HYDROSTATICS. CHAPTER I. 1. WE learn from common experience that such substances as air and water are characterised by the ease with which portions of their mass can be removed, and by their extreme divisibility. These properties are illustrated by various common facts; if, for instance, we consider the ease with which fluids can be made to permeate each other, the extreme tenuity to which one fluid can be reduced by mixture with a large portion of another fluid, the rarefaction of air which can be effected by means of an air-pump, and other facts of a similar kind, it is clear that, practically, the divisibility of fluid is unlimited: we find, moreover, that in separating portions of fluids from each other, the resistance offered to the division is very slight, and in general almost inappreciable. By a generalization from such observations, the conception naturally arises of a substance possessing in the highest degree these properties, which exist, in a greater or less degree, in every fluid with which we are acquainted, and hence we are led to the following Definition of a Perfect Fluid. 2. A perfect fluid is an aggregation of particles which yield at once to the slightest effort made to separate them from each other. If then an indefinitely thin plane be made to divide such a fluid in any direction, no resistance will be offered to the division, and the pressure exerted by the fluid on the plane B. H. 1 will be entirely normal to it; that is, a perfect fluid is assumed to have no "viscosity," no property of the nature of friction. The following fundamental property of a fluid is therefore obtained from the above definition. The pressure of a perfect fluid is always normal to any surface with which it is in contact. As a matter of fact, all fluids do more or less offer a resistance to separation or division, but, just as the idea of a rigid body is obtained from the observation of bodies in nature which only change form slightly on the application of great force, so is the idea of a perfect fluid obtained from our experiences of substances which possess the characteristics of extremely easy separability and apparently unlimited divisibility. The following definition will include fluids of all degrees of viscosity. A fluid is an aggregation of particles which yield to the slightest effort made to separate them from each other, if it be continued long enough. Hence it follows that, in a viscous fluid at rest, there can be no tangential action, or shearing stress, and therefore, as in the case of a perfect fluid, The pressure of a fluid at rest is always normal to any surface with which it is in contact. Thus all propositions in Hydrostatics are true for all fluids whatever be the viscosity. In Hydrodynamics it will be found that the equations of motion are considerably modified by taking account of the viscosity of a fluid. 3. Fluids are divided into Liquids and Gases; the former, such as water and mercury, are not sensibly compressible, except under very great pressures; the latter are easily compressible, and expand freely if permitted to do so. Hence the former are sometimes called inelastic, and the latter elastic fluids. 4. Fluids are acted upon by the force of gravity in the same way as solids; with regard to liquids this is obvious; and that air has weight can be shewn directly by weighing a closed vessel, exhausted as far as possible: moreover, the phenomena of the tides shew that fluids are subject to the attractive forces of the sun and moon as well as of the earth, and it is assumed, from these and other similar facts, that fluids of all kinds are subject to the law of gravitation, that is, that they attract, and are attracted by, all other portions of matter, in accordance with that law. Measure of the Pressure of Fluids. 5. Consider a mass of fluid at rest under the action of any forces, and let A be the area of a plane surface exposed to the action of the fluid, that is, in contact with it, and P the force which is required to counterbalance the action of the fluid upon A. If the action of the fluid upon A be uniform, P then is the pressure on each unit of the area A. If the A pressure be not uniform, it must be considered as varying continuously from point to point of the area A, and if ☎ be the force on a small portion a of the area about a given point, @ then will approximately express the rate of pressure over a. α When a is indefinitely diminished let ultimately=p, then α p is defined to be the measure of the pressure at the point considered, p being the force which would be exerted on an unit of area, if the rate of pressure over the unit were uniform and the same as at the point considered. The force upon any small area a about a point, the pressure at which is p, is therefore pa+y, where y vanishes ultimately in comparison with pa when a (and consequently pa) vanishes. 6. The pressure at any point of a fluid at rest is the same in every direction. This is the most important of the characteristic properties |