HTDKAULICS.] HYDROMECHANICS PART III. HYDRAULICS. I. THE DATA OF HYDRAULICS. [Units. Except where other units are given, the units throughout this article are feet, pounds, pounds per sq. ft., feet per second.] 1. Properties of Fluids. The fluids to which the laws of practical hydraulics relate are substances the parts of which possess very great mobility, or which offer a very small resistance to distortion independently of inertia. Under the general heading Hydromechanics a fluid is defined to be a substance which yields continually to the slightest tangential stress, and hence in a fluid at rest there can be no tangential stress. But, further, in fluids such as water, air, steam, &c., to which the present division of the article relates, the tangential stresses that are called into action between contiguous portions during distortion or change of figure are always small compared with the weight, inertia, pressure, <fec., which produce the visible motions it is the object of hydraulics to estimate. On the other hand, while a fluid passes easily from one form to another, it opposes considerable resistance to change of volume. It is easily deduced from the absence or smallness of the tangential stress that contiguous portions of fluid act on each other with a pressure which is exactly or very nearly normal to the interface which separates them. The stress must be a pressure, not a tension, or the parts would separate. Further, at any point in a fluid the pressure in all directions must be the same ; or, in other words, the ; pressure on any small element of surface is independent of the orientation of the surface. 2. Fluids are divided into liquids, or incompressible fluids, and gases, or compressible fluids. Very great changes of pressure change the volume of liquids only by a small amount, and if the pressure on them is reduced to zero they do not sensibly dilate. In gases or compressible fluids the volume alters sensibly for small changes of pressure, and if the pressure is indefinitely diminished they dilate without limit. In ordinary Hydraulics, liquids are treated as absolutely incompressible. In dealing with gases the changes of volume which accompany changes of pressure must be taken into account. 3. Viscous fluids are those in which change of form under a continued stress proceeds gradually and increases indefinitely. A very viscous fluid opposes great resistance to change of form in a short time, and yet may be deformed considerably by a small stress acting for a long period. A block of pitch is more easily splintered than indented by a hammer, but under the action of the mere weight of its parts acting for a long enough time it flattens out and flows like a liquid. All actual fluids are viscous. They oppose a resistance to the relative motion of their parts. This resistance diminishes with the velocity of the relative motion, and becomes zero in a fluid the parts of which are relatively at rest. When the relative motion of different parts of a fluid is small, the viscosity may be neglected without introduc ing important errors. On the other hand, where there is considerable relative motion, the viscosity may be expected to have an influence too great to be neglected. Measurement of Viscosity. Coefficient of Viscosity. Suppose tlie plane ab, fig. 11, of area a>,to move with the velocity V relatively to the surface cd and parallel to it. Let the space between be filled with liquid. The layers of liquid in contact with ab and cd adhere to them. The intermediate layers all offering an equal resistance to shearing or distortion, the rectangle of fluid abed will take the form of the parallelogram a b cd. Further, the resistance to the motion of ab may be expressed in the form R = cwV (1), where * is a coefficient the nature of which remains to be deter mined.
- <*
/a. ft /& T / / A c / /d If we suppose the liquid between ab and cd divided into layers as. shown in tig. 12, it will be clear that the stress R acts, at each divid ing face, forwards in the direction of mo tion if we consider the upper layer, backwards if we con sider the lower layer. Now suppose the original thickness of the layer T increased to?iT; if the bound ing plane in its new position has the ve locity ?iV, the shear ing at each dividing face will be exactly the same as before, and the resistance must therefore be the same. Hence, But equations (1) and (2) may both be expressed in one equation if K and K are replaced by a constant varying in versely as the thickness of the layer. Putting Fig. 12. or, for an indefinitely thin layer, R = M V . dV "Tt (3), an expression first proposed by Navier. The coefficient jt is termed the coefficient of viscosity. According to Maxwell, the value of /u for air at Fahr. in pounds, when the velocities are expressed in feet per second, is M = -000 000 025 6(461 + 0) ; that is, the coefficient of viscosity is proportional to the absolute - temperature and independent of the pressure. The value of /j. for water at 77 Fahr. is, according to Helmholtz and Piotrowski, jt= O OOO 001 91, the units being the same as before. For water p decreases rapidly with increase of temperature. 4. When a fluid flows in a very regular manner, as for instance when it flows in a capillary tube, the velocities vary gradually at any moment from one point of the fluid to a neighbouring point. The layer adjacent to the sides of the tube adheres to it and is at rest. The layers more interior than this slide on each other. But the resistance developed by these regular movements is very small. If in large pipes and open channels there were a similar regu larity of movement, the neighbouring filaments would acquire, especially near the sides, very great relative veloci ties. Boussinesq has shown that the central filament in a semicircular canal of 1 metre radius, and inclined at a slope of only 0-0001, would have a velocity of 187 metres per second, 1 the layer next the boundary remaining at rest. But before such a difference of velocity can arise, the motion of the fluid becomes much more complicated. Volumes of fluid are detached continually from the boundaries, and, revolving, form eddies traversing the fluid in all directions, and sliding with finite relative velocities against those surrounding them. These slidings develop resistances, incomparably greater than the viscous resistance due to movements varying continuously from point to point. The movements which produce the phenomena commonly ascribed to fluid friction must be regarded as rapidly or even suddenly varying from one point to another. The in ternal resistances to the motion of the fluid do not depend 1 Journal de M. Liouville, t. xiii. , 1868 ; Memoires de I Academic
des Sciences de I lnstitut de France, i. xjdii., xxiv., 1877.
