History of fluid mechanics

The history of fluid mechanics is a fundamental strand of the history of physics and engineering. The study of the movement of fluids (liquids and gases) and the forces that act upon them dates back to pre-history. The field has undergone a continuous evolution, driven by human dependence on water, meteorological conditions, and internal biological processes.

The success of early civilizations, can be attributed to developments in the understanding of water dynamics, allowing for the construction of canals and aqueducts for water distribution and farm irrigation, as well as maritime transport. Due to its conceptual complexity, most discoveries in this field relied almost entirely on experiments, at least until the development of advanced understanding of differential equations and computational methods. Significant theoretical contributions were made by notables figures like Archimedes, Johann Bernoulli and his son Daniel Bernoulli, Leonhard Euler, Claude-Louis Navier and George Stokes, who developed the fundamental equations to describe fluid mechanics. Advancements in experimentation and computational methods have further propelled the field, leading to practical applications in more specialized industries ranging from aerospace to environmental engineering. Fluid mechanics has also been important for the study of astronomical bodies and the dynamics of galaxies.

Antiquity

Pre-history

A pragmatic, if not scientific, knowledge of fluid flow was exhibited by ancient civilizations, such as in the design of arrows, spears, boats, and particularly hydraulic engineering projects for flood protection, irrigation, drainage, and water supply. The earliest human civilizations began near the shores of rivers, and consequently coincided with the dawn of hydrology, hydraulics, and hydraulic engineering.

Buoyancy

Observations of specific gravity and buoyancy were recorded by ancient Chinese philosophers. In the 4th century BCE Mencius describes the weight of the gold is equivalent to the feathers. In 3rd century CE, Cao Chong describes the story of weighing the elephant by observing displacement of the boats loaded with different weights.

thumb|The forces at work in [[buoyancy as discovered by Archimedes. Note that the object is floating because the upward force of buoyancy is equal to the downward force of gravity.]]

The fundamental principles of hydrostatics and dynamics were given by Archimedes 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.

Greco-Roman engineering

In the Greek school at Alexandria, 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. Al-Khazini, in The Book of the Balance of Wisdom (1121), invented a hydrostatic balance.

During his experiments on fluid mechanics, Al-Biruni invented the conical measure.

Islamicate engineers

In the 9th century, Banū Mūsā brothers' Book of Ingenious Devices described a number of early automatic controls in fluid mechanics. Two-step level controls for fluids, an early form of discontinuous variable structure controls, 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 valves as "in-line" components in flow systems, "the first known use of conical valves as automatic controllers". float valve 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".

In 1206, Al-Jazari's Book of Knowledge of Ingenious Mechanical Devices described many hydraulic machines. Of particular importance were his water-raising pumps. 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.

Sixteenth century

During the Renaissance, Leonardo da Vinci was well known for his experimental skills. His notes provide precise depictions of various phenomena, including vessels, jets, hydraulic jumps, eddy formation, tides, as well as designs for both low drag (streamlined) and high drag (parachute) configurations. Da Vinci is also credited for formulating the conservation of mass in one-dimensional steady flow.

In 1586, the Flemish engineer and mathematician Simon Stevin published De Beghinselen des Waterwichts (Principles on the Weight of Water), a study of hydrostatics that, among other things, extensively discussed the hydrostatic paradox.

Seventeenth century

thumb|[[Torricelli's law|Torricelli law established the speed and quantity of flow that exits a barrel..]]

Torricelli's law

Benedetto Castelli, and Evangelista Torricelli, two of the disciples of Galileo Galilei, 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. Torricelli's law 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).

The Euler equations were among the first partial differential equations to be written down, after the wave equation. In Euler's original work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible flow.

Early works on fluid resistance

One of the most successful labourers in the science of hydrodynamics at this period was Pierre-Louis-Georges du Buat. Following in the steps of the Abbé Charles Bossut (Nouvelles Experiences sur la résistance 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. He then began work on generalizing these equations to hydrodynamics.

The first derivations of the Navier-Stokes equation appeared in two memoirs by Navier as a result of this work, Sur les lois des mouvements des fluides, en ayant égard à l'adhésion des molecules in 1822, and Sur Les Lois du Mouvement des Fluides in 1823. Previous to these publications, most equations of fluid flow had been formulated in terms of perfect, frictionless fluids. However, in his memoir, Navier introduced friction into the equations of motion of fluids. As a result, Navier is credited with developing the equation of motion for viscous flows.

Navier's original proof was not widely influential, and was re-derived again by Augustin-Louis Cauchy in 1823, by Siméon Denis Poisson in 1829, by Adhémar Barré de Saint-Venant in 1837, and by George Stokes in 1845, with methodological differences, such as which molecular assumptions to use.

Stokes' 1845 proof, published after Navier's death, was particularly influential because he extended Navier's equations to address different behaviors of fluids under boundary conditions. He explicitly addressed the behavior of incompressible fluids, which are fluids that do not change volume under pressure. He also addressed creeping flow (also called 'Stokes flow'), where very slow velocities allow for the equations to simplify and result in analytical solutions.

The final result of these physicists' work was the Navier–Stokes equations, a set of partial differential equations that describe how the velocity, pressure, temperature, and density of a moving fluid are related.

In honor of Stokes' many contributions to fluid mechanics, the unit for kinematic viscosity is called a stoke in his honor.

Reynolds number and turbulence

The concept of a number to quantify turbulence was introduced by Stokes in 1851, and its use was popularized by Osborne Reynolds in 1883. Arnold Sommerfeld later christened this number the Reynolds number.

Vortex dynamics

thumb|[[Smoke ring demonstration: a vortex ring of smoke is created using a drum. First described by Hermann von Helmholtz.]]

In 1858 Hermann von Helmholtz 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, Horace 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. Henri Poincaré's Théorie des Tourbillons (1893), Henri Villat's Leçons sur la Théorie des Tourbillons (1930), Clifford Truesdell's The Kinematics of Vorticity (1954), and Philip 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.

19th century hydraulics

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. The friction of water, investigated for slow speeds by Charles-Augustin de 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 Reynolds and by Henry Selby Hele-Shaw.

Developments in vorticity and turbulence

From 1894-1910, vortex dynamics achieved more attention and development as a result of the concurrently developing field of aerodynamics. The Kutta-Joucowski theorem, for example, developed from 1902-1906, is a fundamental result in aerodynamics that was proved by considering the fluid flow in the presence of the airfoil as the superposition of a translational flow and a rotating flow. The rotating flow is solved using vortex dynamics by considering it as being induced by a line vortex. Much of the wing-flow problem in aerodynamics was solved in a similar fashion, by taking advantage of vortex dynamics.

As high-speed flight necessitated further developments, the problem of turbulence began to gain increasing relevance in the mid-twentieth century. Turbulent flow describes unpredictable or chaotic changes in pressure and flow velocity. In turbulent flow, unsteady vortices interact with each other and increase drag and friction as a result.

Vorticity and vortex lines have been used to understand how turbulent fluids behave. In the twentieth century, the discovery of turbulent coherent structures allowed for the development of theories of vortex dynamics in turbulence. These turbulent coherent structures are regions of concentrated vorticity that are organized and persist for a long time. 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, tornadoes, waterspouts, aircraft trailing vortices (e.g., wingtip vortices), drainhole vortices (including the bathtub vortex), smoke rings, 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.

The behavior of turbulent flow remains an unsolved problem in fluid mechanics. Methods to address turbulence include direct numerical simulation and large eddy simulation.

Microfluidics

Beginning in the 1950s, advances in semiconductor device fabrication enabled the creation of engraving micro-patterns, sparking interest in miniaturizing sensors, and integrating them with microcomputers to develop portable platforms for scientific research. Stephen C. Terry from Standford University miniaturized gas chromatograph on a silicon wafer in the 1979, considered one of the first lab-on-a-chip. Building on this, Swiss chemist introduced in 1990 the concept of the miniaturized total analysis system (μTAS), capable of performing complete analyses on a microfluidic chip. By the end of the decade, microfluidics was firmly established with applications in capillary electrophoresis, genetic testing, droplet-based microfluidics, polymerase chain reaction (PCR) tests and other technologies of the 21st century. One of which is the fluid mechanics problem of Navier–Stokes existence and smoothness, which deals with the mathematical properties of the Navier-Stokes equations. The statement of the problem, as proposed by Charles Fefferman, goes as follows: <blockquote>Prove or give a counter-example of the following statement:

In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.</blockquote>The prize sparked interest in the problem. In 2016, mathematician Terrence Tao published "Finite time blowup for an averaged three-dimensional Navier–Stokes equation", where he invents an "averaged" version of the equations and shows that it exhibits breakdown. He suggests a program for constructing a blowup solution to the true Navier-Stokes equations. In September 2025, Google DeepMind announced it had used machine learning techniques to uncover "the first systematic discovery of new families of unstable singularities" in the equations of fluid flow.