_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 [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
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
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], _/ _/
§ 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.