§ 10. Moving fluids as commonly observed are conveniently classified
thus:
(1) _Streams_ are moving masses of indefinite length, completely or
incompletely bounded laterally by solid boundaries. When the solid
boundaries are complete, the flow is said to take place in a pipe. When
the solid boundary is incomplete and leaves the upper surface of the
fluid free, it is termed a stream bed or channel or canal.
(2) A stream bounded laterally by differently moving fluid of the same
kind is termed a _current_.
(3) A _jet_ is a stream bounded by fluid of a different kind.
(4) An _eddy_, _vortex_ or _whirlpool_ is a mass of fluid the particles
of which are moving circularly or spirally.
(5) In a stream we may often regard the particles as flowing along
definite paths in space. A chain of particles following each other along
such a constant path may be termed a fluid filament or elementary
stream.
§ 11. _Steady and Unsteady, Uniform and Varying, Motion._--There are
two quite distinct ways of treating hydrodynamical questions. We may
either fix attention on a given mass of fluid and consider its changes
of position and energy under the action of the stresses to which it is
subjected, or we may have regard to a given fixed portion of space,
and consider the volume and energy of the fluid entering and leaving
that space.
If, in following a given path ab (fig. 4), a mass of water a has a
constant velocity, the motion is said to be uniform. The kinetic
energy of the mass a remains unchanged. If the velocity varies from
point to point of the path, the motion is called varying motion. If at
a given point a in space, the particles of water always arrive with
the same velocity and in the same direction, during any given time,
then the motion is termed steady motion. On the contrary, if at the
point a the velocity or direction varies from moment to moment the
motion is termed unsteady. A river which excavates its own bed is in
unsteady motion so long as the slope and form of the bed is changing.
It, however, tends always towards a condition in which the bed ceases
to change, and it is then said to have reached a condition of
permanent regime. No river probably is in absolutely permanent regime,
except perhaps in rocky channels. In other cases the bed is scoured
more or less during the rise of a flood, and silted again during the
subsidence of the flood. But while many streams of a torrential
character change the condition of their bed often and to a large
extent, in others the changes are comparatively small and not easily
observed.
[Illustration: FIG. 4.]
As a stream approaches a condition of steady motion, its regime
becomes permanent. Hence steady motion and permanent regime are
sometimes used as meaning the same thing. The one, however, is a
definite term applicable to the motion of the water, the other a less
definite term applicable in strictness only to the condition of the
stream bed.
§ 12. _Theoretical Notions on the Motion of Water._--The actual motion
of the particles of water is in most cases very complex. To simplify
hydrodynamic problems, simpler modes of motion are assumed, and the
results of theory so obtained are compared experimentally with the
actual motions.
_Motion in Plane Layers._--The simplest kind of motion in a stream is
one in which the particles initially situated in any plane cross
section of the stream continue to be found in plane cross sections
during the subsequent motion. Thus, if the particles in a thin plane
layer ab (fig. 5) are found again in a thin plane layer a´b´ after any
interval of time, the motion is said to be motion in plane layers. In
such motion the internal work in deforming the layer may usually be
disregarded, and the resistance to the motion is confined to the
circumference.
[Illustration: FIG. 5.]
_Laminar Motion._--In the case of streams having solid boundaries, it
is observed that the central parts move faster than the lateral parts.
To take account of these differences of velocity, the stream may be
conceived to be divided into thin laminae, having cross sections
somewhat similar to the solid boundary of the stream, and sliding on
each other. The different laminae can then be treated as having
differing velocities according to any law either observed or deduced
from their mutual friction. A much closer approximation to the real
motion of ordinary streams is thus obtained.
_Stream Line Motion._--In the preceding hypothesis, all the particles
in each lamina have the same velocity at any given cross section of
the stream. If this assumption is abandoned, the cross section of the
stream must be supposed divided into indefinitely small areas, each
representing the section of a fluid filament. Then these filaments may
have any law of variation of velocity assigned to them. If the motion
is steady motion these fluid filaments (or as they are then termed
_stream lines_) will have fixed positions in space.
_Periodic Unsteady Motion._--In ordinary streams with rough
boundaries, it is observed that at any given point the velocity varies
from moment to moment in magnitude and direction, but that the average
velocity for a sensible period (say for 5 or 10 minutes) varies very
little either in magnitude or velocity. It has hence been conceived
that the variations of direction and magnitude of the velocity are
periodic, and that, if for each point of the stream the mean velocity
and direction of motion were substituted for the actual more or less
varying motions, the motion of the stream might be treated as steady
stream line or steady laminar motion.
[Illustration: FIG. 6.]
§ 13. _Volume of Flow._--Let A (fig. 6) be any ideal plane surface, of
area [omega], in a stream, normal to the direction of motion, and let
V be the velocity of the fluid. Then the volume flowing through the
surface A in unit time is