HYDROMECHANICS

is the study of the dynamical properties of fluids and their application as motive power for machinery. Hydrostatics is the study of fluids in equilibrium or at rest. Hydrodynamics is concerned with fluids in motion. For these purposes. a fluid is defined as any substance which can. yield continuously to any force which tends to divide it along any plane. If through any body we draw an imaginary plane surface, the stre (tension, pre ure, or shearing force) acting acro . this plane can be resolved into two components—one perpendicular to, the other along the plane. In a solid, both components may be present; in a fluid at rest, the perpendicular component alone can be present. Fluids, therefore, include both liquids and gases. The basis for the first division of the subject— Hydrostatics—is given by the principle just enunciated, that in a fluid at rest the stre in any plane drawn through the fluid must always be normal to that plane. ‘This stre generally takes the form of a pre ure, which is measured at any point by the thrust or pre ure exerted over unit area including the point. It may be expre ed in atmospheres, pounds weight, or tons weight per square inch or square foot, or Gif the C.G.S. system be adopted) in dynes per square centimetre. The principal theorems of hydrostatics may be summarized as follows, it being understood, of course, that they apply to fluids at rest or in equilibrium: (1) The pre ure at any point of a fluid is the same in all directions. (2) Pre ure is transmi ible from one point to another in the same ma of fluid—i.e,, any additional pre ure applied to an incompre ible fluid will be transmitted equally to every point of the fluid. (8) In liquids acted on by gravity, the pre ure is uniform over all points in the same horizontal plane, and therefore the free surface of a liquid at rest under gravity is a horizontal plane. (4) The pre ure at any point in a homogeneous liq id acted on by gravity is proportional -to the depth of the point below the free surface; hence liquids ‘find their level,’ as the saying is, and where two liquids do not mix, their surface of separation is a horizontal plane. (5) The pre ure on any plane area immersed in a fluid is equal to the weight of a column of the liquid whose cro -sectional area is equal to the ersed area and whose height is equal to the depth below the free surface of the centre of gravity of the immersed area. (6) Any body which is wholly or partially immersed in a fluid acted on by gravity experiences an upward thrust equal in amount to the weight of fluid displaced by the body, and _ this thrust acts vertically upwards _ through the centre of gravity of the displaced fluid (Arch i me des’ principle). ' (7%) In order that a body floating freely a fluid may be in equilibrium, the * condition involved in (6) above must be satisfied, and, further, the centres of ' gravity of the displaced fluid and of the floating body must lie in the same vertical e. These theorems (for the proofs of which the reader is referred to any treatise on the subject) have a very extensive application in the sciences, “arts, and industries, and we have only space to mention a few in illustration. The transmi ibility of fluid pre ure is applied to the conveyance of power from one point to another. At a central station, hydraulic pre es or accumulators apply pre ure to a body of water; _ pipes in communication with the pre es convey the pre ure to cranes, motors, lifts, and other hydraulic machines The principle that water finds its own level is - familiar to all. The variation of pre ure with depth is the principle of Hare’s Hydrometer, and of similar methods of determining the density of liquids. It is also illustrated by the diminishing ' pre ure of the atmosphere as we ascend from sea-level, but in this case the rate of diminution is complicated by the fact that air is compre ible. In the case of the water in an ocean, the proportionate increase of pre ure with depth is more nearly correct, owing to the low compre ibility of water. ‘The expre ion given in (5) above for the total thrust on an immersed surface is constantly in use for calculating the stre es on dock gates, reservoir walls, dams. and other immersed surfaces. Arch i me des’ principle, taken together with the conditions of equilibrium given -in (7), introduces the whole question as to the equilibrium of ships, submarines, diving-bells, cai ons, balloons, and al 1} other bodies which are supported vertically by the upward thrust of the fluid in which they are wholly or partially immersed. In this connection, the question of stability of equilibrium of a: floating body arises, and thus introduces matters of the utmost importance in naval architecture. The principle of Arch i me des also forms the basis of a method for ascertaining the specific gravity of a body. For if the upward thrust of a body when totally immersed in water is equal to the weight of an equal volume of water, the weight of the body when weighed in water will be diminished by this amount. Hence the ratio of the weight of the body to the difference between the weights in water and air gives the specific gravity. Obviously, also, the ratio of the diminutions in weight of the same body when weiohed ‘na given liquid and in water is equal to the specific gravity of the liquid. In the case of a body specifically lighter than water, it floats in water with that proportion of its whole volume immersed which equals the specific gravity of the body. This is the basis for the construction of hydrometers, which are used to give direct readings of the specific gravity of liquids. In Hydrodynamics; we start with the a umption that liquids are perfect— i.e, that the reiative motions of their parts are not impeded by viscous friction, There is no liquid in nature which satisfies this condition, but in many liquids the effects of viscosity are so slight that they may be neglected. Later on in the treatment of the subject; the equations of motion are modified so as to allow for viscosity, but their complicated character has rendered their solution difficult except in a few simpler cases. It is not po ible to give here any explanation of the hydrodynamical equations, but one or two of the leading ideas connected with them may be mentioned. The motions of a fluid may be treated in two ways. We may fix attention on a given volume of space and take into account the amounts of fluid which enter or leave that space. Or we may choose a certain small volume of the liquid and study the changes which, during its motion, it may undergo in shape, position, speed, pre ure, etc. In either case we are led to the equations which describe the motion of the fluid. The whole ma may then be mapped out by lines which at each point have the same direction as the velocity of the fluid at that point. These are termed lines of flow, or stream-lines. Following the gravitational analogy, according to which water, in flowing down a hillside, always takes the steepest po ible course, we can draw a series of surfaces in the fluid such that they are always at right angles to the stream-lines. Such surfaces are termed surfaces of equivelocity potential, and the fluid 1 always move from places of higher’ to places of lower velocity potential. The term irrotational is applied to -those species of fluid motion in which a velocity potential exists, in order to distinguish them from cases of rotational or vortex motion where such potential does not exist. Another way of looking at irrotational motion is to imagine that each stream-line begins at some point where fluid is continuously produced and ends at some point where it is continuuously annihilated—that is, to start from a source and end ata sink. In rotational or vortex motion, a cylindrical portion of the fluid of very small diameter is in rotation about its axis. Such motion is accompanied by a tension between the ends of the vortex and a pre ure on its cylindrical surface or boundary. It can be proved that, in a perfect fluid, vortex motion cannot be created, and if fin existence it cannot be destroyed. In a@ viscous fluid, however, vortices can be produced; but if left to itself without a supply.of energy from without, the vortex is destroyed in time by the viscous forces in the fluid,