Hydromechanics
Encyclopedia
The history of fluid mechanics
, the study of how fluid
s move and the force
s on them, dates back to the Ancient Greeks.
in his work On Floating Bodies, around 250 BC. In it, Archimedes develops the law of buoyancy
, also known as Archimedes' Principle
. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. 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.
, which flourished under the auspices of the Ptolemies
, attempts were made at the construction of hydraulic machinery, and about 120 BC the fountain of compression, the siphon
, and the forcing-pump were invented by Ctesibius
and Hero
. The siphon is a simple instrument; but the forcing-pump 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 forcing-pump.
, 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 (tubes), 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.
(fl. 11151–130), were the first to apply experiment
al scientific method
s to fluid mechanics, especially in the field of fluid statics
, such as for determining specific weight
s. They applied the mathematical theories of ratio
s and infinitesimal
techniques, and introduced algebra
ic and fine calculation
techniques into the field of fluid statics.
In fluid statics, Biruni discovered that there is a correlation between the specific gravity of an object and the volume of water it displaces. He also introduced the method of checking tests
during experiments and measured the weight
s of various liquids. He also recorded the differences in weight between freshwater
and saline water
, and between hot water and cold water. During his experiments on fluid mechanics, Biruni invented the conical measure
, in order to find the ratio between the weight
of a substance in air and the weight of water displaced.
Al-Khazini, in The Book of the Balance of Wisdom (1121), invented a hydrostatic balance
.
brothers' Book of Ingenious Devices
described a number of early automatic control
s in fluid mechanics. Two-step level controls for fluids, an early form of discontinuous variable structure control
s, was developed by the Banu Musa brothers. They also described an early feedback controller
for fluids. According to Donald Routledge Hill
, the Banu Musa brothers were "masters in the exploitation of small variations" in hydrostatic
pressures and in using conical valve
s as "in-line" components in flow systems, "the first known use of conical valves as automatic controllers." They also described the use of other valves, including a plug valve
, float valve and tap
. The Banu Musa also developed an early fail-safe
system where "one can withdraw small quantities of liquid repeatedly, but if one withdraws a large quantity, no further extractions are possible." The double-concentric siphon
and the funnel
with bent end for pouring in different liquids, neither of which appear in any earlier Greek works, were also original inventions by the Banu Musa brothers. Some of the other mechanisms they described include a float chamber
and an early differential pressure
.
In 1206, Al-Jazari
's Book of Knowledge of Ingenious Mechanical Devices described many hydraulic machines. Of particular importance were his water-raising pump
s. The first known use of a crankshaft
in a chain pump was in one of al-Jazari's saqiya
machines. The concept of minimizing intermittent working
is also first implied in one of al-Jazari's saqiya chain pumps, which was for the purpose of maximising the efficiency of the saqiya chain pump. Al-Jazari also invented a twin-cylinder
reciprocating
piston
suction pump, which included the first suction
pipes, suction pumping, double-action pumping, and made early uses of valves and a crankshaft
-connecting rod
mechanism. This pump is remarkable for three reasons: the first known use of a true suction pipe (which sucks fluids into a partial vacuum
) in a pump, the first application of the double-acting principle, and the conversion of rotary
to reciprocating motion
, via the crankshaft-connecting rod mechanism.
In 1551, Taqi al-Din invented a six-cylinder pump incorporating lead weights and pistons.
, and Evangelista Torricelli
, 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 canal
s; 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).
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.
(1620–1684), whose Traité du mouvement des eaux, published after his death in the year 1686, is founded on a great variety of well-conducted 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. But as Mariotte observed similar obstructions even in glass pipes where no transverse currents could exist, the cause assigned 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.
-born French
engineer Henri Pitot
afterwards showed that the retardations arising from friction are inversely as the diameters of the pipes in which the fluid moves.
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.
.
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 depending-solely on the fundamental laws of mechanics. Colin Maclaurin
and John Bernoulli, 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.
. When generalizing the theory of pendulum
s of Jacob Bernoulli 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 Traité 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 equation
s 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
.
'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.
(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 the 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 1780 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.
published his seminal paper "Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen," in Journal für die reine und angewandte Mathematik, vol. 55, pp. 25–55. So important was the paper that a few years later P. G. Tait published an English translation, "On integrals of the hydrodynamical equations which express vortex motion", in Philosophical Magazine
, vol. 33, pp. 485–512 (1867). In his paper Helmholtz established his three "laws of vortex motion" in much the same way one finds them in any advanced textbook of fluid mechanics
today. This work established the significance of vorticity to fluid mechanics and science in general.
For the next century or so vortex dynamics matured as a subfield of fluid mechanics, always commanding at least a major chapter in treatises on the subject. Thus, H. Lamb's
well known Hydrodynamics (6th ed., 1932) devotes a full chapter to vorticity and vortex dynamics as does G. K. Batchelor's
Introduction to Fluid Dynamics (1967). In due course entire treatises were devoted to vortex motion. H. Poincaré's
Théorie des Tourbillons (1893), H. Villat's Leçons sur la Théorie des Tourbillons (1930), C. Truesdell's
The Kinematics of Vorticity (1954), and P. G. Saffman's Vortex Dynamics (1992) may be mentioned. Early on individual sessions at scientific conferences were devoted to vortices, vortex motion, vortex dynamics and vortex flows. Later, entire meetings were devoted to the subject.
The range of applicability of Helmholtz's work grew to encompass atmospheric and oceanographic flows, to all branches of engineering
and applied science
and, ultimately, to superfluid
s (today including Bose–Einstein condensate
s). In modern fluid mechanics the role of vortex dynamics in explaining flow phenomena is firmly established. Well known vortices have acquired names and are regularly depicted in the popular media: hurricanes, tornado
es, waterspout
s, aircraft trailing vortices (e.g., wingtip vortices
), drainhole vortices (including the bathtub vortex), smoke ring
s, underwater bubble air rings, cavitation vortices behind ship propellers, and so on. In the technical literature a number of vortices that arise under special conditions also have names: the Kármán vortex street
wake behind a bluff body, Taylor vortices between rotating cylinders, Görtler vortices
in flow along a curved wall, etc.
, who published in 1801 a valuable compendium of hydraulics entitled Handbuch der Mechanik und 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 water wheel
will have its maximum effect when its circumference moves with half the velocity of the stream.
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 JV Poncelet (1788–1867) and JA Lesbros (1790–1860).
PP Boileau (1811–1891) discussed their results and added experiments of his own (Traité de la mesure des eaux courantes, Paris, 1854). KR Bornemann re-examined all these results with great care, and gave formulae expressing the variation of the coefficients of discharge in different conditions (Civil Ingenieur, 1880). Julius Weisbach
(1806–1871) also made many experimental investigations on the discharge of fluids.
The experiments of JB 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 HGP Darcy (1803–1858) and continued by H Bazin, at the expense of the French government (Recherches hydrauliques, Paris, 1866).
made for the United States government by AA Humphreys and HL Abbot
, by Robert Gordon
's gaugings of the Ayeyarwady River
, and by Allen JC 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 HS Hele-Shaw
.
A curious diversion in the history of vortex dynamics was the vortex atom theory
of William Thomson
, later Lord Kelvin. His basic idea was that atoms were to be represented as vortex motions in the ether. This theory predated the quantum theory
by several decades and because of the scientific standing of its originator received considerable attention. Many profound insights into vortex dynamics were generated during the pursuit of this theory. Other interesting corollaries were the first counting of simple knots by P. G. Tait, today considered a pioneering effort in graph theory
, topology
and knot theory
. Ultimately, Kelvin's vortex atom was seen to be wrong-headed but the many results in vortex dynamics that it precipitated have stood the test of time. Kelvin himself originated the notion of circulation and proved that in an inviscid fluid
circulation around a material contour would be conserved. This result singled out by Einstein
as one of the most significant results of Kelvin's work provided an early link between fluid dynamics and topology.
The history of vortex dynamics seems particularly rich in discoveries and re-discoveries of important results, because results obtained were entirely forgotten after their discovery and then were re-discovered decades later. Thus, the integrability of the problem of three point vortices on the plane was solved in the 1877 thesis of a young Swiss applied mathematician named Walter Gröbli. In spite of having been written in Göttingen
in the general circle of scientists surrounding Helmholtz
and Kirchhoff
, and in spite of having been mentioned in Kirchhoff's well known lectures on theoretical physics
and in other major texts such as Lamb's Hydrodynamics, this solution was largely forgotten. A 1949 paper by the noted applied mathematician J. L. Synge
created a brief revival, but Synge's paper was in turn forgotten. A quarter century later a 1975 paper by E. A. Novikov and a 1979 paper by H. Aref
on chaotic advection finally brought this important earlier work to light. The subsequent elucidation of chaos in the four-vortex problem, and in the advection of a passive particle by three vortices, made Gröbli's work part of "modern science".
Another example of this kind is the so-called "localized induction approximation" (LIA) for three-dimensional vortex filament motion, which gained favor in the mid-1960s through the work of Arms, Hama, Betchov and others, but turns out to date from the early years of the 20th century in the work of Da Rios, a gifted student of the noted Italian mathematician T. Levi-Civita
. Da Rios published his results in several forms but they were never assimilated into the fluid mechanics literature of his time. In 1972 H. Hasimoto used Da Rios' "intrinsic equations" (later re-discovered independently by R. Betchov) to show how the motion of a vortex filament under LIA could be related to the non-linear Schrödinger equation
. This immediately made the problem part of "modern science" since it was then realized that vortex filaments can support solitary twist waves of large amplitude.
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
, the study of how fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s move and the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s on them, dates back to the Ancient Greeks.
Archimedes
The fundamental principles of hydrostatics were given by ArchimedesArchimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
in his work On Floating Bodies, around 250 BC. In it, Archimedes develops the law of buoyancy
Buoyancy
In physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...
, also known as Archimedes' Principle
Archimedes' principle
Archimedes' principle relates buoyancy to displacement. It is named after its discoverer, Archimedes of Syracuse.-Principle:Archimedes' treatise On floating bodies, proposition 5, states that...
. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. 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.
The Alexandrian school
In the Greek school at AlexandriaAlexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
, which flourished under the auspices of the Ptolemies
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
, attempts were made at the construction of hydraulic machinery, and about 120 BC the fountain of compression, the siphon
Siphon
The word siphon is sometimes used to refer to a wide variety of devices that involve the flow of liquids through tubes. But in the English language today, the word siphon usually refers to a tube in an inverted U shape which causes a liquid to flow uphill, above the surface of the reservoir,...
, and the forcing-pump were invented by Ctesibius
Ctesibius
Ctesibius or Ktesibios or Tesibius was a Greek inventor and mathematician in Alexandria, Ptolemaic Egypt. He wrote the first treatises on the science of compressed air and its uses in pumps...
and Hero
Hero of Alexandria
Hero of Alexandria was an ancient Greek mathematician and engineerEnc. Britannica 2007, "Heron of Alexandria" who was active in his native city of Alexandria, Roman Egypt...
. The siphon is a simple instrument; but the forcing-pump 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
Noria
A noria is a machine for lifting water into a small aqueduct, either for the purpose of irrigation or, in at least one known instance, to feed seawater into a saltern....
, 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 forcing-pump.
Sextus Julius Frontinus
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 FrontinusSextus Julius Frontinus
Sextus Julius Frontinus was one of the most distinguished Roman aristocrats of the late 1st century AD, but is best known to the post-Classical world as an author of technical treatises, especially one dealing with the aqueducts of Rome....
, inspector of the public fountains at Rome
Rome
Rome is the capital of Italy and the country's largest and most populated city and comune, with over 2.7 million residents in . The city is located in the central-western portion of the Italian Peninsula, on the Tiber River within the Lazio region of Italy.Rome's history spans two and a half...
in the reigns of Nerva
Nerva
Nerva , was Roman Emperor from 96 to 98. Nerva became Emperor at the age of sixty-five, after a lifetime of imperial service under Nero and the rulers of the Flavian dynasty. Under Nero, he was a member of the imperial entourage and played a vital part in exposing the Pisonian conspiracy of 65...
and Trajan
Trajan
Trajan , was Roman Emperor from 98 to 117 AD. Born into a non-patrician family in the province of Hispania Baetica, in Spain Trajan rose to prominence during the reign of emperor Domitian. Serving as a legatus legionis in Hispania Tarraconensis, in Spain, in 89 Trajan supported the emperor against...
. 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 (tubes), and the mode of distributing the waters of an aqueduct
Aqueduct
An aqueduct is a water supply or navigable channel constructed to convey water. In modern engineering, the term is used for any system of pipes, ditches, canals, tunnels, and other structures used for this purpose....
or a fountain
Fountain
A fountain is a piece of architecture which pours water into a basin or jets it into the air either to supply drinking water or for decorative or dramatic effect....
. 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.
Islamicate physicists
Islamicate scientists, particularly Abu Rayhan Biruni (973–1048) and later Al-KhaziniAl-Khazini
Abu al-Fath Abd al-Rahman Mansour al-Khāzini or simply Abu al-Fath Khāzini was a Muslim astronomer of Greek ethnicity from Merv, then in the Khorasan province of Persia .-References:...
(fl. 11151–130), were the first to apply experiment
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...
al scientific method
Scientific method
Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...
s to fluid mechanics, especially in the field of fluid statics
Fluid statics
Fluid statics is the science of fluids at rest, and is a sub-field within fluid mechanics. The term usually refers to the mathematical treatment of the subject. It embraces the study of the conditions under which fluids are at rest in stable equilibrium...
, such as for determining specific weight
Specific weight
The specific weight is the weight per unit volume of a material. The symbol of specific weight is γ ....
s. They applied the mathematical theories of ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s and infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
techniques, and introduced algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
ic and fine calculation
Calculation
A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition...
techniques into the field of fluid statics.
In fluid statics, Biruni discovered that there is a correlation between the specific gravity of an object and the volume of water it displaces. He also introduced the method of checking tests
Test method
A test method is a definitive procedure that produces a test result.A test can be considered as technical operation that consists of determination of one or more characteristics of a given product, process or service according to a specified procedure. Often a test is part of an experiment.The test...
during experiments and measured the weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...
s of various liquids. He also recorded the differences in weight between freshwater
Freshwater
Fresh water is naturally occurring water on the Earth's surface in ice sheets, ice caps, glaciers, bogs, ponds, lakes, rivers and streams, and underground as groundwater in aquifers and underground streams. Fresh water is generally characterized by having low concentrations of dissolved salts and...
and saline water
Saline water
Saline water is a general term for water that contains a significant concentration of dissolved salts . The concentration is usually expressed in parts per million of salt....
, and between hot water and cold water. During his experiments on fluid mechanics, Biruni invented the conical measure
Conical measure
A conical measure is a type of laboratory glassware which consists of a conical cup with a notch on the top to allow for the easy pouring of liquids, and graduated markings on the side to allow easy and accurate measurement of volumes of liquid....
, in order to find the ratio between the weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...
of a substance in air and the weight of water displaced.
Al-Khazini, in The Book of the Balance of Wisdom (1121), invented a hydrostatic balance
Hydrostatic equilibrium
Hydrostatic equilibrium or hydrostatic balance is the condition in fluid mechanics where a volume of a fluid is at rest or at constant velocity. This occurs when compression due to gravity is balanced by a pressure gradient force...
.
Islamicate engineers
In the 9th century, Banū MūsāBanu Musa
The Banū Mūsā brothers , namely Abū Jaʿfar Muḥammad ibn Mūsā ibn Shākir , Abū al‐Qāsim Aḥmad ibn Mūsā ibn Shākir and Al-Ḥasan ibn Mūsā ibn Shākir , were three 9th-century Persian scholars of Baghdad who are known for their Book of Ingenious Devices on automata and mechanical devices...
brothers' Book of Ingenious Devices
Book of Ingenious Devices
The Book of Ingenious Devices was a large illustrated work on mechanical devices, including automata, published in 850 by the three Persian brothers known as the Banu Musa working at the House of Wisdom in Baghdad, Iraq, under the Abbasid Caliphate...
described a number of early automatic control
Automatic control
Automatic control is the application of concepts derived from the research area of modern control theory. Automatic control is also a technology for application of control strategies. The implementing requires prior of analyzing and modeling of the subject to be controlled...
s in fluid mechanics. Two-step level controls for fluids, an early form of discontinuous variable structure control
Variable structure control
Variable structure control, or VSC, is a form of discontinuous nonlinear control. The method alters the dynamics of a nonlinear system by application of a high-frequency switching control. The state-feedback control law is not a continuous function of time; it switches from one smooth condition to...
s, was developed by the Banu Musa brothers. They also described an early feedback controller
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
for fluids. According to Donald Routledge Hill
Donald Routledge Hill
Donald Routledge Hill was an English engineer and historian of science and technology.Alongside more general works on the history of technology, he wrote works on the history of medieval Arabic science and technology, and translated The Book of Knowledge of Ingenious Mechanical Devices of the...
, the Banu Musa brothers were "masters in the exploitation of small variations" in hydrostatic
Fluid statics
Fluid statics is the science of fluids at rest, and is a sub-field within fluid mechanics. The term usually refers to the mathematical treatment of the subject. It embraces the study of the conditions under which fluids are at rest in stable equilibrium...
pressures and in using conical valve
Valve
A valve is a device that regulates, directs or controls the flow of a fluid by opening, closing, or partially obstructing various passageways. Valves are technically pipe fittings, but are usually discussed as a separate category...
s as "in-line" components in flow systems, "the first known use of conical valves as automatic controllers." They also described the use of other valves, including a plug valve
Plug valve
Plug valves are valves with cylindrical or conically-tapered "plugs" which can be rotated inside the valve body to control flow through the valve. The plugs in plug valves have one or more hollow passageways going sideways through the plug, so that fluid can flow through the plug when the valve is...
, float valve and tap
Tap (valve)
A tap is a valve controlling release of liquids or gas. In the British Isles and most of the Commonwealth, the word is used for any everyday type of valve, particularly the fittings that control water supply to bathtubs and sinks. In the U.S., the term "tap" is more often used for beer taps,...
. The Banu Musa also developed an early fail-safe
Fail-safe
A fail-safe or fail-secure device is one that, in the event of failure, responds in a way that will cause no harm, or at least a minimum of harm, to other devices or danger to personnel....
system where "one can withdraw small quantities of liquid repeatedly, but if one withdraws a large quantity, no further extractions are possible." The double-concentric siphon
Siphon
The word siphon is sometimes used to refer to a wide variety of devices that involve the flow of liquids through tubes. But in the English language today, the word siphon usually refers to a tube in an inverted U shape which causes a liquid to flow uphill, above the surface of the reservoir,...
and the funnel
Funnel
A funnel is a pipe with a wide, often conical mouth and a narrow stem. It is used to channel liquid or fine-grained substances into containers with a small opening. Without a funnel, spillage would occur....
with bent end for pouring in different liquids, neither of which appear in any earlier Greek works, were also original inventions by the Banu Musa brothers. Some of the other mechanisms they described include a float chamber
Float chamber
A float chamber is a device for automatically regulating the supply of a liquid to a system. It is most typically found in the carburettor of a normally aspirated internal combustion engine, where it automatically meters the fuel supply to the engine...
and an early differential pressure
Pressure sensor
A pressure sensor measures pressure, typically of gases or liquids. Pressure is an expression of the force required to stop a fluid from expanding, and is usually stated in terms of force per unit area. A pressure sensor usually acts as a transducer; it generates a signal as a function of the...
.
In 1206, Al-Jazari
Al-Jazari
Abū al-'Iz Ibn Ismā'īl ibn al-Razāz al-Jazarī was a Muslim polymath: a scholar, inventor, mechanical engineer, craftsman, artist, mathematician and astronomer from Al-Jazira, Mesopotamia, who lived during the Islamic Golden Age...
's Book of Knowledge of Ingenious Mechanical Devices described many hydraulic machines. Of particular importance were his water-raising pump
Pump
A pump is a device used to move fluids, such as liquids, gases or slurries.A pump displaces a volume by physical or mechanical action. Pumps fall into three major groups: direct lift, displacement, and gravity pumps...
s. The first known use of a crankshaft
Crankshaft
The crankshaft, sometimes casually abbreviated to crank, is the part of an engine which translates reciprocating linear piston motion into rotation...
in a chain pump was in one of al-Jazari's saqiya
Sakia
A sakia , tympanum or tablia is a water wheel, somewhat similar to a noria, and used primarily in Egypt. It is a large hollow wheel, normally made of galvanized sheet steel, with scoops or buckets at the periphery...
machines. The concept of minimizing intermittent working
Intermittency
In dynamical systems, intermittency is the irregular alternation of phases of apparently periodic and chaotic dynamics , or different forms of chaotic dynamics ....
is also first implied in one of al-Jazari's saqiya chain pumps, which was for the purpose of maximising the efficiency of the saqiya chain pump. Al-Jazari also invented a twin-cylinder
Straight-two
A straight-two engine, is a two-cylinder piston engine that has its cylinders arranged side by side....
reciprocating
Reciprocating engine
A reciprocating engine, also often known as a piston engine, is a heat engine that uses one or more reciprocating pistons to convert pressure into a rotating motion. This article describes the common features of all types...
piston
Piston
A piston is a component of reciprocating engines, reciprocating pumps, gas compressors and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-tight by piston rings. In an engine, its purpose is to transfer force from...
suction pump, which included the first suction
Suction
Suction is the flow of a fluid into a partial vacuum, or region of low pressure. The pressure gradient between this region and the ambient pressure will propel matter toward the low pressure area. Suction is popularly thought of as an attractive effect, which is incorrect since vacuums do not...
pipes, suction pumping, double-action pumping, and made early uses of valves and a crankshaft
Crankshaft
The crankshaft, sometimes casually abbreviated to crank, is the part of an engine which translates reciprocating linear piston motion into rotation...
-connecting rod
Connecting rod
In a reciprocating piston engine, the connecting rod or conrod connects the piston to the crank or crankshaft. Together with the crank, they form a simple mechanism that converts linear motion into rotating motion....
mechanism. This pump is remarkable for three reasons: the first known use of a true suction pipe (which sucks fluids into a partial vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
) in a pump, the first application of the double-acting principle, and the conversion of rotary
Rotary engine
The rotary engine was an early type of internal-combustion engine, usually designed with an odd number of cylinders per row in a radial configuration, in which the crankshaft remained stationary and the entire cylinder block rotated around it...
to reciprocating motion
Reciprocating motion
Reciprocating motion, also called reciprocation, is a repetitive up-and-down or back-and-forth motion. It is found in a wide range of mechanisms, including reciprocating engines and pumps. The two opposite motions that comprise a single reciprocation cycle are called strokes...
, via the crankshaft-connecting rod mechanism.
In 1551, Taqi al-Din invented a six-cylinder pump incorporating lead weights and pistons.
Castelli and Torricelli
Benedetto CastelliBenedetto Castelli
Benedetto Castelli , born Antonio Castelli, was an Italian mathematician. He took the name "Benedetto" upon entering the Benedictine Order in 1595....
, and Evangelista Torricelli
Evangelista Torricelli
Evangelista Torricelli was an Italian physicist and mathematician, best known for his invention of the barometer.-Biography:Evangelista Torricelli was born in Faenza, part of the Papal States...
, 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 canal
Canal
Canals are man-made channels for water. There are two types of canal:#Waterways: navigable transportation canals used for carrying ships and boats shipping goods and conveying people, further subdivided into two kinds:...
s; 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
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
of the head
Hydraulic head
Hydraulic head or piezometric head is a specific measurement of water pressure above a geodetic datum. It is usually measured as a water surface elevation, expressed in units of length, at the entrance of a piezometer...
, apart from the resistance of the air and the friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of 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).
Blaise Pascal
In the hands of Blaise PascalBlaise Pascal
Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...
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.
Mariotte and Guglielmini
The theorem of Torricelli was employed by many succeeding writers, but particularly by Edme MariotteEdme Mariotte
Edme Mariotte was a French physicist and priest.- Biography :Edme Mariotte was the youngest son of Simon Mariotte, administrator at the district Til-Châtel , and Catherine Denisot . His parents lived in Til-Châtel and had 4 other children: Jean, Denise, Claude, and Catharine...
(1620–1684), whose Traité du mouvement des eaux, published after his death in the year 1686, is founded on a great variety of well-conducted experiments on the motion of fluids, performed at Versailles
Versailles
Versailles , a city renowned for its château, the Palace of Versailles, was the de facto capital of the kingdom of France for over a century, from 1682 to 1789. It is now a wealthy suburb of Paris and remains an important administrative and judicial centre...
and Chantilly
Chantilly, Oise
Chantilly is a small city in northern France. It is designated municipally as a commune in the department of Oise.It is in the metropolitan area of Paris 38.4 km...
. 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
Bologna
Bologna is the capital city of Emilia-Romagna, in the Po Valley of Northern Italy. The city lies between the Po River and the Apennine Mountains, more specifically, between the Reno River and the Savena River. Bologna is a lively and cosmopolitan Italian college city, with spectacular history,...
, had ascribed this diminution of velocity in rivers to transverse motions arising front inequalities in their bottom. But as Mariotte observed similar obstructions even in glass pipes where no transverse currents could exist, the cause assigned 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.
Friction and viscosity
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, ItalianItaly
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
-born French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
engineer Henri Pitot
Henri Pitot
Henri Pitot was a French hydraulic engineer and the inventor of the Pitot tube.He became interested in studying the flow of water at various depths and was responsible for disproving the prevailing belief that speed of water increases with depth.In a Pitot tube the height of the fluid column is...
afterwards showed that the retardations arising from friction are inversely as the diameters of the pipes in which the fluid moves.
Orifices
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 hyperbolaHyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
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
Vena contracta
Vena contracta is the point in a fluid stream where the diameter of the stream is the least, such as in the case of a stream issuing out of a nozzle,...
) 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.
Waves
Newton was also the first to investigate the difficult subject of the motion of wavesWAVES
The WAVES were a World War II-era division of the U.S. Navy that consisted entirely of women. The name of this group is an acronym for "Women Accepted for Volunteer Emergency Service" ; the word "emergency" implied that the acceptance of women was due to the unusual circumstances of the war and...
.
Daniel Bernoulli
In 1738 Daniel BernoulliDaniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
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 depending-solely on the fundamental laws of mechanics. Colin Maclaurin
Colin Maclaurin
Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
and John Bernoulli, 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
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
, defective in clearness and precision.
Jean le Rond d'Alembert
The theory of Daniel Bernoulli was opposed also by Jean le Rond d'AlembertJean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...
. When generalizing the theory of pendulum
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
s of Jacob Bernoulli 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
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
. 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 Traité 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 equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
s 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
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
.
Leonhard Euler
The resolution of the questions concerning the motion of fluids was effected by means of Leonhard EulerLeonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
's partial differential coefficients
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
. 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.
Pierre Louis Georges Dubuat
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 Abbé Charles BossutCharles Bossut
Charles Bossut was a French mathematician and confrère of the Encyclopaedists. He was born at Tartaras, Loire, and died in Paris.His works include...
(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 the 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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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 1780 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.
Hermann von Helmholtz
In 1858 Hermann von HelmholtzHermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
published his seminal paper "Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen," in Journal für die reine und angewandte Mathematik, vol. 55, pp. 25–55. So important was the paper that a few years later P. G. Tait published an English translation, "On integrals of the hydrodynamical equations which express vortex motion", in Philosophical Magazine
Philosophical Magazine
The Philosophical Magazine is one of the oldest scientific journals published in English. Initiated by Alexander Tilloch in 1798, in 1822 Richard Taylor became joint editor and it has been published continuously by Taylor & Francis ever since; it was the journal of choice for such luminaries as...
, vol. 33, pp. 485–512 (1867). In his paper Helmholtz established his three "laws of vortex motion" in much the same way one finds them in any advanced textbook of fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
today. This work established the significance of vorticity to fluid mechanics and science in general.
For the next century or so vortex dynamics matured as a subfield of fluid mechanics, always commanding at least a major chapter in treatises on the subject. Thus, H. Lamb's
Horace Lamb
Sir Horace Lamb FRS was a British applied mathematician and author of several influential texts on classical physics, among them Hydrodynamics and Dynamical Theory of Sound...
well known Hydrodynamics (6th ed., 1932) devotes a full chapter to vorticity and vortex dynamics as does G. K. Batchelor's
George Batchelor
George Keith Batchelor FRS was an Australian applied mathematician and fluid dynamicist. He was for many years the Professor of Applied Mathematics in the University of Cambridge, and was founding head of the Department of Applied Mathematics and Theoretical Physics...
Introduction to Fluid Dynamics (1967). In due course entire treatises were devoted to vortex motion. H. Poincaré's
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
Théorie des Tourbillons (1893), H. Villat's Leçons sur la Théorie des Tourbillons (1930), C. Truesdell's
Clifford Truesdell
Clifford Ambrose Truesdell III was an American mathematician, natural philosopher, historian of science, and polemicist.-Life:...
The Kinematics of Vorticity (1954), and P. G. Saffman's Vortex Dynamics (1992) may be mentioned. Early on individual sessions at scientific conferences were devoted to vortices, vortex motion, vortex dynamics and vortex flows. Later, entire meetings were devoted to the subject.
The range of applicability of Helmholtz's work grew to encompass atmospheric and oceanographic flows, to all branches of engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
and applied science
Applied science
Applied science is the application of scientific knowledge transferred into a physical environment. Examples include testing a theoretical model through the use of formal science or solving a practical problem through the use of natural science....
and, ultimately, to superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
s (today including Bose–Einstein condensate
Bose–Einstein condensate
A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...
s). In modern fluid mechanics the role of vortex dynamics in explaining flow phenomena is firmly established. Well known vortices have acquired names and are regularly depicted in the popular media: hurricanes, tornado
Tornado
A tornado is a violent, dangerous, rotating column of air that is in contact with both the surface of the earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. They are often referred to as a twister or a cyclone, although the word cyclone is used in meteorology in a wider...
es, waterspout
Waterspout
A waterspout is an intense columnar vortex that occurs over a body of water and is connected to a cumuliform cloud. In the common form, it is a non-supercell tornado over water. While it is often weaker than most of its land counterparts, stronger versions spawned by mesocyclones do occur...
s, aircraft trailing vortices (e.g., wingtip vortices
Wingtip vortices
Wingtip vortices are tubes of circulating air that are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure...
), drainhole vortices (including the bathtub vortex), smoke ring
Smoke ring
A smoke ring is a visible vortex ring formed by sudden release of smoke. It can be created by blowing smoke from the mouth, quickly lighting a cigarette lighter and putting it out or holding a burning incense stick or a cigarette vertically, pushing it with the burning side up and suddenly pulling...
s, underwater bubble air rings, cavitation vortices behind ship propellers, and so on. In the technical literature a number of vortices that arise under special conditions also have names: the Kármán vortex street
Von Kármán vortex street
A Kármán vortex street is a term in fluid dynamics for a repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid over bluff bodies...
wake behind a bluff body, Taylor vortices between rotating cylinders, Görtler vortices
Görtler vortices
In fluid dynamics, Görtler vortices are secondary flows that appears in a boundary layer flow along a concave wall. If the boundary layer is thin compared to the radius of curvature of the wall, the pressure remains constant across the boundary layer...
in flow along a curved wall, etc.
Gaspard Riche de Prony
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 eighty-two (fifty-one on the velocity of water in conduit pipes, and thirty-one 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.Johann Albert Eytelwein
JA Eytelwein of BerlinBerlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
, who published in 1801 a valuable compendium of hydraulics entitled Handbuch der Mechanik und 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 water wheel
Water wheel
A water wheel is a machine for converting the energy of free-flowing or falling water into useful forms of power. A water wheel consists of a large wooden or metal wheel, with a number of blades or buckets arranged on the outside rim forming the driving surface...
will have its maximum effect when its circumference moves with half the velocity of the stream.
Jean Nicolas Pierre Hachette and others
JNP HachetteJean Nicolas Pierre Hachette
Jean Nicolas Pierre Hachette , French mathematician, was born at Mézières, where his father was a bookseller.For his early education he proceeded first to the college of Charleville, and afterwards to that of Reims...
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 JV Poncelet (1788–1867) and JA Lesbros (1790–1860).
PP Boileau (1811–1891) discussed their results and added experiments of his own (Traité de la mesure des eaux courantes, Paris, 1854). KR Bornemann re-examined all these results with great care, and gave formulae expressing the variation of the coefficients of discharge in different conditions (Civil Ingenieur, 1880). Julius Weisbach
Julius Weisbach
Julius Ludwig Weisbach was a German mathematician and engineer.-Life and work:...
(1806–1871) also made many experimental investigations on the discharge of fluids.
The experiments of JB 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 HGP Darcy (1803–1858) and continued by H Bazin, at the expense of the French government (Recherches hydrauliques, Paris, 1866).
Andreas Rudolf Harlacher and others
German engineers have also devoted special attention to the measurement of the flow in rivers; the Beiträge zur Hydrographie des Königreiches Bohmen (Prague, 1872–1875) of AR Harlacher contained valuable measurements of this kind, together with a comparison 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 MississippiMississippi
Mississippi is a U.S. state located in the Southern United States. Jackson is the state capital and largest city. The name of the state derives from the Mississippi River, which flows along its western boundary, whose name comes from the Ojibwe word misi-ziibi...
made for the United States government by AA Humphreys and HL Abbot
Henry Larcom Abbot
Henry Larcom Abbot was a military engineer and officer in the United States Army.-Early life:Henry Larcom Abbot was born in Beverly, Massachusetts. Abbot attended West Point and graduated second in his class with a degree in military engineering in 1854...
, by Robert Gordon
Robert Gordon
Robert Gordon , a 17th century merchant and philanthropist, was born in Aberdeen. He was the only son of Arthur Gordon who married Isabella Menzies of Balgownie...
's gaugings of the Ayeyarwady River
Ayeyarwady River
The Irrawaddy River or Ayeyarwady River is a river that flows from north to south through Burma . It is the country's largest river and most important commercial waterway. Originating from the confluence of the N'mai and Mali rivers, it flows relatively straight North-South before emptying through...
, and by Allen JC Cunningham's experiments on the Ganges canal. The friction of water, investigated for slow speeds by Coulomb
Charles-Augustin de Coulomb
Charles-Augustin de Coulomb was a French physicist. He is best known for developing Coulomb's law, the definition of the electrostatic force of attraction and repulsion. The [SI unit] of charge, the coulomb, was named after him....
, was measured for higher speeds by William Froude
William Froude
William Froude was an English engineer, hydrodynamicist and naval architect. He was the first to formulate reliable laws for the resistance that water offers to ships and for predicting their stability....
(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
Osborne Reynolds
Osborne Reynolds FRS was a prominent innovator in the understanding of fluid dynamics. Separately, his studies of heat transfer between solids and fluids brought improvements in boiler and condenser design.-Life:...
and by Professor HS Hele-Shaw
Henry Selby Hele-Shaw
Henry Selby Hele-Shaw FRS was an English mechanical and automobile engineer. He was the inventor of the variable-pitch propeller, which contributed to British success in the Battle of Britain in 1940, and he experimented with flows through thin cells. Flows through such configurations are named in...
.
Developments in vortex dynamics
Vortex dynamics is a vibrant subfield of fluid dynamics, commanding attention at major scientific conferences and precipitating workshops and symposia that focus fully on the subject.A curious diversion in the history of vortex dynamics was the vortex atom theory
Aether theories
Aether theories in early modern physics proposed the existence of a medium, the aether , a space-filling substance or field, thought to be necessary as a transmission medium for the propagation of electromagnetic waves...
of William Thomson
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...
, later Lord Kelvin. His basic idea was that atoms were to be represented as vortex motions in the ether. This theory predated the quantum theory
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...
by several decades and because of the scientific standing of its originator received considerable attention. Many profound insights into vortex dynamics were generated during the pursuit of this theory. Other interesting corollaries were the first counting of simple knots by P. G. Tait, today considered a pioneering effort in graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
. Ultimately, Kelvin's vortex atom was seen to be wrong-headed but the many results in vortex dynamics that it precipitated have stood the test of time. Kelvin himself originated the notion of circulation and proved that in an inviscid fluid
Euler equations
In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass , momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically,...
circulation around a material contour would be conserved. This result singled out by Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
as one of the most significant results of Kelvin's work provided an early link between fluid dynamics and topology.
The history of vortex dynamics seems particularly rich in discoveries and re-discoveries of important results, because results obtained were entirely forgotten after their discovery and then were re-discovered decades later. Thus, the integrability of the problem of three point vortices on the plane was solved in the 1877 thesis of a young Swiss applied mathematician named Walter Gröbli. In spite of having been written in Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
in the general circle of scientists surrounding Helmholtz
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
and Kirchhoff
Gustav Kirchhoff
Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...
, and in spite of having been mentioned in Kirchhoff's well known lectures on theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
and in other major texts such as Lamb's Hydrodynamics, this solution was largely forgotten. A 1949 paper by the noted applied mathematician J. L. Synge
John Lighton Synge
John Lighton Synge was an Irish mathematician and physicist.-Background:Synge was born 1897 in Dublin, Ireland, in a Protestant family and educated at St. Andrew's College, Dublin. He entered Trinity College, Dublin in 1915...
created a brief revival, but Synge's paper was in turn forgotten. A quarter century later a 1975 paper by E. A. Novikov and a 1979 paper by H. Aref
Hassan Aref
Dr. Hassan Aref was the Reynolds Metals Professor in the Department of at Virginia Tech, and the Niels Bohr Visiting Professor at the Technical University of Denmark. Prior to joining Virginia Tech as Dean of Engineering in 2003-2005 Aref was Head of the Department of Theoretical and Applied...
on chaotic advection finally brought this important earlier work to light. The subsequent elucidation of chaos in the four-vortex problem, and in the advection of a passive particle by three vortices, made Gröbli's work part of "modern science".
Another example of this kind is the so-called "localized induction approximation" (LIA) for three-dimensional vortex filament motion, which gained favor in the mid-1960s through the work of Arms, Hama, Betchov and others, but turns out to date from the early years of the 20th century in the work of Da Rios, a gifted student of the noted Italian mathematician T. Levi-Civita
Tullio Levi-Civita
Tullio Levi-Civita, FRS was an Italian mathematician, most famous for his work on absolute differential calculus and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus...
. Da Rios published his results in several forms but they were never assimilated into the fluid mechanics literature of his time. In 1972 H. Hasimoto used Da Rios' "intrinsic equations" (later re-discovered independently by R. Betchov) to show how the motion of a vortex filament under LIA could be related to the non-linear Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. This immediately made the problem part of "modern science" since it was then realized that vortex filaments can support solitary twist waves of large amplitude.