**HYDRAULICS** (Gr. ὕδωρ, water, and αὐλός, a pipe), the branch
of engineering science which deals with the practical applications
of the laws of hydromechanics.

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§ 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.

Fig. 1. |

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 ω, 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

where κ 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 *n*T; if the bounding
plane in its new position has the velocity *n*V, the shearing at each
dividing face will be exactly the same as before, and the resistance
must therefore be the same. Hence,

*n*V).

But equations (1) and (2) may both be expressed in one equation if
κ and κ′ are replaced by a constant varying inversely as the thickness
of the layer. Putting κ = μ/T, κ′ = μ/*n*T,

or, for an indefinitely thin layer,

*d*V/

*dt*,

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

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

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

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

the units being the same as before. For water μ decreases rapidly with increase of temperature.

Fig. 2. |

§ 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.