticity and Viscosity._--Every solid substance is found to be
plastic more or less, as exemplified by punching, shearing and cutting;
but the plastic solid is distinguished from the viscous fluid in that a
plastic solid requires a certain magnitude of stress to be exceeded to
make it flow, whereas the viscous liquid will yield to the slightest
stress, but requires a certain length of time for the effect to be
appreciable.
According to Maxwell (_Theory of Heat_) "When a continuous alteration of
form is produced only by a stress exceeding a certain value, the
substance is called a solid, however soft and plastic it may be. But
when the smallest stress, if only continued long enough, will cause a
perceptible and increasing change of form, the substance must be
regarded as a viscous fluid, however hard it may be." Maxwell
illustrates the difference between a soft solid and a hard liquid by a
jelly and a block of pitch; also by the experiment of supporting a
candle and a stick of sealing-wax; after a considerable time the
sealing-wax will be found bent and so is a fluid, but the candle remains
straight as a solid.
4. _Definition of a Fluid._--A fluid is a substance which yields
continually to the slightest tangential stress in its interior; that is,
it can be divided very easily along any plane (given plenty of time if
the fluid is viscous). It follows that when the fluid has come to rest,
the tangential stress in any plane in its interior must vanish, and the
stress must be entirely normal to the plane. This mechanical axiom of
the _normality of fluid pressure_ is the foundation of the mathematical
theory of hydrostatics.
The theorems of hydrostatics are thus true for all stationary fluids,
however viscous they may be; it is only when we come to hydrodynamics,
the science of the motion of a fluid, that viscosity will make itself
felt and modify the theory; unless we begin by postulating the perfect
fluid, devoid of viscosity, so that the principle of the _normality of
fluid pressure_ is taken to hold when the fluid is in movement.
5. _The Measurement of Fluid Pressure._--The pressure at any point of
a plane in the interior of a fluid is the intensity of the normal
thrust estimated per unit area of the plane.
Thus, if a thrust of P lb. is distributed uniformly over a plane area
of A sq. ft., as on the horizontal bottom of the sea or any reservoir,
the pressure at any point of the plane is P/A lb. per sq. ft., or
P/144A lb. per sq. in. (lb./ft.² and lb./in.², in the Hospitalier
notation, to be employed in the sequel). If the distribution of the
thrust is not uniform, as, for instance, on a vertical or inclined
face or wall of a reservoir, then P/A represents the average pressure
over the area; and the actual pressure at any point is the average
pressure over a small area enclosing the point. Thus, if a thrust
[Delta]P lb. acts on a small plane area [Delta]A ft.² enclosing a
point B, the pressure p at B is the limit of [Delta]P/[Delta]A; and
p = lt([Delta]P/[Delta]A) = dP/dA, (1)
in the notation of the differential calculus.
6. _The Equality of Fluid Pressure in all Directions._--This
fundamental principle of hydrostatics follows at once from the
principle of the normality of fluid pressure implied in the definition
of a fluid in § 4. Take any two arbitrary directions in the plane of
the paper, and draw a small isosceles triangle abc, whose sides are
perpendicular to the two directions, and consider the equilibrium of a
small triangular prism of fluid, of which the triangle is the cross
section. Let P, Q denote the normal thrust across the sides bc, ca,
and R the normal thrust across the base ab. Then, since these three
forces maintain equilibrium, and R makes equal angles with P and Q,
therefore P and Q must be equal. But the faces bc, ca, over which P
and Q act, are also equal, so that the pressure on each face is equal.
A scalene triangle abc might also be employed, or a tetrahedron.
[Illustration: FIG. 1a.]
It follows that the pressure of a fluid requires to be calculated in
one direction only, chosen as the simplest direction for convenience.
7. _The Transmissibility of Fluid Pressure._--Any additional pressure
applied to the fluid will be transmitted equally to every point in the
case of a liquid; this principle of the _transmissibility of pressure_
was enunciated by Pascal, 1653, and applied by him to the invention of
the _hydraulic press_.
This machine consists essentially of two communicating cylinders (fig.
1a), filled with liquid and closed by pistons. If a thrust P lb. is
applied to one piston of area A ft.², it will be balanced by a thrust
W lb. applied to the other piston of area B ft.², where
p = P/A = W/B, (1)
the pressure p of the liquid being supposed uniform; and, by making
the ratio B/A sufficiently large, the mechanical advantage can be
increased to any desired amount, and in the simplest manner possible,
without the intervention of levers and machinery.
Fig. 1b shows also a modern form of the hydraulic press, applied to
the operation of covering an electric cable with a lead coating.
8. _Theorem._--In a fluid at rest under gravity the pressure is the
same at any two points in the same horizontal plane; in other words, a
surface of equal pressure is a horizontal plane.
This is proved by taking any two points A and B at the same level, and
considering the equilibrium of a thin prism of liquid AB, bounded by
planes at A and B perpendicular to AB. As gravity and the fluid
pressure on the sides of the prism act at right angles to AB, the
equilibrium requires the equality of thrust on the ends A and B; and
as the areas are equal, the pressure must be equal at A and B; and so
the pressure is the same at all points in the same horizontal plane.
If the fluid is a liquid, it can have a free surface without diffusing
itself, as a gas would; and this free surface, being a surface of zero
pressure, or more generally of uniform atmospheric pressure, will also
be a surface of equal pressure, and therefore a horizontal plane.
[Illustration: FIG. 1b.]
Hence the _theorem_.--The free surface of a liquid at rest under
gravity is a horizontal plane. This is the characteristic
distinguishing between a solid and a liquid; as, for instance, between
land and water. The land has hills and valleys, but the surface of
water at rest is a horizontal plane; and if disturbed the surface
moves in waves.
9. _Theorem._--In a homogeneous liquid at rest under gravity the
pressure increases uniformly with the depth.
This is proved by taking the two points A and B in the same vertical
line, and considering the equilibrium of the prism by resolving
vertically. In this case the thrust at the lower end B must exceed the
thrust at A, the upper end, by the weight of the prism of liquid; so
that, denoting the cross section of the prism by [alpha] ft.², the
pressure at A and By by p0 and p lb./ft.², and by w the density of the
liquid estimated in lb./ft.³,
p[alpha] - p0[alpha] = w[alpha]·AB, (1)
p = w·AB + p0. (2)
Thus in water, where w = 62.4lb./ft.³, the pressure increases 62.4
lb./ft.², or 62.4 ÷ 144 = 0.433 lb./in.² for every additional foot of
depth.
10. _Theorem._--If two liquids of different density are resting in
vessels in communication, the height of the free surface of such
liquid above the surface of separation is inversely as the density.
For if the liquid of density [sigma] rises to the height h and of
density [rho] to the height k, and p0 denotes the atmospheric
pressure, the pressure in the liquid at the level of the surface of
separation will be [sigma]h + p0 and [rho]k + p0, and these being
equal we have
[sigma]h = [rho]k. (1)
The principle is illustrated in the article BAROMETER, where a column
of mercury of density [sigma] and height h, rising in the tube to the
Torricellian vacuum, is balanced by a column of air of density [rho],
which may be supposed to rise as a homogeneous fluid to a height k,
called the height of the homogeneous atmosphere. Thus water being
about 800 times denser than air and mercury 13.6 times denser than
water,
k/h = [sigma]/[rho] = 800 × 13.6 = 10,880; (2)
and with an average barometer height of 30 in. this makes k 27,200
ft., about 8300 metres.
11. _The Head of Water or a Liquid._--The pressure [sigma]h at a depth
h ft. in liquid of density [sigma] is called the pressure due to a
_head_ of h ft. of the liquid. The atmospheric pressure is thus due to
an average head of 30 in. of mercury, or 30 × 13.6 ÷ 12 = 34 ft. of
water, or 27,200 ft. of air. The pressure of the air is a convenient
unit to employ in practical work, where it is called an "atmosphere";
it is made the equivalent of a pressure of one kg/cm²; and one
ton/inch², employed as the unit with high pressure as in artillery,
may be taken as 150 atmospheres.
12. _Theorem._--A body immersed in a fluid is buoyed up by a force
equal to the weight of the liquid displaced, acting vertically upward
through the centre of gravity of the displaced liquid.
For if the body is removed, and replaced by the fluid as at first,
this fluid is in equilibrium under its own weight and the thrust of
the surrounding fluid, which must be equal and opposite, and the
surrounding fluid acts in the same manner when the body replaces the
displaced fluid again; so that the resultant thrust of the fluid acts
vertically upward through the centre of gravity of the fluid
displaced, and is equal to the weight.
When the body is floating freely like a ship, the equilibrium of this
liquid thrust with the weight of the ship requires that the weight of
water displaced is equal to the weight of the ship and the two centres
of gravity are in the same vertical line. So also a balloon begins to
rise when the weight of air displaced is greater than the weight of
the balloon, and it is in equilibrium when the weights are equal. This
theorem is called generally the _principle of Archimedes_.
It is used to determine the density of a body experimentally; for if W
is the weight of a body weighed in a balance in air (strictly _in
vacuo_), and if W´ is the weight required to balance when the body is
suspended in water, then the upward thrust of the liquid or weight of
liquid displaced is W - W´, so that the _specific gravity_ (S.G.),
defined as the ratio of the weight of a body to the weight of an equal
volume of water, is W/(W - W´).
As stated first by Archimedes, the principle asserts the obvious fact
that a body displaces its own volume of water; and he utilized it in
the problem of the determination of the adulteration of the crown of
Hiero. He weighed out a lump of gold and of silver of the same weight
as the crown; and, immersing the three in succession in water, he
found they spilt over measures of water in the ratio 1/14 : 4/77 :
2/21 or 33 : 24 : 44; thence it follows that the gold : silver alloy
of the crown was as 11 : 9 by weight.
13. _Theorem._--The resultant vertical thrust on any portion of a
curved surface exposed to the pressure of a fluid at rest under
gravity is the weight of fluid cut out by vertical lines drawn round
the boundary of the curved surface.
_Theorem._--The resultant horizontal thrust in any direction is
obtained by drawing parallel horizontal lines round the boundary, and
intersecting a plane perpendicular to their direction in a plane
curve; and then investigating the thrust on this plane area, which
will be the same as on the curved surface.
The proof of these theorems proceeds as before, employing the
normality principle; they are required, for instance, in the
determination of the liquid thrust on any portion of the bottom of a
ship.
In casting a thin hollow object like a bell, it will be seen that the
resultant upward thrust on the mould may be many times greater than
the weight of metal; many a curious experiment has been devised to
illustrate this property and classed as a hydrostatic paradox (Boyle,
_Hydrostatical Paradoxes_, 1666).
[Illustration: FIG. 2.]
Consider, for instance, the operation of casting a hemispherical bell,
in fig. 2. As the molten metal is run in, the upward thrust on the
outside mould, when the level has reached PP´, is the weight of metal
in the volume generated by the revolution of APQ; and this, by a
theorem of Archimedes, has the same volume as the cone ORR´, or 1/3
[pi]y³, where y is the depth of metal, the horizontal sections being
equal so long as y is less than the radius of the outside hemisphere.
Afterwards, when the metal has risen above B, to the level KK´, the
additional thrust is the weight of the cylinder of diameter KK´ and
height BH. The upward thrust is the same, however thin the metal may
be in the interspace between the outer mould and the core inside; and
this was formerly considered paradoxical.
_Analytical Equations of Equilibrium of a Fluid at rest under any
System of Force._
14. Referred to three fixed coordinate axes, a fluid, in which the
pressure is p, the density [rho], and X, Y, Z the components of
impressed force per unit mass, requires for the equilibrium of the
part filling a fixed surface S, on resolving parallel to Ox,
_ _ _ _ _
/ / / / /
| | lp dS = | | | [rho]X dx dy dz, (1)
_/ _/ _/ _/ _/
where l, m, n denote the direction cosines of the normal drawn outward
of the surface S.
But by Green's transformation
_ _ _ _ _
/ / / / / dp
| | lp dS = | | | -- dx dy dz, (2)
_/ _/ _/ _/ _/ dx
thus leading to the differential relation at every point
dp dp dp
-- = [rho]X, -- = [rho]Y, -- = [rho]Z. (3)
dx dy dz
The three equations of equilibrium obtained by taking moments round
the axes are then found to be satisfied identically.
Hence the space variation of the pressure in any direction, or the
_pressure-gradient_, is the resolved force per unit volume in that
direction. The resultant force is therefore in the direction of the
steepest pressure-gradient, and this is normal to the surface of equal
pressure; for equilibrium to exist in a fluid the lines of force must
therefore be capable of being cut orthogonally by a system of
surfaces, which will be surfaces of equal pressure.
Ignoring temperature effect, and taking the density as a function of
the pressure, surfaces of equal pressure are also of equal density,
and the fluid is stratified by surfaces orthogonal to the lines of
force;
1 dp 1 dp 1 dp
----- --, ----- --, ----- --, or X, Y, Z (4)
[rho] dx [rho] dy [rho] dz
are the partial differential coefficients of some function P, =
[int]dp/[rho], of x, y, z; so that X, Y, Z must be the partial
differential coefficients of a potential -V, such that the force in
any direction is the downward gradient of V; and then
dP dV
-- + -- = 0, or P + V = constant, (5)
dx dx
in which P may be called the hydrostatic head and V the head of
potential.
With variation of temperature, the surfaces of equal pressure and
density need not coincide; but, taking the pressure, density and
temperature as connected by some relation, such as the gas-equation,
the surfaces of equal density and temperature must intersect in lines
lying on a surface of equal pressure.
15. As an example of the general equations, take the simplest case of
a uniform field of gravity, with Oz directed vertically downward;
employing the gravitation unit of force,
1 dp 1 dp 1 dp
----- -- = 0, ----- -- = 0, ----- -- = 1, (1)
[rho] dx [rho] dy [rho] dz
_
/
P = | dp/[rho] = z + a constant. (2)
_/