HYDROMECHANICS (Gr. ubpops avuca), the science of the mechanics of water and fluids in general, including hydrostatics or the mathematical theory of fluids in equilibrium, and hydromechanics, the theory of fluids in motion. The practical application of hydromechanics forms the province of hydraulics (q.v.).
Historical.—The fundamental principles of hydrostatics were first given by Archimedes in his work IIepi r%uv oxouuiewv, or De its quae vehuntur in humido, about 250 B.C., and were afterwards applied to experiments by Marino Ghetaldi (1566–1627) in his Promotus Archimedes (1603). Archimedes maintained that each particle of a fluid mass, when in equilibrium, is equally pressed in every direction ; and he inquired into the conditions according to which a solid body floating in a fluid should assume and preserve a position of equilibrium.
In the Greek school at Alexandria, which flourished tinder the auspices of the Ptolemies, the first attempts were made at the construction of hydraulic machinery, and about 120 B.C. the fountain of compression, the siphon, and the forcingpump were invented by Ctesibius and Hero. The siphon is a simple instrument; but the forcingpump is a complicated invention, which could scarcely have been expected in the infancy of hydraulics. It was probably suggested to Ctesibius by the Egyptian Wheel or Noria, which was common at that time, and which was a kind of chain pump, consisting of a number of earthen pots carried• round by a wheel. In some of these machines the pots have a 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 early as the time of Ctesibius, it is not difficult to perceive how such a machine might have led to the invention of the forcingpump.
Notwithstanding these inventions of the Alexandrian school, its attention does not seem to have been directed to the motion of fluids; and the first attempt to investigate this subject was made by Sextus Julius Frontinus, inspector of the public fountains at Rome in the reigns of Nerva and 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 waters of an 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 reservoir; and that a 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 original direction of the current. But as he was unacquainted with the law of the velocities of running water as depending upon the depth of the orifice, the want of precision which appears in his results is not surprising.
Benedetto Castelli (1577–1644), and Evangelista Torricelli (16o8–1647), two of the disciples of Galileo, applied the discoveries of their master to the science of hydrodynamics. In 1628 Castelli published a small work, Della misura dell' acque correnti, in which he satisfactorily explained several phenomena in the motion of fluids in rivers and canals; but he committed a great paralogism in supposing the velocity of the water proportional to the depth of the orifice below the surface of the vessel. Torricelli, observing that in a jet where the water rushed through a small ajutage it 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 root of the head, apart from the resistance of the air and the friction of the orifice. This theorem was published in 1643, at the end of his treatise De motu gravium projectorum, and it was confirmed by the experiments of Raffaello Magiotti on the quantities of water discharged from different ajutages under different pressures (1648).
In the hands of Blaise Pascal (1623–1662) hydrostatics assumed the dignity of a science, and in a treatise on the equilibrium of liquids (Sur l'equilibre des liqueurs), found among his manuscripts after his death and published in 1663, the 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 Mariotte (162o–1684), whose Traite du mouvement des eaux, published after his death in the year 1686, is founded on a great variety of wellconducted experiments on the motion of fluids, performed at Versailles and 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 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 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 glass pipes where no transverse currents could exist, the causeassigned by Guglielmini seemed destitute of foundation. The 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 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 axis of the pipe. In this way the 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 Sir Isaac Newton, who threw much light upon several branches of hydromechanics. At a time when the Cartesian system of vortices universally prevailed, he found it necessary to investigate that 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 advantage of these results, 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 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 column of water to be divided into two parts,—the first, which he called the " cataract," being an hyperboloid generated by the revolution of an hyperbola of the fifth degree around the axis of the cylinder which should pass through the orifice, and the second the remainder of the water in the cylindrical vessel. He considered the horizontal strata of this hyperboloid as always in motion, while the remainder of the water was in a state of rest, and imagined that there was a kind of cataract in the 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 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 diameter of the aperture, the 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 Daniel Bernoulli (1700–1782) published his Hydrodynamica seu de viribus et motibus fluidorum 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 Academy of St 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 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 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 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 dependingsolely on the fundamental laws of mechanics. Colin Maclaurin (1698–1746) and John Bernoulli (1667–1748), who were of this opinion, resolved the problem by more direct methods, the one in his Fluxions, published in 1742, and the other in his Hydraulica nunc primum detecta, et demonstrata directe ex furulamentis pure mechanicis, which forms the fourth volume of his works. The method employed by Maclaurin has been thought not sufficiently rigorous; and that of John Bernoulli is, in the opinion of Lagrange, defective in clearness and precision. The theory of Daniel Bernoulli was opposed also by Jean le Rond d'Alembert. When generalizing the theory of pendulums of Jacob Bernoulli (1634–1705) he discovered a principle of dynamics so simple and general that it reduced the laws of the motions of bodies to that of their equilibrium. He applied this
principle to the motion of fluids, and gave a specimen of its application at the end of his Dynamics in 1743. It was more fully developed in his Traite des fluides, published in 1744, in which he gave simple and elegant solutions of problems 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 representing the motion of the fluid. It remained a desideratum to 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 canal, taken in a mass of fluid in equilibrium, is itself in equilibrium, and that a portion of the fluid, in passing from one 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 Euler.
The resolution of the questions concerning the motion of fluids was effected by means of Euler's partial 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 hydrodynamics at this period was Pierre Louis Georges Dubuat (1734–1809). Following in the steps of the Abbe Charles Bossut (Nouvelles Experiences sur la resistance des fluides, 1777), he published, in 1786, a revised edition of his Principes d'hydraulique, which contains a satisfactory theory of the motion of fluids, founded solely upon experiments. Dubuat considered that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane. But as the motion of rivers is not continually accelerated,and soon arrives at a state of uniformity,it is evident that the viscosity of the water, and the friction of the channel in which it descends, must equal tke accelerating force. Dubuat, therefore, assumed it as a proposition of fundamental importance that, when water flows in any channel or 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 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 researches of Gaspard Riche de Prony (1755–1839). From a collection of the best experiments by previous workers he selected eightytwo (fiftyone on the velocity of water in conduit pipes, and thirtyone on its velocity in open canals) ; and, discussing these on physical and mechanical principles, he succeeded in drawing up general formulae, which afforded a simple expression for the velocity of running water.
J. A. Eytelwein (1764–1848) of Berlin, who published in 18o1 a valuable compendium of hydraulics entitled hand buck der Mechanik and der Hydraulik, investigated the subject of the discharge of water by compound pipes, the motions of jets and their impulses against plane and oblique surfaces; and he showed theoretically that a waterwheel will have its maximum effect when its circumference moves with half the velocity of the stream.
J. N. P. Hachette (1769–1834) in 1816–1817 published memoirs containing the results of experiments on the spouting of fluids and the discharge of vessels. His object was to measure the contracted part of a fluid vein, to examine the phenomena attendant on additional tubes, and to investigate the 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, Paris, 1832) were conducted under the direction of the French government by J. V. 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 reexamined all these results with great care, and gave formulae expressing the variation of the coefficients of discharge in different conditions (Civil Ingenieur, 188o). Julius Weisbach (1806–1871) also made many experimental investigations on the discharge of fluids. The experiments of J. B. Francis (Lowell Hydraulic Experiments, Boston, Mass., 1855) led him to propose variations in the accepted formulae for the discharge over weirs, and a generation later a very complete investigation of this subject was carried out by H. Bazin. An elaborate inquiry on the flow of water in pipes and channels was conducted by H. G. P. Darcy (1803–1858) and continued by H.Bazin, at the expense of the French government (Recherches hydrauliques, Paris, i866). German
engineers have also devoted special attention to the measurement
of the flow in rivers; the Beitrage zur Hydrographie des Konig
reiches Bohmen (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 Mississippi made for the United
States government by A. A. Humphreys and H. L. Abbot, by Robert
Gordon's gaugings of the Irrawaddy, and by Allen J. C. Cunningham's
experiments on the Ganges canal. The friction of water, investigated
for slow speeds by Coulomb, was measured for higher speeds by
William Froude (1810–1879), whose work is of great value in the
theory of ship resistance (Brit. Assoc. Report., 1869), and stream line
motion was studied by Professor Osborne Reynolds and by Professor
H. S. Hele Shaw. (X.)
HYDROSTATICS
Hydrostatics is a science which 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 logic with which the fundamental theorems may be established, and by the elegance of its mathematical operations, insomuch that hydrostatics may be considered as the Euclidean pure geometry of mechanical science.
1. The Different States of a Substance or Matter.—All substance in nature falls into one of the two classes, solid and fluid; a solid substance, the 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 earth.
Fluids again are divided into two classes, termed a liquid and a gas, of which water and air are the chief examples.
A liquid is a fluid which is incompressible or practically so, i.e. it does not 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 substance or states of matter, the solid, liquid and gas, are summarized thus in O. Lodge's Mechanics:
A solid has both 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 ice can be melted into the liquid state of water, and the water again can be boiled off into the gaseous state as steam. Again, by raising the temperature, a metal in the solid state can be melted and liquefied, and poured into a 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 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 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 Maxwell (Theory of 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 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 block of pitch; also by the experiment of supporting a candle and a stick of sealingwax; after a considerable time the sealingwax will be found bent and so is a fluid, but the candle remains straight as a solid.
4. 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 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 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 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 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 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 distribution of the thrust is not uniform, as, for instance, on a vertical or inclined face or wall of a reservoir, then P/A represents the 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 paper, and draw a small isosceles triangle abc, whose sides are perpendicular to the two directions, and consider the equilibrium of a small triangular prism of fluid, of which the triangle is the cross section. Let P, Q denote the normal thrust across the sides be, ca, and R the normal thrust across the 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 act, are also equal, so that the pressure on each face is equal. A
scalene triangle abc might also be employed, or a W 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 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 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 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 modern form of the hydraulic press, applied to the operation of covering an electric'cable with a 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 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 instance, 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 lower end B must exceed the thrust at A, the upper, end, by the 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 density of the liquid estimated in lb/ft.',
papoa=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 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 article BAROMETER, where a column of mercury of density o and height h, rising in the 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 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 equivalent of a pressure of one kg/cm2; and one ton/inch2, employed as the unit with high pressure as in 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 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 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 WW', 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/(WW )
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 adulteration of the crown of Hiero. He wqighed out a lump of gold and of silver of the same weightas the crown; and, immersing the three in succession in water, he found they spilt over 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 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 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 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 property and classed as 'a hydrostatic paradox (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 cone ORR', or *7rya, where y is the depth of metal, the horizontal sections being equal so long as y is less than the 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 outer mould and the core inside; and this was formerly considered paradoxical.
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 pressuregradient, is the resolved force per unit volume in that direction. The resultant force is therefore in the direction of the steepest pressuregradient, 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 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 downward gradient of v; and then
dP dV o; or P+V =constant, (3)I2= idp Idp
pdx pO'paz=l
P=fdp/p=z+a constant. (2)
When the density p is uniform, this becomes, as before in (2) § 9
P =pz+Po• (3)
Suppose the density p varies as some nth power of the depth below 0, then
dp/dz=p=, z"
pz p (/ p' p=11n1 1 n+I =—'– µ/ supposing p and p to vanish together.
These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gasequation
p=pk=Rpp, (6)
where ® denotes the absolute temperature; and then
de
Rde a d  ( \p) (7)
n+I'
so that the temperaturegradient d6/da is constant, as in convective equilibrium in (ii).
From the gasequation. in general, in the atmosphere
I dp_i,dp d _p 1 de=t I de (g)
p dz p dz B dz p—e k—® dz'
which is positive, and the density p diminishes with the ascent, provided the temperaturegradient de/dz does not exceed 0/k.
With uniform temperature, taking k constant in the gasequation,
dp/dz=p=pik, p=poet'' (9)
so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures pi and p2 at heights z1 and z2
(ziz:)/k =log,(pz/pl) =2,3 logio(p2/p,). (Io)
In the convective equilibrium of the atmosphere, the air is supposed to change in density and pressure without exchange of heat by conduction; and then
pips = (0/Bo)", p/po = (e/®o)n+I, (II)
de p
ire= p4J=(n+1) ® =(n+1)R, y=1+;,
where y is the ratio of the specific heat at constant pressure and constant volume.
In the more gerferal ease of the convective equilibrium of a spherical atmosphere surrounding the earth, of radius a,
dp=(n+t).=
po Bo rzdr, (12)
gravity "varying inversely as the square of the distance r from the centre; so that, ,kps/po, denoting the height of the homogeneous atmosphere at the surface,,6 is given by
(n+1)k(r O/9o) a(I air), (13)
or if c denotes the distance where 0 = o,
O a c—r
6o=r
When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as
p—ps=k(e—es), or p/po=li(p—po)/X, x=kpo, : (15)
so that h is the pressure due to a head k of the liquidat densitypo under atmoipherrc pressure po; and it is the gauge pressure required on this law to double the density. Then
dp/dz=kdp/dz=p, p=poe='k, ppo=kpo(esik_1); (,6) and if the liquid was incompressible, the depth at pressure p would be (p  po) 1po, so that the lowering of the surface due tocompression rs
kern kz =10/1e, when k is large. (17)
For sea water, x is about 25,000 atmospheres, and k is then 25,000 times the height of the water barometer, about 250,000 metres, so that in an ocean to kilometres deep the level is lowered about 200 metres by the compressibility of the water; and the density at the bottom is increased 4 %.
On another physical assumption of constant cubical elasticity h. dp adp/p, (p'e NV). =log( A/po), (,8)
(IS dp =x x/1_1\ ..
z+ 1fL. 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 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 gasequation, 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 field of gravity, with Oz directed vertically downward; employing the gravitation unit of force,
(I)
(4) (5)
(14),
.zPo 'z (20)
and the lowering of the surface is
log z.._klog(t—i) as before in 17).
16. Centre of Pressure.—A plane area exposed to fluid pressure on one side experiences a, single resultant thrust, the integrated
for equilibrium these two forces must be equal and opposite in the same line.
The conditions of equilibrium of a body, floating like a ship on the surface of a liquid, are therefore:
(i,) the weight of the body must be less than the weight of the total volume of liquid it can displace; or else the body will sink
pressure over the area, acting through a definite point called to the bottom of the liquid; the difference of the weights is the centre of pressure (C.P.) of the area. I called the " reserve of buoyancy."
Thus if the plane is normal to Os, the resultant,thrust (ii.) the weight of liquid which the body displaces in the
R= f rpdxdy, (I) position• of equilibrium is equal to the weight W of the body; and and the coordinates i, y of the C.P. are given by
(.,) the C.G.s B of the liquid displaced and G of the body, zR = f f xpdxdy, 5,R = f f yydxdy. (2) 1..11
The C.P. is thus the C.G. of a plane lamina bonnded by the areh, must lie in the saute vertical line GB.
in which the surface density is p. i8. In addition to. satisfying these conditions of equilibrium,
If p is uniform, the C.P. and C.G. of the area coincide. a ship must fulfil the further condition of stability, so as to keep For a homogeneous liquid at rest under gravity, p is proportional upright; if displaced slightly from this position, the forces
to the depth below the surface, i.e. to the perpendicular distance called into must be such as to restore the ship to the upright
from the line of intersection of the plane of the area with the free play surface of the liquid. again. 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 deck and the angle is plane area, is written observed of the heel produced.
xcos a+ysin a—h=o, (3)
R= f f p(h—xcos a—y 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 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, 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'sina, ,(1o) }would give the vertical in a new position of equilibrium. Thus in
where H.M.S. "Achilles of, goon tons displacement it was found that
ffx'dA=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 semiaxes of the momenta] 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 locus of the antipodes with respect to the momental ellipse of water lines which touch the boundary of the area. Thus the core of a circle or an ellipse is a concentric circle or ellipse of one 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 design of a structure such as a tall reservoir 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 MetacentreThe principle of Archimedes in § 1.2 leads immediatelyto the conditions of equilibrium of a body supported freely in fluid,
like a fish in water or a balloon in the air or like a ship (fig. 3i floating partly 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 diagram it is conducive. to clearness to draw the. ship in one position, and to incline the waterline; and the page can be turned if it is desired to bring the new waterline horizontal.
Suppose the ship turns about an axis through F in the waterline area, perpendicular to the plane of the paper; denoting by y the distance of an element dA if the waterline 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 waterline area, which we denote by V' and call the centre of flotation.
The righting couple of the wedges of 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 waterline 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 principal axis at F of the momental ellipse of the waterline area A. For if the ship turns through a small anglea about the line FF', then b,, b2,the C.G. of the wedge ofimmereion and emersion, will be the C.P. with respect ,to FF' of the two partsof the waterline area, so that bile will be conjugate to FF' with respect to the momental ellipse at F.
The 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 float upright at the smallest draft with no load on board, the stability at any other draft of water can be arranged by the stowage of the weight, high or low.
19. Proceeding as in § 16 for the determination of the C.P. of an area, the same 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 waterline area A, but a confocal to it, of squared semiaxes
a2hV/A, b2hV/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 crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on 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 waterline 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=(k2hV/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 waterline area. Burning the coal on a voyage has the reverse effect on a steamer.
HYDRODYNAMICS
20. In considering the motion of a fluid we shall suppose it nonviscous, 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 analysis when viscosity is taken into account, are excluded from treatment here, as constituting a separate branch called "hydraulics" (q.v.). Two methods are employed in hydrodynamics, called the Eulerian and Lagrangian, although both are due originally to Leonhard Euler. Inthe 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 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 outwarddrawn normal makes with the velocity q at that point,
dM/dt = rate of increase of fluid inside the surface, (I)
= 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 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 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 timerate 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 =xPa ,
with the two others
dt +ud 'vdywdz Y d ' dw dw dw dw i dp dt +u— x+vdy +w _
dz Z p dz.
23. As a rule these equations are established immediately by determining the component 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 pressuregradient, will according to d'Alembert's principle form a system in equilibrium.
To determine the component acceleration of a particle, supposeF.
,..
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 outwarddrawn 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+udi+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 =
aydz=2 , ddx=2v' rdy 21'
dn+dr=o, Tx dy dz the equations of motion may be written
2ef+2wn+dx =0,
de dH
dt  2wH+2ur+ _ °,
dl tun+21)t+dH0,
where
H =fdp/p+V +1q2, (7)
42 = u2+v2+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 'dxndx'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)pdxpdxp d =°, with two similar equations.
Putting
End of Article: HYDROMECHANICS (Gr. ubpops avuca) 

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