Fig. 4. |

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.

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.

Fig. 5. |

*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.

*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.

Fig. 6. |

*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.

§ 13. *Volume of Flow.*—Let A (fig. 6) be any ideal plane surface,
of area ω, 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

Thus, if the motion is rectilinear, all the particles at any instant in the surface A will be found after one second in a similar surface A′, at a distance V, and as each particle is followed by a continuous thread of other particles, the volume of flow is the right prism AA′ having a base ω and length V.

If the direction of motion makes an angle θ with the normal to the surface, the volume of flow is represented by an oblique prism AA′ (fig. 7), and in that case

Fig. 7. |

If the velocity varies at different points of the surface, let the surface
be divided into very small portions, for each of which the
velocity may be regarded as constant. If *d*ω is the area and *v*, or
*v* cos θ, the normal velocity for this element of the surface, the
volume of flow is

*v d*ω, or ∫

*v*cos θ

*d*ω,

as the case may be.

§ 14. *Principle of Continuity.*—If we consider any completely
bounded fixed space in a moving liquid initially and finally filled
continuously with liquid, the inflow must be equal to the outflow.
Expressing the inflow with a positive and the outflow with a negative
sign, and estimating the volume of flow Q for all the boundaries,

In general the space will remain filled with fluid if the pressure at every point remains positive. There will be a break of continuity, if at any point the pressure becomes negative, indicating that the stress at that point is tensile. In the case of ordinary water this statement requires modification. Water contains a variable amount of air in solution, often about one-twentieth of its volume. This air is disengaged and breaks the continuity of the liquid, if the pressure falls below a point corresponding to its tension. It is for this reason that pumps will not draw water to the full height due to atmospheric pressure.

*Application of the Principle of Continuity to the case of a Stream.*—If
A_{1}, A_{2} are the areas of two normal cross sections of a stream,
and V_{1}, V_{2} are the velocities of the stream at those sections, then
from the principle of continuity,

_{1}A

_{1}= V

_{2}A

_{2};

_{1}/V

_{2}= A

_{2}/A

_{1}

that is, the normal velocities are inversely as the areas of the cross sections. This is true of the mean velocities, if at each section the velocity of the stream varies. In a river of varying slope the velocity varies with the slope. It is easy therefore to see that in parts of large cross section the slope is smaller than in parts of small cross section.

If we conceive a space in a liquid bounded by normal sections at
A_{1}, A_{2} and between A_{1}, A_{2} by stream lines (fig. 8), then, as there
is no flow across the stream lines,

_{1}/V

_{2}= A

_{2}/A

_{1},

as in a stream with rigid boundaries.

Fig. 8. |

In the case of compressible fluids the variation of volume due to
the difference of pressure at the two sections must be taken into
account. If the motion is steady the weight of fluid between two
cross sections of a stream must remain constant. Hence the weight
flowing in must be the same as the weight flowing out. Let *p*_{1}, *p*_{2}
be the pressures, *v*_{1}, *v*_{2} the velocities, G_{1}, G_{2} the weight per cubic foot
of fluid, at cross sections of a stream of areas A_{1}, A_{2}. The volumes
of inflow and outflow are

_{1}

*v*

_{1}and A

_{2}

*v*

_{2},

and, if the weights of these are the same,

_{1}A

_{1}

*v*

_{1}= G

_{2}A

_{2}

*v*

_{2};

and hence, from (5*a*) § 9, if the temperature is constant,

*p*

_{1}A

_{1}

*v*

_{1}=

*p*

_{2}A

_{2}

*v*

_{2}.