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 [omega] and length V.
If the direction of motion makes an angle [theta] with the normal to
the surface, the volume of flow is represented by an oblique prism AA´
(fig. 7), and in that case
Q = [omega]V cos [theta].
[Illustration: 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[omega] is the area and v,
or v cos [theta], the normal velocity for this element of the surface,
the volume of flow is
_ _
/ /
Q = | v d[omega], or | v cos [theta] d[omega],
_/ _/
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,
[Sigma]Q = 0.
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 A1, A2 are the areas of two normal cross sections of a
stream, and V1, V2 are the velocities of the stream at those sections,
then from the principle of continuity,