HYDRAULICS.] HYDROMECHANICS 461 II. KINEMATICS OF FLUIDS. 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 ivhirlpool 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 buch a constant path may be termed a fluid filament or elementary stream. 11. Stead y 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. 14), a mass of water a has a con stant 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 givt-n 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 a, steady motion. On the Q ____ contrary, if at the point a James500 (talk) 05:01, 2 January 2020 (UTC) _^^^ 5 the velocity or direction varies from moment to * 1 S- 14. moment the motion is termed unsteady. Steady motion is sometimes termed permanent motion. 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 dur ing the subsidence of the flood. But while many streams of a tor rential 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 which would be strictly de fined as one of steady motion, its regime becomes permanent. Hence steady motion and permanent regime are sometimes used as mean ing the same thing. The one, however, is a definite term appli cable 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 experi mentally 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. 15) 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 inter nal work in deforming Fig. 15. the layer may usually be disregarded, and the resistance to the motion is confined to the circumference. Laminar Mo ion. In the case of streams having solid boundaries, it is obrscrved that the central p:irts move faster than the lateral parts. To take account of these differences of velocity, the stream may be conceived to be divided into thin lamina , "having cross Sections somewhat similar to the solid boundary of the stream, and sliding on each other. The different lamiiue can then be treated as having differing velocities according to any law either observed a or deduced from their mutual friction. A much closer approxi mation to the real motion of ordinary streams is thus obtained. Stream Line Motion. In the preceding hypothesis, all the par ticles in each lamina have the same velocity at any given cross sec tion 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 ac tual 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. 16) be any ideal plane surface, of area u, in a stream, normal to the direction of motion, and let V A Fig. 16. be the velocity of the fluid. Then the volume flowing through the- surface A in unit time is Q = o>Y (1). 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 ca and length V. If the direction of motion makes an angle 6 with the normal to the surface, the volume of flow is represented by an oblique prism AA (fig. 17), and in that case Q = o>V cos 6 . If the velocity varies at different points of the surface, let the sur face be divided into very small portions, for each of which the velocity may be regarded as constant. If du is the area and v, or v cos 6, the normal velocity for this element of the surface, the volume of flow is Q =yVd&>, orjv cos 6 dca , 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 nega tive sign, and estimating the volume of flow Q for all the boundaries, 2Q-0. In general the space will remain filled with fluid if the pressure at every point remains positive. There will be a break of con tinuity, 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 AJ , A., are the areas of two normal cross sections of a stream, and V, , VjTare the velocities of the stream at those sections, then,
from the principle of continuity,
