TheInfoList

Fluid mechanics is the branch of
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 of
ancient 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 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 ...
, 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 after
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 ...

) 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 $\mathbf$, the Navier–Stokes equations are : $\frac + \left(\mathbf \cdot \nabla\right) \mathbf = - \frac\nabla P + \nu \nabla^2 \mathbf$. 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, ...
$\nu$ 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 no
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, ...

, $\nu=0$. 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 after
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 ...

) 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 the
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, ...

. A simple equation to describe incompressible Newtonian fluid behavior is :$\tau = -\mu\frac$ where :$\tau$ 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 ...
") :$\mu$ is the fluid viscosity—a constant of proportionality :$\frac$ 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 :$\tau_ = \mu\left\left(\frac+\frac \right\right) = \mu\partial_v_$ where :$\tau_$ 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 :$\tau_ = \mu \left\left( \frac + \frac - \frac \delta_ \nabla \cdot \mathbf \right\right) + \kappa \delta_ \nabla \cdot \mathbf$ where $\kappa$ 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 $\mathbf$ in the Navier–Stokes equation vanishes. The equation reduced in this form is called the Euler equation.

*
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