Online Encyclopedia
Search over 40,000 articles from the original, classic Encyclopedia Britannica, 11th Edition.
HYDROMECHANICS (Gr. ubpops avuca)
Online EncyclopediaEncyclopedia Home :: HOR-I25
del.icio.us it!
|
See also:HYDROMECHANICS (Gr. ubpops avuca) , the See also:science of the See also:mechanics of See also:water and fluids in See also:general, including See also:hydrostatics or the mathematical theory of fluids in See also:equilibrium, and hydro-mechanics, the theory of fluids in See also:motion. The See also:practical application of hydromechanics forms the See also:province of See also:hydraulics (q.v.). See also:Historical.—The fundamental principles of hydrostatics were first given by See also:Archimedes in his See also:work IIepi r%uv oxouuiewv, or De its quae vehuntur in humido, about 250 B.C., and were afterwards applied to experiments by See also:Marino Ghetaldi (1566–1627) in his Promotus Archimedes (1603). Archimedes maintained that each particle of a fluid See also:mass, when in equilibrium, is equally pressed in every direction ; and he inquired into the conditions according to which a solid See also:body floating in a fluid should assume and preserve a position of equilibrium. In the See also:Greek school at See also:Alexandria, which flourished See also:tinder the auspices of the See also:Ptolemies, the first attempts were made at the construction of See also:hydraulic machinery, and about 120 B.C. the See also:fountain of See also:compression, the See also:siphon, and the forcing-See also:pump were invented by Ctesibius and See also:Hero. The siphon is a See also:simple See also:instrument; but the forcing-pump is a complicated invention, which could scarcely have been expected in the See also:infancy of hydraulics. It was probably suggested to Ctesibius by the See also:Egyptian See also:Wheel or Noria, which was See also:common at that See also:time, and which was a See also:kind of See also:chain pump, consisting of a number of earthen pots carried• See also:round by a wheel. In some of these See also:machines the pots have a See also:valve in the bottom which enables them to descend without much resistance, and diminishes greatly the load upon the wheel; and, if we suppose that this valve was introduced so See also:early as the time of Ctesibius, it is not difficult to perceive how such a See also:machine might have led to the invention of the forcing-pump. Notwithstanding these inventions of the Alexandrian school, its See also:attention does not seem to have been directed to the motion of fluids; and the first See also:attempt to investigate this subject was made by Sextus See also:Julius See also:Frontinus, inspector of the public fountains at See also:Rome in the reigns of See also:Nerva and See also:Trajan. In his work De aquaeductibus urbis Romae commentaries, he considers the methods which were at that time employed for ascertaining the quantity of water discharged from ajutages, and the mode of distributing the See also:waters of an See also:aqueduct or a fountain. He remarked that the flow of water from an orifice depends not only on the magnitude of the orifice itself, but also on the height of the water in the See also:reservoir; and that a See also:pipe employed to carry off a portion of water from an aqueduct should, as circumstances required, have a position more or less inclined to the See also:original direction of the current. But as he was unacquainted with the See also:law of the velocities of See also:running water as depending upon the See also:depth of the orifice, the want of precision which appears in his results is not surprising. Benedetto See also:Castelli (1577–1644), and Evangelista See also:Torricelli (16o8–1647), two of the disciples of Galileo, applied the discoveries of their See also:master to the science of See also:hydrodynamics. In 1628 Castelli published a small work, Della misura dell' acque See also:correnti, in which he satisfactorily explained several phenomena in the motion of fluids in See also:rivers and canals; but he committed a See also:great paralogism in sup-posing the velocity of the water proportional to the depth of the orifice below the See also:surface of the See also:vessel. Torricelli, observing that in a See also:jet where the water rushed through a small ajutage it See also:rose to nearly the same height with the reservoir from which it was supplied, imagined that it ought to move with the same velocity as if it had fallen through that height by the force of gravity, and hence he deduced the proposition that the velocities of liquids are as the square See also:root of the See also:head, apart from the resistance of the See also:air and the See also:friction of the orifice. This theorem was published in 1643, at the end of his See also:treatise De motu gravium projectorum, and it was See also:con-firmed by the experiments of Raffaello Magiotti on the quantities of water discharged from different ajutages under different pressures (1648). In the hands of Blaise See also:Pascal (1623–1662) hydrostatics assumed the dignity of a science, and in a treatise on the equilibrium of liquids (Sur l'equilibre See also:des See also:liqueurs), found among his See also:manuscripts after his See also:death and published in 1663, the See also:laws of the equilibrium of liquids were demonstrated in the most simple manner, and amply confirmed by experiments. The theorem of Torricelli was employed by many succeeding writers, but particularly by Edme See also:Mariotte (162o–1684), whose Traite du mouvement des eaux, published after his death in the See also:year 1686, is founded on a great variety of well-conducted experiments on the motion of fluids, performed at See also:Versailles and See also:Chantilly. In the discussion of some points he committed considerable mistakes. Others he treated very superficially, and in none of his experiments apparently did he attend to the diminution of efflux arising from the contraction of the liquid vein, when the orifice is merely a perforation in a thin See also:plate; but he appears to have been the first who attempted to ascribe the discrepancy between theory and experiment to the retardation of the water's velocity through friction. His contemporary Domenico Guglielmini (1655–1710), who was inspector of the rivers and canals at See also:Bologna, had ascribed this diminution of velocity in rivers to transverse motions arising front inequalities in their bottom. Rut as Mariotte observed similar obstructions even in See also:glass pipes where no transverse currents could exist, the causeassigned by Guglielmini seemed destitute of See also:foundation. The See also:French philosopher, therefore, regarded these obstructions as the effects of friction. He supposed that the filaments of water which graze along the sides of the pipe lose a portion of their velocity; that the contiguous filaments, having on this See also:account a greater velocity, rub upon the former, and suffer a diminution of their celerity; and that the other filaments are affected with similar retardations proportional to their distance from the See also:axis of the pipe. In this way the See also:medium velocity of the current may be diminished, and consequently the quantity of water discharged in a given time must, from the effects of friction, be considerably less than that which is computed from theory. The effects of friction and viscosity in diminishing the velocity of running water were noticed in the Principia of See also:Sir See also:Isaac See also:Newton, who threw much See also:light upon several branches of hydromechanics. At a time when the Cartesian See also:system of vortices universally prevailed, he found it necessary to investigate that See also:hypothesis, and in the course of his investigations he showed that the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclose it; and from this it evidently follows that the velocity of a filament of water moving in a pipe is an arithmetical mean between the velocities of the filaments which surround it. Taking See also:advantage of these results, See also:Henri Pitot (1695–1771) afterwards showed that the retardations arising from friction are inversely as the diameters of the pipes in which the fluid moves. The attention of Newton was also directed to the See also:discharge of water from orifices in the bottom of vessels. He supposed a cylindrical vessel full of water to be perforated in its bottom with a small hole by which the water escaped, and the vessel to be supplied with water in such a manner that it always remained full at the same height. He then supposed this cylindrical See also:column of water to be divided into two parts,—the first, which he called the " See also:cataract," being an hyperboloid generated by the revolution of an See also:hyperbola of the fifth degree around the axis of the See also:cylinder which should pass through the orifice, and the second the See also:remainder of the water in the cylindrical vessel. He considered the See also:horizontal strata of this hyperboloid as always in motion, while the remainder of the water was in a See also:state of See also:rest, and imagined that there was a kind of cataract in the See also:middle of the fluid. When the results of this theory were compared with the quantity of water actually discharged, Newton concluded that the velocity with which the water issued from the orifice was equal to that which a falling body would receive by descending through See also:half the height of water in the reservoir. This conclusion, however, is absolutely irreconcilable with the known fact that jets of water rise nearly to the same height as their reservoirs, and Newton seems to have been aware of this objection. Accordingly, in the second edition of his Principia, which appeared in 1713, he reconsidered his theory. He had discovered a contraction in the vein of fluid (vena contracta) which issued from the orifice, and found that, at the distance of about a See also:diameter of the See also:aperture, the See also:section of the vein was contracted in the subduplicate ratio of two to one. He regarded, therefore, the section of the contracted vein as the true orifice from which the discharge of water ought to be deduced, and the velocity of the effluent water as due to the whole height of water in the reservoir; and by this means his theory became more conformable to the results of experience, though still open to serious objections. Newton was also the first to investigate the difficult subject of the motion of waves (q.v.).
In 1738 See also:Daniel See also:Bernoulli (1700–1782) published his Hydrodynamica seu de viribus et motibus fluidorum See also:commentarii. His theory of the motion of fluids, the germ of which was first published in his memoir entitled Theoria nova de motu aquarum per canales quocunque fluentes, communicated to the See also:Academy of St See also:Petersburg as early as 1726, was founded on two suppositions, which appeared to him conformable to experience. He supposed that the surface of the fluid, contained in a vessel which is emptying itself by an orifice, remains always horizontal; and, if the fluid mass is conceived to be divided into an See also:infinite number of horizontal strata of the same bulk, that these strata remain contiguous to each other, and that all their points descend vertically, with velocities inversely proportional to their breadth, or to the horizontal sections of the reservoir. In See also:order to determine the motion of each stratum, he employed the principle of the conservatio virium vivarum, and obtained very elegant solutions. But in the See also:absence of a general demonstration of that principle, his results did not command the .confidence which they would otherwise have deserved, and it became desirable to have a theory more certain, and depending-solely on the fundamental laws of mechanics. See also:Colin See also:Maclaurin (1698–1746) and See also: It was more fully See also:developed in his Traite des fluides, published in 1744, in which he gave simple and elegant solutions of problems See also:relating to the equilibrium and motion of fluids. He made use of the same suppositions as Daniel Bernoulli, though his calculus was established in a very different manner. He considered, at every instant, the actual motion of a stratum as composed of a motion which it had in the preceding instant and of a motion which it had lost; and the laws of equilibrium between the motions lost furnished him with equations re-presenting the motion of the fluid. It remained a desideratum to See also:express by equations the motion of a particle of the fluid in any assigned direction. These equations were found by d'Alembert from two principles—that a rectangular See also:canal, taken in a mass of fluid in equilibrium, is itself in equilibrium, and that a portion of the fluid, in passing from one See also:place to another, preserves the same volume when the fluid is incompressible, or dilates itself according to a given law when the fluid is elastic. His ingenious method, published in 1752, in his Essai sur la resistance des fluides, was brought to perfection in his Opuscules mathematiques, and was adopted by Leonhard See also:Euler.
The See also:resolution of the questions concerning the motion of fluids was effected by means of Euler's partial See also:differential coefficients. This calculus was first applied to the motion of water by d'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.
One of the most successful labourers in the science of hydro-dynamics at this See also:period was See also:Pierre See also: Dubuat, therefore, assumed it as a proposition of fundamental importance that, when water flows in any channel or See also:bed, the accelerating force which obliges it to move is equal to the sum of all the resistances which it meets with, whether they arise from its own viscosity or from the friction of its bed. This principle was employed by him in the first edition of his work, which appeared in 1779. The theory contained in that edition was founded on the experiments of others, but he soon saw that a theory so new, and leading to results so different from the See also:ordinary theory, should be founded on new experiments more direct than the former, and he was employed in the performance of these from 178o to 1783. The experiments of Bossut were made only on pipes of a moderate declivity, but Dubuat used declivities of every kind, and made his experiments upon channels of various sizes. The theory of running water was greatly advanced by the re-searches of Gaspard Riche de See also:Prony (1755–1839). From a collection of the best experiments by previous workers he selected eighty-two (fifty-one on the velocity of water in conduit pipes, and See also:thirty-one on its velocity in open canals) ; and, discussing these on See also:physical and See also:mechanical principles, he succeeded in See also:drawing up general formulae, which afforded a simple expression for the velocity of running water. J. A. Eytelwein (1764–1848) of See also:Berlin, who published in 18o1 a valuable compendium of hydraulics entitled See also:hand See also:buck der Mechanik and der Hydraulik, investigated the subject of the discharge of water by See also:compound pipes, the motions of jets and their impulses against plane and oblique surfaces; and he showed theoretically that a water-wheel will have its maximum effect when its circumference moves with half the velocity of the stream. J. N. P. See also:Hachette (1769–1834) in 1816–1817 published See also:memoirs containing the results of experiments on the spouting of fluids and the discharge of vessels. His See also:object was to measure the contracted See also:part of a fluid vein, to examine the phenomena attendant on additional tubes, and to investigate the See also:form of the fluid vein and the results obtained when different forms of orifices are employed. Extensive experiments on the discharge of water from orifices (Experiences hydrauliques, See also:Paris, 1832) were conducted under the direction of the French See also:government by J. V. See also:Poncelet (1788–1867) and J. A. Lesbros . (179o–186o). P. P. Boileau (1811–1891) discussed their results and added experiments of his own (Traite de la mesure des eaux courantes, Paris, 1854). K. R. Bornemann re-examined all these results with great care, and gave formulae expressing the variation of the coefficients of discharge in different conditions (See also:Civil Ingenieur, 188o). Julius Weisbach (1806–1871) also made many experimental investigations on the discharge of fluids. The experiments of J. B. See also:Francis (See also:Lowell Hydraulic Experiments, See also:Boston, Mass., 1855) led him to propose See also:variations in the accepted formulae for the discharge over weirs, and a See also:generation later a very See also:complete investigation of this subject was carried out by H. See also:Bazin. An elaborate inquiry on the flow of water in pipes and channels was conducted by H. G. P. See also:Darcy (1803–1858) and continued by H.Bazin, at the expense of the French government (Recherches hydrauliques, Paris, i866). See also:German See also:engineers have also devoted See also:special attention to the measurement of the flow in rivers; the Beitrage zur Hydrographie des See also:Konig- reiches Bohmen (See also:Prague, 1872–1875) of A. R. Harlacher (1842–1890)
contained valuable measurements of this kind, together with a com-
parison of the experimental results with the formulae of flow that had
been proposed up to the date of its publication, and important data
were yielded by the gaugings of the See also:Mississippi made for the See also:United
States government by A. A. See also:Humphreys and H. L. See also: Hele See also:Shaw. (X.) HYDROSTATICS Hydrostatics is a science which See also:grew originally out of a number of isolated practical problems; but it satisfies the requirement of perfect accuracy in its application to phenomena, the largest and smallest, of the behaviour of a fluid. At the same time, it delights the pure theorist by the simplicity of the See also:logic with which the fundamental theorems may be established, and by the elegance of its mathematical operations, insomuch that hydro-See also:statics may be considered as the Euclidean pure See also:geometry of mechanical science. 1. The Different States of a Substance or See also:Matter.—All substance in nature falls into one of the two classes, solid and fluid; a solid substance, the See also:land, for instance, as contrasted with a fluid, like water, being a substance which does not flow of itself. A fluid, as the name implies, is a substance which flows, or is capable of flowing; water and air are the two fluids distributed most universally over the surface of the See also:earth. Fluids again are divided into two classes, termed a liquid and a See also:gas, of which water and air are the See also:chief examples. A liquid is a fluid which is incompressible or practically so, i.e. it does not See also:change in volume sensibly with change of pressure. A gas is a compressible fluid, and the change in volume is considerable with moderate variation of pressure. Liquids, again, can be poured from one open vessel into another, and can be kept in an uncovered vessel, but a gas tends to diffuse itself indefinitely and must be preserved in a closed reservoir. The distinguishing characteristics of the three kinds of sub-stance or states of matter, the solid, liquid and gas, are summarized thus in O. See also:Lodge's Mechanics: A solid has both See also:size and shape. A liquid has size but not shape. A' gas has neither size nor shape. 2. The Change of State of Matter.—By a change of temperature and pressure combined, a substance can in general be made to pass from one state into another; thus by gradually increasing the temperature a solid piece of See also:ice can be melted into the liquid state of water, and the water again can be boiled off into the gaseous state as See also:steam. Again, by raising the temperature, a See also:metal in the solid state can be melted and liquefied, and poured into a See also:mould to assume any form desired, which is retained when the metal cools and solidifies again; the gaseous state of a metal is revealed by the spectroscope. Conversely, a See also:combination of increased pressure and lowering of temperature will, if carried far enough, reduce a gas to a liquid, and afterwards to the solid state; and nearly every gaseous substance has now undergone this operation. A certain See also:critical temperature is observed in a gas, above which the liquefaction is impossible; so that the gaseous state has two subdivisions into (i.)a true gas, which cannot be liquefied, because its temperature is above the critical temperature, (ii.) a vapour, where the temperature is below the critical, and which can ultimately be liquefied by further lowering of temperature or increase of pressure. 3. Plasticity and Viscosity.—Every solid substance is found to be plastic more or less, as exemplified by punching, shearing and cutting; but the plastic solid is distinguished from the viscous fluid in that a plastic solid requires a certain magnitude of stress to be exceeded to make it flow, whereas the viscous liquid will yield to the slightest stress, but requires a certain length of time for the effect to be appreciable. According to See also:Maxwell (Theory of See also:Heat) " When a continuous alteration of form is produced only by a stress exceeding a certain value, the substance is called a solid, however soft and plastic it may be. But when the smallest stress, if only continued See also:long enough, will cause a perceptible and increasing change of form, the substance must be regarded as a viscous fluid, however hard it may be." Maxwell illustrates the difference between a soft solid and a hard liquid by a jelly and a See also:block of See also:pitch; also by the experiment of supporting a See also:candle and a stick of sealing-See also:wax; after a considerable time the sealing-wax will be found See also:bent and so is a fluid, but the candle remains straight as a solid. 4. See also:Definition of a Fluid.—A fluid is a substance which yields continually to the slightest tangential stress in its interior; that is, it can be divided very easily along any plane (given plenty of time if the fluid is viscous). It follows that when the fluid has come to rest, the tangential stress in any plane in its interior must vanish, and the stress must be entirely normal to the plane. This mechanical See also:axiom of the normality of fluid pressure is the foundation of the mathematical theory of hydrostatics. The theorems of hydrostatics are thus true for all stationary fluids, however, viscous they may be; it is only when we come to hydrodynamics, the science of the motion of a fluid, that viscosity will make itself See also:felt and modify the theory; unless we begin by postulating the perfect fluid, devoid of viscosity, so that the principle of the normality of fluid pressure is taken to hold when the fluid is in See also:movement. 5. The Measurement of Fluid Pressure.—The pressure at any point of a plane in the interior of a fluid is the intensity of the normal thrust estimated per unit See also:area of the plane. Thus, if a thrust of P lb is distributed uniformly over a plane area of A sq. ft., as on the horizontal bottom of the See also:sea or any reservoir, the pressure at any point of the plane is P/A lb per sq. ft., or P/144A lb per sq. in. (lb/ft.2 and lb/in.2, in the Hospitalier notation, to be employed in the sequel). If the See also:distribution of the thrust is not See also:uniform, as, for instance, on a See also:vertical or inclined See also:face or See also:wall of a reservoir, then P/A represents the See also:average pressure over the area ; and the actual pressure at any point is the average pressure over a small area enclosing the point. Thus, if a thrust OP lb acts on a small plane area AA ft.2 enclosing a point B, the pressure p at B is the limit of AP/AA; and p= It (AP/AA) =dP/dA, (I) in the notation of the differential calculus. 6. The Equality of Fluid Pressure in all Directions.—This fundamental principle of hydrostatics follows at once from the principle of the normality of fluid pressure implied in the definition of a fluid in § 4. Take any two arbitrary directions in the plane of the See also:paper, and draw a small isosceles triangle See also:abc, whose sides are perpendicular to the two directions, and consider the equilibrium of a small triangular See also:prism of fluid, of which the triangle is the See also:cross section. Let P, Q denote the normal thrust across the sides be, ca, and R the normal thrust across the See also:base ab. Then, since these three forces maintain equilibrium, and R makes equal angles with P and Q, therefore P and Q must be equal. But the faces bc, ca, over which P and Q See also:act, are also equal, so that the pressure on each face is equal. A scalene triangle abc might also be employed, or a W See also:tetrahedron. It follows that the pressure of a fluid requires to be calculated in one direction only, chosen as the simplest direction for convenience. 7. The Transmissibility of Fluid Pressure. Any additional pressure applied to the fluid' will be transmitted equally to every point in the See also:case of a liquid; this principle of the transmissibility of pressure was enunciated by Pascal, 1653, and applied by him to the invention of the hydraulic See also:press. This machine consists essentially of two communicating cylinders (fig. fa), filled with liquid and closed by pistons. If a thrust P lb is applied to one See also:piston of area A ft.2, it will be balanced by a thrust W lb applied to the other piston of area B ft?, where p=P/A=W/B, (I) the pressure p of the liquid being supposed uniform; and, by making the ratio B/A sufficiently large, the mechanical advantage can be increased to any desired amount, and in the simplest manner possible, without the intervention of levers and machinery. Fig. ib shows also a See also:modern form of the hydraulic press, applied to the operation of covering an electric'See also:cable with a See also:lead coating. 8. Theorem.—In a fluid at rest under gravity the pressure is the same at any two points in the same horizontal plane; in other words, a surface of equal pressure is a horizontal plane. This is proved by taking any two points A and B at the samelevel, and considering the equilibrium of a thin prism of liquid AB, bounded by planes at A and B perpendicular to AB. As gravity and the fluid pressure on the sides of the prism act at right angles to AB, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B; and so the pressure is the same at all points in the same horizontal plane. If the fluid is a liquid, it can have a See also:free surface without diffusing itself, as a gas would; and this free surface, being a surface of zero pressure, or more generally of uniform atmospheric pressure, will also be a surface of equal pressure, and therefore a horizontal plane. Hence the theorem.—The free surface of a liquid at rest under gravity is a horizontal plane. This is the characteristic distinguishing between a solid and a liquid; as, for in-stance, between land and water. The land has hills and valleys, but the surface of water at rest is a horizontal plane; and if disturbed the surface moves in waves. 9. Theorem.—In a homogeneous liquid at rest under gravity the pressure increases uniformly with the depth. This is proved by taking the two points A and B in the same vertical line, and considering the equilibrium of the prism by resolving vertically. In this case the thrust at the See also:lower end B must exceed the thrust at A, the upper, end, by the See also:weight of the prism of liquid; so that, denoting the cross section of the prism by a ft.2, ,the pressure at A and By by ph and p lb/ft.2, and by w the See also:density of the liquid estimated in lb/ft.', pa-poa=wa. AB, (I) p =ce. AB +po. (2) Thus in water, where w=62.4lb/ft.', the pressure increases 62.4 lb/ft.2, or 62.4=144=0.433 lb/in.2 for every additional See also:foot of depth. to. Theorem.—If two liquids of different density are resting in vessels in communication, the height of the free surface of such liquid above the surface of separation is inversely as the density. For if the liquid of density o rises to the height h and of density p to the height k, and Po denotes the atmospheric pressure, the pressure in the liquid at the level of the surface of separation will be oh+po and pk+Po, and these being equal we have eh = pk. (I) The principle is illustrated in the See also:article See also:BAROMETER, where a column of See also:mercury of density o and height h, rising in the See also:tube to the Torricellian vacuum, is balanced by a column of air of density p, which may be supposed to rise as a homogeneous fluid to a height k, called the height of the homogeneous See also:atmosphere. Thus water being about 800 times denser than air and mercury 13.6 times denser than water, k/h=e/p=800X13.6=Io,88o; (2) and with an average barometer height of 30 in. this makes k 27,200 ft., about 8300 metres. I I. The Head of Water or a Liquid.—The pressure eh at a depth h ft. in liquid of density a is called the pressure due to a head of h ft. of the liquid. The atmospheric pressure is thus due to an average head of 30 in. of mercury, or 30)03.6+12 =34 ft. of water, or 27,200 ft. of air. The pressure of the air is a convenient unit to employ in practical work, where it is called an " atmosphere "; it is made the See also:equivalent of a pressure of one kg/cm2; and one ton/inch2, employed as the unit with high pressure as in See also:artillery, may be taken as 15o atmospheres. 12. Theorem.—A body immersed in a fluid is buoyed up by a force equal to the weight of the liquid displaced, acting vertically upward through the centre of gravity of the displaced Iiquid. For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acct$ vertically upward through the centre of gravity of the fluid displaced; and is equal to the weight. When the body is floating freely like a ship, the equilibrium of this liquid thrust with the weight of the ship requires that the weight of water displaced is equal to the weight of the ship and the two centres of gravity are in the same vertical line. So also a See also:balloon begins to rise when the weight of air displaced is greater than the weight of the balloon, and it is in equilibrium when the weights are equal. This theorem is called generally the principle of Archimedes. It is used to determine the density of a body experimentally; for if W is the weight of a body weighed in a See also:balance in air (strictly in vacuo), and if W' is the weight required to balance when the body is suspended in water, then the upward thrust of the liquid or weight of liquid displaced is W-W', so that thespeeifc gravity (S.G.),{defined, as the ratio of the weight of a body to the weight if 'an equal volume of water', is W/(W-W ) As stated first by Archimedes, the principle asserts the obvious fact that a body displaces its overt volume of water; and he utilized it in the problem of the determination of the See also:adulteration of the See also:crown of See also:Hiero. He wqighed out a lump of See also:gold and of See also:silver of the same weightas the crown; and, immersing the three in See also:succession in water, he found they spilt over See also:measures of water in the ratio is :A; or 33: 24': 44; thence it follows that the gold : silver alloy of the grown was as I I : 9 by weight. 13. Theprem: The resultant vertical thrust on any portion of a curved surface eXposed to the pressure of a fluid at rest under gravity is the weight f fluid cut out by vertical lines See also:drawn round the boundary of the curved surface. Theorem.—The 'resultant horizontal thrust in any direction is obt*ipedby de veing parallel horizontal lines round the boundary, and. iintetsecting a plane perpendicular to their direction in a plane See also:curve; asd then investigating the thrust on this plane area, which will, he the same as on the curved surface. The $eOof- of these theorems proceeds as before, employing the normality, principle; they are required, for instance, in the determination of the liquid thrust on any portion of the bottom of a ship. In casting a thinhollow object like a See also:bell, it will be seen that the resultant upward thrust on the mould may be many times greater than the weight ofmetal; many a curious' experiment has been devised 'to illustrate this See also:property and classed as 'a hydrostatic See also:paradox (See also:Boyle, Hydrostatical Paradoxes, 1666). Consider,_ for instance, the operation of casting a hemispherical bell, in fig. 2. As the molten metal is run in, the upward thrust on the outside mould, when IC' 4i the level has reached PP', is the weight of metal in the volume generated by the revolution of APQ; and this, by a theorem of Archimedes, has the same volume as the See also:cone ORR', or *7rya, where y is the depth of metal, the horizontal sections being equal so long as y is less than the See also:radius of the outside hemisphei'e. Afterwards, when the metal has risen above B, to the level KK', the additional thrust is the weight of the cylinder of diameter KK' and height BH. The upward thrust is the same, however thin the metal may be in the interspace between the See also:outer mould and the core inside; and this was formerly considered paradoxical. See also:Analytical Equations of Equilibrium of .a Fluid at rest under any System pf Force. 14. Referred to three fixed coordinate axes, . a fluid, in which. the pressure is p, the density p, and X, Y, Z the components of impressed force per unit mass, requires for the equilibrium of the part filling a fixed surface S, on resolving parallel to Ox, thus leading to the differential relation at every point =PX, d ='pY, d =pZ. (3) The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically. Hence the space variation of the pressure in any direction, or the pressure-gradient, is the resolved force per unit volume in that direction. The resultant force is therefore in the direction of the steepest pressure-gradient, and this is normal to the surface of equal pressure; for equilibrium to exist in a fluid the lines of force must therefore be capable of being cut orthogonally by a system of surfaces, which will be surfaces of equal pressure. Ignoring temperature effect, and taking the density as a See also:function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force;
t
slat
, I c (I t , or X, Y, Z (4)
x are p z
are the partial differential coefficients of some function P,=fdp/p, of x, y, s;so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force m any direction is the down-See also: Xkpi, (19)
p p' Po p l p ,
f JlpdS =f f f pXdxdydz, (I)
where 1, in, n denote the direction cosines of the normal drawn outward of the surface S.
But by See also:Green's transformation
fftpdS =fff azdxdydz,
(2)
in which P may be 'called the hydrostatic head and V the head of potential.
With variation of temperature, the surfaces of equal pressure and density need not coincide; but, taking the pressure, density and temperature as connected by some relation, such as the gas-equation, the surfaces ofequal density and temperature must intersect in lines lying on a surface of equal pressure.
15. As an example of the general equations, take the simplest case of a uniform See also: The stability of a ship is Investigated practically by If the equation of this line, referred to new coordinate axes in the inclining it; a weight is moved across the See also:deck and the See also:angle is plane area, is written observed of the See also:heel produced. xcos a+ysin a—h=o, (3) R= f f p(h—xcos a—y See also:sin a)dxdy, (4) Suppose l' tons is moved c ft. across the deck of a ship of W tons £R= px(h-±x cgs a—y sin a)dxdy, (5) displacement; the C.G. will move from G to GI the reduced distance G5Cx,=c (P/W) ; and if . B, called the centre of buoyancy, moves = f py(h—x See also:cos a —y sin a)dxdy. to Bialong the curve of buoyancy BBL, the normal of this curve at Placing the new origin at the C.G. of the atea A, BI will be the new vertical BiGi, See also:meeting the old vertical in a point f fxdxdy=o, f 1ydxdy=o, '(6) M, the centre of curvature of BB,, called the metacentre. R=pn , (7) ' If the ship'heelsWthrough an angle 0 or a slope of r in et, ihA= —cos of fxydA- sin affxydA, (8) GM=GG' cote=mc(P/W), (1) yhA = -cos a ff xydA —sin o f f yadA. (9) and GM is called the metecentric height; and the ship must be Turning the axes to make them coincide with the prtnaipal axes • ballasted, so that G lies below M. If G was above M, the tangent of the area A, thus making f f xydA =o, drawn from G to the svolute of B, and normal to the curve of buoyancy, zh=-atcos•a,9h=—b'sin-a, ,(1o) }would give the vertical in a new position of equilibrium. Thus in where H.M.S. "See also:Achilles of, goon tons displacement it was found that ffx'dA=See also:Aar, f fydA=Ab', (t t) moving 20 tons 'across the deck, a distance of 42 ft., caused the bob a and b denoting the semi-axes of the momenta] See also:ellipse of the area, of a pendulum 2d ft. long to move through .10 in., so that This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental ellipse at the C.G. of the area. Thus the C.P. of a rectangle or parallelogram with a side in the surface is at of the depth of the lower side; of a triangle with a vertex in the surface and base horizontal is i of the depth of the base; but if the base is in the surface, the C.P. is at half the depth of the vertex; as on the faces of atetrahedron, with one edge in the surface. The core of an area is the name given to ,the limited area round its C.G. within which the C.P. must lie when the area is immersed completely; the boundary of the core is therefore the See also:locus of the See also:antipodes with respect to the momental ellipse of water lines which See also:touch the boundary of the area. Thus the core of a circle or an ellipse is a concentric circle or ellipse of one See also:quarter the size. The C.P. of water lines passing through a fixed point lies on a straight line, the antipolar of the point; and thus the core of a triangle is a similar triangle of one quarter the size, and the core of a parallelogram is another parallelogram, the diagonals of which are the middle third of the median lines. In the See also:design of a structure such as a tall reservoir See also:dam it is important that the line of thrust in the material should pass inside the core of a section, so that' the material should not be in a state of tension anywhere and so liable to open and admit the water. 17. Equilibrium and Stability of a Ship or Floating Body. The See also:Meta-centre--The principle of Archimedes in § 1.2 leads immediatelyto the conditions of equilibrium of a body sup-ported freely in fluid, like a See also:fish in water or a balloon in the air or like a ship (fig. 3i floating partly See also:im- mersed in water and the rest in air. The body is in equilibrium under two forces:—(i.) its weight W acting vertically downward through G, the C.G. of the body, and (ii.) the buoyancy of the fluid, equal to the weight of the displaced fluid, and acting vertically upward through B, the C.G. of the displaced fluid; also In a See also:diagram it is conducive. to clearness to draw the. ship in one position, and to incline the water-line; and the See also:page can be turned if it is desired to bring the new water-line horizontal. Suppose the ship turns about an axis through F in the water-line area, perpendicular to the plane of the paper; denoting by y the distance of an See also:element dA if the water-line area from the axis of rotation; the change of displacement is EydA tan 0, so that there is no change of displacement if EydA =0, that is, if the axis passes through the C.G. of the water-line area, which we denote by V' and See also:call the centre of flotation. The righting couple of the wedges of See also:immersion and emersion will be MwydA tan 0.y -w tan B2y'dA=w tan B.Ak' ft. tons, (4) w denoting the density of water in tons/ft.', and W =wV, for a displacement of V ft.' This couple, combined with the original buoyancy W through B, is equivalent to the new buoyancy through B, so that W.BBI =wAk' tan 8, (5) BM=BBicot0=Ak'/V, (6) giving the radius of curvature BM of the curve of buoyancy B, in terms of the displacement V, and Ak' the moment of inertia of the water-line area about an axis through F, perpendicular to the• plane of displacement, , . An inclining couple due to moving a weight about in a ship •will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a See also:principal axis at F of the momental ellipse of the water-line area A. For if the ship turns through a small angle-a about the line FF', then b,, b2,the C.G. of the See also:wedge ofimmereion and emersion, will be the C.P. with respect ,to FF' of the two partsof the water-line area, so that bile will be conjugate to FF' with respect to the momental ellipse at F. The See also:naval architect distinguishes between the stability of, form, represented by the righting couple W.BM, and the stability of batlastleg, represented by W.BG. Ballasted with G at B, the righting couple when the ship is heeled through B is given by W.BM. tan B; but if weights inside the ship are raised to bring G above B, the righting couple is diminished by W.BG. tan 0, so that the resultant righting couple is W,GM. tan O. Provided the ship is designed to See also:float upright at the smallest draft with no load on See also:board, the stability at any other draft of water can be arranged by the stowage of the weight, high or See also:low. 19. Proceeding as in § 16 for the determination of the C.P. of an area, the same See also:argument will show that an inclining couple due to GM=2IO X42Xg000224ft.; ' cot =24, 0 = 2° 24'. the movement of a weight P through a distance a will cause the ship to heel through an angle B about an axis FF' through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes See also:a2-hV/A, b2-hV/A, (I) h denoting the vertical height BG between C.G. and centre of buoyancy. The varying direction of the inclining couple Pc may be realized by swinging the weight P from a See also:crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on See also:shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of k with respect to the confocal ellipse, parallel to FF', at a distance F from F FK=(k2-hV/A)/FQ sin QFF' (2) through an angle 0 or a slope of one in m, given by ' sin 0=m -= QFF', (3) where k denotes the radius of gyration about FF' of the water-line area. Burning the See also:coal on a voyage has the See also:reverse effect on a steamer. HYDRODYNAMICS 20. In considering the motion of a fluid we shall suppose it non-viscous, so that whatever the state of motion the stress across any section is normal, and the principle of the normality and thence of the equality of fluid pressure can be employed, as in hydrostatics. The practical problems of fluid motion, which are amenable to mathematical See also:analysis when viscosity is taken into account, are excluded from treatment here, as constituting a See also:separate See also:branch called "hydraulics" (q.v.). Two methods are employed in hydrodynamics, called the Eulerian and Lagrangian, although both are due originally to Leonhard Euler. In-the Eulerian method the attention is fixed on a particular point of space, and the change is observed there of pressure, density and velocity, which takes place during the motion; but in the Lagrangian method we follow up a particle of fluid and observe how it changes. The first may be called the statistical method, and the second the historical, according to J. C. Maxwell. The Lagrangian method being employed rarely, we shall confine ourselves to the Eulerian treatment. The Eulerian Form of the Equations of Motion. 21. The first equation to be established is the equation of continuity, which expresses the fact that the increase of matter within a fixed surface is due to the flow of fluid across the surface into its interior. In a straight uniform current of fluid of density p, flowing with velocity q, the flow in See also:units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle 0 with the velocity, is oAq cos 0, the product of the density p, the area A, and q cos 0 the component velocity normal to the plane. Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and 0 the angle which the outward-drawn normal makes with the velocity q at that point, dM/dt = See also:rate of increase of fluid inside the surface, (I) = See also:flux across the surface into the interior _ -f f pq cos 6dS, the integral equation of continuity. In the Eulerian notation u, v, w denote the components of the velocity g parallel to the coordinate axes at any point (x, y, z) at the time t; u, v, w are functions of x, y, z, t, the See also:independent variables; and d is used here to denote partial differentiation with respect to any one of these four independent variables, all capable of varying one at a time. To See also:transfer the integral equation into the differential equation of continuity, Green's transformation is required again, namely, J fe J (§1+2") dxdydz =ff (l +mn+nf)dS, (2) or individuallywhich becomes by Green's transformation ( J J J \dl+ddx +dd +da dxdydz=o, (5) y leading to the differential equation of continuity when the integration is removed. 22. The equations of motion can be established in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the force acting throughout the space S, and by the pressure acting over the surface S. Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is - Jf f puq cos odS = - f f (lpu2+mpuv+npuw)dS, (t) which by Green's transformation is f J J +d d uv+a(dz w) dxdydz. (2) y The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is f f f pXdxdydz, and by the pressure at the surface S is `f f 1pds - -f f f dxdydz, by Green's transformation. The time rate of increase of momentum of the fluid inside S is f f f d(:t )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that f f ( dpudpusdpuv+dpuw_p +dpl dxdydz=o, (6) ) ( dx dy dz X dx/ leading to the differential equation of motion dpu dpu2 dpuv dpuw_ x_dp dt + dx ++ p dx' with two similar equations. The absolute unit of force is employed here, and not the gravitation unit of hydrostatics; in a numerical application it is assumed that C.G.S. units are intended. These equations may be simplified slightly, using the equation of continuity (5) § 21; for dpu dpu2 dpuv dpuw -t+dx+ dy+ dz =p (ac +dx+dy du , az +u (dp , dpudpv , dpw\ . dt'-x' dy dz / reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual form di+ax } vdy, du =x-Pa , with the two others dt +ud 'vdy-wdz Y d ' See also:dw dw dw dw i dp dt +u— x+vdy +w _ dz -Z- p dz. 23. As a See also:rule these equations are established immediately by determining the component See also:acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d'Alembert's principle form a system in equilibrium. To determine the component acceleration of a particle, suppose-F. ,.. to denote any function of x, y, z, t, and investigate the time rate of F for a moving particle; denoting the change by DF/dt, DF_it F(x+uat, y+vat, z+wat, t+bt)-F(x, y, z, t) at = a +udx +v y+w a and D/dt is called particle differentiation, because it follows the rate of change of a particle as it leaves the point x, y, z; but dF/dt, dF/dx, dF/dy, dF/dz (2) represent the rate of change of F at the time t, at the point, x, y, z. fixed in space. J J J azdxdydz=ff1 dS,..., (3) where the integrations extend throughout the volume and over the surface of a closed space S; 1, m, it denoting the direction cosines of the outward-drawn normal at the surface element dS, and E, n, iany continuous functions of x, y, z. The integral equation of continuity (t) may now be written ( J J adxdydz+ f. f (Ipu+mpv+npw)dS =o, (4) (3) (4) (7) (8) (9) (io) (II) The components of acceleration of a particle of fluid are consequently Du du du du du dt = dt +udx+vdy+, Dv dv dv dv dv dt =di+ud-i+vdy+wdz' Dw dw dw, dw dw dt =d+udx+vdy+2' leading to the equations of motion above. If F (x, y, z, t) =o represents the equation of a surface containing always the same particles of fluid, = o, or dt u dx +v +w dz = o, (6) which is called the differential equation of the bounding surface. A bounding surface is such that there is no flow'of fluid across it, as expressed by equation (6). The surface always contains the same fluid inside it, and condition (6) is satisfied over the complete surface, as well as any part of it. But turbulence in the motion will vitiate the principle that a bounding surface will always consist of the same fluid particles, as we see on the surface of turbulent water. 24. To integrate the equations of motion, suppose the impressed force is due to a potential V, such that the force in any direction is the rate of diminution of V, or its downward gradient; and then X=-dV/dx, Y=-dV/dy, Z=-dV/dz; (1) and putting dw dv du dw dv du = ay-dz=2 , d-dx=2v' r-dy 21' dn+dr=o, Tx dy dz the equations of motion may be written 2ef+2wn+dx =0, de dH dt - 2wH+2ur+ _ °, dl -See also:tun+21)t+dH0, where H =fdp/p+V +1q2, (7) 42 = u2+v2+See also:w2, (8) and the three terms in H may be called the pressure head, potential head, and head of velocity, when the gravitation unit is employed and Zq2 is replaced by ;q2/g. Eliminating H between (5) and (6) DE du dv dw (du dv dw\ dt 'dx-ndx-'dx+ dx+dy+"di/ =o, and combining this with the equation of continuity I Dp du dv dw p dt +dx+dy+dz =°' D (f f du n dv dw di \P)-pdx-pdx-p d =°, with two similar equations. Additional information and CommentsThere are no comments yet for this article.
» Add information or comments to this article.
Please link directly to this article:
Highlight the code below, right click, and select "copy." Then paste it into your website, email, or other HTML. Site content, images, and layout Copyright © 2006 - Net Industries, worldwide. |
|
|
[back] HYDROLYSIS (Gr. 6Swp, water, Anew, to loosen) |
[next] HYDROMEDUSAE |