Fluid mechanics is the branch of

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{{DEFAULTSORT:Fluid Mechanics Fluid mechanics, Civil engineering

physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

concerned with the mechanics
Mechanics (Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ...

of fluid
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

s (liquid
A liquid is a nearly incompressible
In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, ...

s, gas
Gas is one of the four fundamental states of matter
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

es, and plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral) or heliotrope, a mineral aggregate
* Quark–gluon plasma, a state of matter in quantum chromodynamics
Biology
* Blood plasma ...

s) and the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

s on them.
It has applications in a wide range of disciplines, including mechanical
Mechanical may refer to:
Machine
* Mechanical system
A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine
A molecular machine, nan ...

, civil
Civil may refer to:
*Civic virtue, or civility
*Civil action, or lawsuit
*Civil affairs
*Civil and political rights
*Civil disobedience
*Civil engineering
*Civil (journalism), a platform for independent journalism
*Civilian, someone not a member ...

, chemical
A chemical substance is a form of matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which ...

and biomedical engineering, geophysics
Geophysics () is a subject of natural science
Natural science is a branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a b ...

, oceanography
Oceanography (from the Ancient Greek
Ancient Greek includes the forms of the Greek language
Greek ( el, label=Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Gr ...

, meteorology
Meteorology is a branch of the (which include and ), with a major focus on . The study of meteorology dates back , though significant progress in meteorology did not begin until the 18th century. The 19th century saw modest progress in the f ...

, astrophysics
Astrophysics is a science that employs the methods and principles of physics in the study of astronomical objects and phenomena. Among the subjects studied are the Sun, other stars, galaxy, galaxies, extrasolar planets, the interstellar medium and ...

, and biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

.
It can be divided into fluid statics
Fluid statics or hydrostatics is the branch of fluid mechanics
Fluid mechanics is the branch of physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that de ...

, the study of fluids at rest; and fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

, the study of the effect of forces on fluid motion.
It is a branch of continuum mechanics
Continuum mechanics is a branch of mechanics
Mechanics (Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a ''macroscopic'' viewpoint rather than from ''microscopic''. Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by numerical methods
Numerical analysis is the study of algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as ...

, typically using computers. A modern discipline, called computational fluid dynamics#REDIRECT Computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid dynamics, fluid flows. Computers are used ...

(CFD), is devoted to this approach. Particle image velocimetryParticle image velocimetry (PIV) is an optical method of flow visualization used in education and research. It is used to obtain instantaneous velocity
The velocity of an object is the rate of change of its position with respect to a fram ...

, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
Brief history

The study of fluid mechanics goes back at least to the days ofancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era wa ...

, when Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

investigated fluid statics and buoyancy
Buoyancy (), or upthrust, is an upward exerted by a that opposes the of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bo ...

and formulated his famous law known now as the Archimedes' principle
Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid
In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external f ...

, which was published in his work ''On Floating Bodies
''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek language, Greek-language work consisting of two books written by Archimedes of Syracuse, Sicily, Syracuse (287 – c. 212 BC), one of the most impo ...

''—generally considered to be the first major work on fluid mechanics. Rapid advancement in fluid mechanics began with Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian of the who was active as a painter, , engineer, scientist, theorist, sculptor and architect. While his fame initially rested on his achievements as a painter, he als ...

(observations and experiments), Evangelista Torricelli
Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian
Italian may refer to:
* Anything of, from, or related to the country and nation of Italy
** Italians, an ethnic group or simply a citizen of the Italian Republic
...

(invented the barometer
A barometer is a scientific instrument that is used to measure air pressure in a certain environment. Pressure tendency can forecast short term changes in the weather. Many measurements of air pressure are used within surface weather analysis to ...

), Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

(investigated viscosity
The viscosity of a fluid
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

) and Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, writer and Catholic Church, Catholic theologian.
He was a child prodigy who was educated by his father, a tax collector i ...

(researched hydrostatics
Fluid statics or hydrostatics is the branch of fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical object ...

, formulated Pascal's law
Pascal's law (also Pascal's principle or the principle of transmission of fluid-pressure) is a principle in fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the are ...

), and was continued by Daniel Bernoulli
Daniel Bernoulli FRS
FRS may also refer to:
Government and politics
* Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States
* Family Resour ...

with the introduction of mathematical fluid dynamics in ''Hydrodynamica'' (1739).
Inviscid flow was further analyzed by various mathematicians (Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanics, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Enc ...

, Joseph Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia

, Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar
A scholar is a person who pursues academic and intellectual activities, particularly those that develop expertise in an area of Studying, study. A ...

, Siméon Denis Poisson
Baron
Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

) and viscous flow was explored by a multitude of engineers
Engineers, as practitioners of engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of enginee ...

including Jean Léonard Marie Poiseuille and Gotthilf Hagen
Gotthilf Heinrich Ludwig Hagen (3 March 1797 – 3 February 1884) was a Germany, German civil engineer who made important contributions to fluid dynamics, hydraulic engineering and probability theory.
Life and work
Hagen was born in Königsberg, E ...

. Further mathematical justification was provided by Claude-Louis Navier
Claude-Louis Navier (born Claude Louis Marie Henri Navier; ; 10 February 1785 – 21 August 1836), was a French mechanical engineer, affiliated with the French government, and a physicist whose work was specialized in continuum mechanics.
The N ...

and George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish English physicist
A physicist is a scientist
A scientist is a person who conducts Scientific method, scientific research to advance knowledge ...

in the Navier–Stokes equations
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

, and boundary layers
In physics and fluid mechanics, a boundary layer is the layer of fluid in the immediate vicinity of a Boundary (thermodynamic), bounding surface where the effects of viscosity are significant. The liquid or gas in the boundary layer tends to clin ...

were investigated (Ludwig Prandtl
Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of Germany, se ...

, Theodore von Kármán
Theodore von Kármán ( hu, (szőlőskislaki) Kármán Tódor ; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fields of aeronautics and astronautics. He was respon ...

), while various scientists such as Osborne Reynolds
Osborne Reynolds FRS (23 August 1842 – 21 February 1912) was an innovator in the understanding of fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids ...

, Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovie ...

, and Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor Order of Merit, OM Royal Society of London, FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time studen ...

advanced the understanding of fluid viscosity and turbulence
In fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other ...

.
Main branches

Fluid statics

Fluid statics
Fluid statics or hydrostatics is the branch of fluid mechanics
Fluid mechanics is the branch of physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that de ...

or hydrostatics is the branch of fluid mechanics that studies fluid
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

s at rest. It embraces the study of the conditions under which fluids are at rest in stable
A stable is a building in which livestock
Livestock are the domesticated
Domestication is a sustained multi-generational relationship in which one group of organisms assumes a significant degree of influence over the reproduction and c ...

equilibrium
List of types of equilibrium, the condition of a system in which all competing influences are balanced, in a wide variety of contexts.
Equilibrium may also refer to:
Film and television
* Equilibrium (film), ''Equilibrium'' (film), a 2002 scien ...

; and is contrasted with fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

, the study of fluids in motion. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure
Atmospheric pressure, also known as barometric pressure (after the barometer
A barometer is a scientific instrument that is used to measure air pressure
Atmospheric pressure, also known as barometric pressure (after the barometer), is the ...

changes with altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum
Data (; ) are individual facts
A fact is something that is truth, true. The usual t ...

, why wood and oil
An oil is any nonpolar
In chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound ...

float on water, and why the surface of water is always level whatever the shape of its container. Hydrostatics is fundamental to hydraulics
Hydraulics (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is a ...

, the engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

of equipment for storing, transporting and using fluids
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

. It is also relevant to some aspects of geophysics
Geophysics () is a subject of natural science
Natural science is a branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a b ...

and astrophysics
Astrophysics is a science that employs the methods and principles of physics in the study of astronomical objects and phenomena. Among the subjects studied are the Sun, other stars, galaxy, galaxies, extrasolar planets, the interstellar medium and ...

(for example, in understanding plate tectonics
Plate tectonics (from the la, label=Late Latin
Late Latin ( la, Latinitas serior) is the scholarly name for the written Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. L ...

and anomalies in the Earth's gravitational field), to meteorology
Meteorology is a branch of the (which include and ), with a major focus on . The study of meteorology dates back , though significant progress in meteorology did not begin until the 18th century. The 19th century saw modest progress in the f ...

, to medicine
Medicine is the science
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (proced ...

(in the context of blood pressure
Blood pressure (BP) is the pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mas ...

), and many other fields.
Fluid dynamics

''Fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

'' is a subdiscipline of fluid mechanics that deals with ''fluid flow''—the science of liquids and gases in motion. Fluid dynamics offers a systematic structure—which underlies these practical disciplines
Applied science is the use of the scientific method
The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at least the 17th century. It involves caref ...

—that embraces empirical and semi-empirical laws derived from flow measurement
Flow measurement is the quantification of bulk fluid
In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter a ...

and used to solve practical problems. The solution to a fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

problem typically involves calculating various properties of the fluid, such as velocity
The velocity of an object is the rate of change of its position with respect to a frame of reference
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

, pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

, density
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

, and temperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal energy refers to several distinct physical concept ...

, as functions of space and time. It has several subdisciplines itself, including ''aerodynamics
Aerodynamics, from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

'' (the study of air and other gases in motion) and ''hydrodynamics'' (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

s and movement
Movement may refer to:
Common uses
* Movement (clockwork), the internal mechanism of a timepiece
* Motion (physics), commonly referred to as movement
Arts, entertainment, and media
Literature
* Movement (short story), "Movement", a shor ...

s on aircraft
An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...

, determining the mass flow rate
In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit of measurement, unit is kilogram per second in SI units, and Slug (unit), slug per second or pound (mass), pound per second in US custo ...

of petroleum
Petroleum, also known as crude oil and oil, is a naturally occurring, yellowish-black liquid
A liquid is a nearly incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isoc ...

through pipelines, predicting evolving weather
Weather is the state of the atmosphere
An atmosphere (from the greek words ἀτμός ''(atmos)'', meaning 'vapour', and σφαῖρα ''(sphaira)'', meaning 'ball' or 'sphere') is a layer or a set of layers of gases surrounding a p ...

patterns, understanding nebula
A nebula (Latin for 'cloud' or 'fog'; pl. nebulae, nebulæ or nebulas) is a distinct body of interstellar clouds (which can consist of cosmic dust, hydrogen, helium, molecular clouds; possibly as Plasma (physics), ionized gases). Originally, th ...

e in interstellar space
Outer space, commonly shortened to space, is the expanse that exists beyond Earth and Earth atmosphere, its atmosphere and between astronomical object, celestial bodies. Outer space is not completely empty—it is a hard vacuum containing a ...

and modeling explosions
An explosion is a rapid expansion in volume
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance ( solid, liquid, gas, or plasma) or shape occupies or contains. Volume ...

. Some fluid-dynamical principles are used in traffic engineering and crowd dynamics.
Relationship to continuum mechanics

Fluid mechanics is a subdiscipline ofcontinuum mechanics
Continuum mechanics is a branch of mechanics
Mechanics (Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

, as illustrated in the following table.
In a mechanical view, a fluid is a substance that does not support shear stress
Shear stress, often denoted by (Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popu ...

; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.
Assumptions

The assumptions inherent to a fluid mechanical treatment of a physical system can be expressed in terms of mathematical equations. Fundamentally, every fluid mechanical system is assumed to obey: *Conservation of mass
In and , the law of conservation of mass or principle of mass conservation states that for any to all transfers of and , the of the system must remain constant over time, as the system's mass cannot change, so quantity can neither be added n ...

* Conservation of energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

* Conservation of momentum
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a direction. If is an object's ma ...

* The continuum assumption
For example, the assumption that mass is conserved means that for any fixed control volume
In continuum mechanics and thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of the ...

(for example, a spherical volume)—enclosed by a control surface—the of the mass contained in that volume is equal to the rate at which mass is passing through the surface from ''outside'' to ''inside'', minus the rate at which mass is passing from ''inside'' to ''outside''. This can be expressed as an equation in integral form over the control volume.
The is an idealization of continuum mechanics
Continuum mechanics is a branch of mechanics
Mechanics (Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

under which fluids can be treated as continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

, even though, on a microscopic scale, they are composed of molecules
A molecule is an electrically
Electricity is the set of physical phenomena associated with the presence and motion
Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position
In physics, motion is the phenomenon ...

. Under the continuum assumption, macroscopic (observed/measurable) properties such as density, pressure, temperature, and bulk velocity are taken to be well-defined at "infinitesimal" volume elements—small in comparison to the characteristic length scale of the system, but large in comparison to molecular length scale. Fluid properties can vary continuously from one volume element to another and are average values of the molecular properties. The continuum hypothesis can lead to inaccurate results in applications like supersonic speed flows, or molecular flows on nano scale. Those problems for which the continuum hypothesis fails can be solved using statistical mechanics
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

. To determine whether or not the continuum hypothesis applies, the Knudsen number
The Knudsen number (Kn) is a dimensionless number
In dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quant ...

, defined as the ratio of the molecular mean free path
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

to the characteristic length scale
Scale or scales may refer to:
Mathematics
* Scale (descriptive set theory)In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set (mathematics), set of point (mathematics), points in some Poli ...

, is evaluated. Problems with Knudsen numbers below 0.1 can be evaluated using the continuum hypothesis, but molecular approach (statistical mechanics) can be applied to find the fluid motion for larger Knudsen numbers.
Navier–Stokes equations

The Navier–Stokes equations (named afterClaude-Louis Navier
Claude-Louis Navier (born Claude Louis Marie Henri Navier; ; 10 February 1785 – 21 August 1836), was a French mechanical engineer, affiliated with the French government, and a physicist whose work was specialized in continuum mechanics.
The N ...

and George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish English physicist
A physicist is a scientist
A scientist is a person who conducts Scientific method, scientific research to advance knowledge ...

) are differential equations
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...

that describe the force balance at a given point within a fluid. For an incompressible fluid
In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among ...

with vector velocity field $\backslash mathbf$, the Navier–Stokes equations are
: $\backslash frac\; +\; (\backslash mathbf\; \backslash cdot\; \backslash nabla)\; \backslash mathbf\; =\; -\; \backslash frac\backslash nabla\; P\; +\; \backslash nu\; \backslash nabla^2\; \backslash mathbf$.
These differential equations are the analogues for deformable materials to Newton's equations of motion for particles – the Navier–Stokes equations describe changes in momentum
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass
Mass is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

(force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

) in response to pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

$P$ and viscosity, parameterized by the kinematic viscosity
The viscosity of a fluid
In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter and include liquids, Gas, ...

$\backslash nu$ here. Occasionally, body force
A body force is a force that acts throughout the volume of a body. Forces due to gravity
Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—including planets, stars, galax ...

s, such as the gravitational force or Lorentz force are added to the equations.
Solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

. In practical terms, only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow in which the Reynolds number
The Reynolds number () helps predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be turbulent
In fluid dynam ...

is small. For more complex cases, especially those involving turbulence
In fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other ...

, such as global weather systems, aerodynamics, hydrodynamics and many more, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This branch of science is called computational fluid dynamics#REDIRECT Computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid dynamics, fluid flows. Computers are used ...

.
Inviscid and viscous fluids

An inviscid fluid has noviscosity
The viscosity of a fluid
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

, $\backslash nu=0$. In practice, an inviscid flow is an idealization, one that facilitates mathematical treatment. In fact, purely inviscid flows are only known to be realized in the case of superfluidity
File:Liquid helium Rollin film.jpg, The liquid helium is in the superfluid phase. A thin invisible film creeps up the inside wall of the bowl and down on the outside. A drop forms. It will fall off into the liquid helium below. This will repeat unt ...

. Otherwise, fluids are generally viscous, a property that is often most important within a boundary layer
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

near a solid surface, where the flow must match onto the no-slip conditionIn fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in ...

at the solid. In some cases, the mathematics of a fluid mechanical system can be treated by assuming that the fluid outside of boundary layers is inviscid, and then matching its solution onto that for a thin laminar
Laminar means "flat". Laminar may refer to:
Terms in science and engineering:
*Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules
* Laminar armour or "banded ...

boundary layer.
For fluid flow over a porous boundary, the fluid velocity can be discontinuous between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition). Further, it is useful at low subsonic
Subsonic may refer to:
Motion through a medium
* Any speed lower than the speed of sound within a sound-propagating medium
* Subsonic aircraft, a flying machine that flies at air speeds lower than the speed of sound
* Subsonic ammunition, a type of ...

speeds to assume that gas is incompressible
In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among ...

—that is, the density of the gas does not change even though the speed and static pressure
In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among for ...

change.
Newtonian versus non-Newtonian fluids

A Newtonian fluid (named afterIsaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

) is defined to be a fluid
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

whose shear stress
Shear stress, often denoted by (Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popu ...

is linearly proportional to the velocity
The velocity of an object is the rate of change of its position with respect to a frame of reference
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

in the direction perpendicular
In elementary geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

to the plane of shear. This definition means regardless of the forces acting on a fluid, it ''continues to flow''. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the drag
Drag or The Drag may refer to:
Places
* Drag, Norway, a village in Tysfjord municipality, Nordland, Norway
* ''Drág'', the Hungarian name for Dragu Commune in Sălaj County, Romania
* Drag (Austin, Texas), the portion of Guadalupe Street adja ...

of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare friction
Friction is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, st ...

). Important fluids, like water as well as most gases, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth.
By contrast, stirring a non-Newtonian fluid
A non-Newtonian fluid is a fluid
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, ...

can leave a "hole" behind. This will gradually fill up over time—this behavior is seen in materials such as pudding, oobleck, or sand
Sand is a granular material composed of finely divided rock (geology), rock and mineral particles. Sand has various compositions but is defined by its grain size. Sand grains are smaller than gravel and coarser than silt. Sand can also refer ...

(although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip paint
Paint is any pigmented liquid
A liquid is a nearly incompressible
In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned wi ...

s). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property—for example, most fluids with long molecular chains can react in a non-Newtonian manner.
Equations for a Newtonian fluid

The constant of proportionality between the viscous stress tensor and the velocity gradient is known as theviscosity
The viscosity of a fluid
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

. A simple equation to describe incompressible Newtonian fluid behavior is
:$\backslash tau\; =\; -\backslash mu\backslash frac$
where
:$\backslash tau$ is the shear stress exerted by the fluid ("drag
Drag or The Drag may refer to:
Places
* Drag, Norway, a village in Tysfjord municipality, Nordland, Norway
* ''Drág'', the Hungarian name for Dragu Commune in Sălaj County, Romania
* Drag (Austin, Texas), the portion of Guadalupe Street adja ...

")
:$\backslash mu$ is the fluid viscosity—a constant of proportionality
:$\backslash frac$ is the velocity gradient perpendicular to the direction of shear.
For a Newtonian fluid, the viscosity, by definition, depends only on temperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal energy refers to several distinct physical concept ...

and pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

, not on the forces acting upon it. If the fluid is incompressible
In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among ...

the equation governing the viscous stress (in Cartesian coordinates
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

) is
:$\backslash tau\_\; =\; \backslash mu\backslash left(\backslash frac+\backslash frac\; \backslash right)\; =\; \backslash mu\backslash partial\_v\_$
where
:$\backslash tau\_$ is the shear stress on the $i^$ face of a fluid element in the $j^$ direction
:$v\_i$ is the velocity in the $i^$ direction
:$x\_j$ is the $j^$ direction coordinate.
If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is
:$\backslash tau\_\; =\; \backslash mu\; \backslash left(\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; \backslash delta\_\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; \backslash right)\; +\; \backslash kappa\; \backslash delta\_\; \backslash nabla\; \backslash cdot\; \backslash mathbf$
where $\backslash kappa$ is the second viscosity coefficient (or bulk viscosity). If a fluid does not obey this relation, it is termed a non-Newtonian fluid
A non-Newtonian fluid is a fluid
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, ...

, of which there are several types. Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic.
In some applications, another rough broad division among fluids is made: ideal and non-ideal fluids. An ideal fluid is non-viscous and offers no resistance whatsoever to a shearing force. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. One example of this is the flow far from solid surfaces. In many cases, the viscous effects are concentrated near the solid boundaries (such as in boundary layers) while in regions of the flow field far away from the boundaries the viscous effects can be neglected and the fluid there is treated as it were inviscid (ideal flow). When the viscosity is neglected, the term containing the viscous stress tensor $\backslash mathbf$ in the Navier–Stokes equation vanishes. The equation reduced in this form is called the Euler equation.
See also

*Aerodynamics
Aerodynamics, from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

*Applied mechanics
Applied mechanics is a branch of the physical science
Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical ...

*Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure, static pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bern ...

*Communicating vessels
Communicating vessels or vasesMario Bunge, ''Philosophy of Science: From Explanation to Justification'', 1998, , p. 369 are a set of containers containing a homogeneous fluid and connected sufficiently far below the top of the liquid: when the liq ...

*Computational fluid dynamics
*Compressor map
*Secondary flow
*Different types of boundary conditions in fluid dynamics
References

Further reading

* * * * *External links

Free Fluid Mechanics books

Annual Review of Fluid Mechanics

CFDWiki

– the Computational Fluid Dynamics reference wiki.

Educational Particle Image Velocimetry – resources and demonstrations

{{DEFAULTSORT:Fluid Mechanics Fluid mechanics, Civil engineering