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{{short description|Branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas)}}
{{Continuum mechanics|cTopic=fluid}}

'''Fluid mechanics''' is that branch of [[Physics:Physics|physics]] concerned with the [[Physics:Mechanics|mechanics]] of [[Physics:Fluid|fluid]]s ([[Physics:Liquid|liquid]]s, [[Physics:Gas|gas]]es, and [[Physics:Plasma|plasma]]s) and the [[Physics:Force|force]]s on them.{{r|White2011|p=3}}
It has applications in a wide range of disciplines, including [[Engineering:Mechanical engineering|mechanical]], [[Engineering:Aerospace engineering|aerospace]], [[Engineering:Civil engineering|civil]], [[Chemistry:Chemical engineering|chemical]], and [[Medicine:Biomedical engineering|biomedical engineering]], as well as [[Physics:Geophysics|geophysics]], [[Physics:Oceanography|oceanography]], [[Physics:Meteorology|meteorology]], [[Physics:Astrophysics|astrophysics]], and [[Biology:Biology|biology]].

It can be divided into fluid statics, the study of fluids at rest; and [[Physics:Fluid dynamics|fluid dynamics]], the study of the effect of forces on fluid motion.{{r|White2011|p=3}}
It is a branch of [[Physics:Continuum mechanics|continuum mechanics]], 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, typically using computers.  A modern discipline, called [[Computational fluid dynamics|computational fluid dynamics]] (CFD), is devoted to this approach.<ref>{{cite book |last1=Tu |first1=Jiyuan |last2=Yeoh |first2=Guan Heng |last3=Liu |first3=Chaoqun |title=Computational Fluid Dynamics: A Practical Approach |date=Nov 21, 2012 |publisher=Butterworth-Heinemann |isbn=978-0080982434}}</ref>  [[Particle image velocimetry]], an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.

==History==
{{main|Physics:History of fluid mechanics}}
{{For timeline|Timeline of fluid and continuum mechanics}}
The study of fluid mechanics goes back at least to the days of {{wipe|Ancient Greece}}, when [[Biography:Archimedes|Archimedes]] investigated fluid statics and [[Physics:Buoyancy|buoyancy]] and formulated his famous law known now as the [[Physics:Archimedes' principle|Archimedes' principle]], which was published in his work ''On Floating Bodies''—generally considered to be the first major work on fluid mechanics. Iranian scholar Abu Rayhan Biruni and later [[Biography:Al-Khazini|Al-Khazini]]  applied [[Experiment|experiment]]al [[Scientific method|scientific method]]s to fluid mechanics.<ref>Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642,</ref> Rapid advancement in fluid mechanics began with [[Biography:Leonardo da Vinci|Leonardo da Vinci]] (observations and experiments), [[Biography:Evangelista Torricelli|Evangelista Torricelli]] (invented the [[Engineering:Barometer|barometer]]), [[Biography:Isaac Newton|Isaac Newton]] (investigated [[Physics:Viscosity|viscosity]]) and [[Biography:Blaise Pascal|Blaise Pascal]] (researched [[Physics:Hydrostatics|hydrostatics]], formulated [[Physics:Pascal's law|Pascal's law]]), and was continued by [[Biography:Daniel Bernoulli|Daniel Bernoulli]] with the introduction of mathematical fluid dynamics in ''Hydrodynamica'' (1739).

Inviscid flow was further analyzed by various mathematicians ([[Biography:Jean le Rond d'Alembert|Jean le Rond d'Alembert]], Joseph Louis Lagrange, [[Biography:Pierre-Simon Laplace|Pierre-Simon Laplace]], [[Biography:Siméon Denis Poisson|Siméon Denis Poisson]]) and viscous flow was explored by a multitude of engineers including [[Biography:Jean Léonard Marie Poiseuille|Jean Léonard Marie Poiseuille]] and [[Biography:Gotthilf Hagen|Gotthilf Hagen]]. Further mathematical justification was provided by [[Biography:Claude-Louis Navier|Claude-Louis Navier]] and George Gabriel Stokes in the [[Physics:Navier–Stokes equations|Navier–Stokes equations]], and boundary layers were investigated ([[Biography:Ludwig Prandtl|Ludwig Prandtl]], [[Biography:Theodore von Kármán|Theodore von Kármán]]), while various scientists such as [[Biography:Osborne Reynolds|Osborne Reynolds]], [[Biography:Andrey Kolmogorov|Andrey Kolmogorov]], and Geoffrey Ingram Taylor advanced the understanding of fluid viscosity  and [[Physics:Turbulence|turbulence]].

==Main branches==

===Fluid statics===
Fluid statics or '''hydrostatics''' is the branch of fluid mechanics that studies [[Physics:Fluid|fluid]]s at rest.  It embraces the study of the conditions under which fluids are at rest in [[Physics:Mechanical equilibrium|stable]] [[Astronomy:Hydrostatic equilibrium|equilibrium]]; and is contrasted with [[Physics:Fluid dynamics|fluid dynamics]], the study of fluids in motion. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why [[Physics:Atmospheric pressure|atmospheric pressure]] changes with [[Altitude|altitude]], why wood and [[Chemistry:Oil|oil]] float on water, and why the surface of water is always level whatever the shape of its container. Hydrostatics is fundamental to [[Physics:Hydraulics|hydraulics]], the [[Engineering:Engineering|engineering]] of equipment for storing, transporting and using fluids.  It is also relevant to some aspects of [[Physics:Geophysics|geophysics]] and [[Physics:Astrophysics|astrophysics]] (for example, in understanding [[Earth:Plate tectonics|plate tectonics]] and anomalies in the [[Physics:Gravity of Earth|Earth's gravitational field]]), to [[Physics:Meteorology|meteorology]], to [[Medicine:Medicine|medicine]] (in the context of [[Physics:Blood pressure|blood pressure]]), and many other fields.

===Fluid dynamics===
{{main|Physics:Fluid dynamics}}
''[[Physics:Fluid dynamics|Fluid dynamics]]'' is a subdiscipline of fluid mechanics that deals with ''fluid flow''—the science of liquids and gases in motion.<ref>Batchelor, C. K., & Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge University Press.</ref> Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from [[Flow measurement|flow measurement]] and used to solve practical problems.  The solution to a [[Physics:Fluid dynamics|fluid dynamics]] problem typically involves calculating various properties  of the fluid, such as [[Physics:Velocity|velocity]], [[Physics:Pressure|pressure]], [[Physics:Density|density]], and [[Physics:Temperature|temperature]], as functions of space and time. It has several subdisciplines itself, including ''[[Physics:Aerodynamics|aerodynamics]]''<ref>Bertin, J. J., & Smith, M. L. (1998). Aerodynamics for engineers (Vol. 5). Upper Saddle River, NJ: Prentice Hall.</ref><ref>Anderson Jr, J. D. (2010). Fundamentals of aerodynamics. Tata McGraw-Hill Education.</ref><ref>Houghton, E. L., & Carpenter, P. W. (2003). Aerodynamics for engineering students. Elsevier.</ref><ref>Milne-Thomson, L. M. (1973). Theoretical aerodynamics. Courier Corporation.</ref> (the study of air and other gases in motion) and ''hydrodynamics''<ref>Milne-Thomson, L. M. (1996). Theoretical hydrodynamics. Courier Corporation.</ref><ref>Birkhoff, G. (2015). Hydrodynamics. Princeton University Press.</ref> (the study of liquids in motion).  Fluid dynamics has a wide range of applications, including calculating [[Physics:Force|force]]s and [[Physics:Moment|movement]]s on [[Engineering:Aircraft|aircraft]], determining the [[Physics:Mass flow rate|mass flow rate]] of [[Chemistry:Petroleum|petroleum]] through pipelines, predicting evolving [[Earth:Weather|weather]] patterns, understanding [[Astronomy:Nebula|nebula]]e in interstellar space and modeling explosions.  Some fluid-dynamical principles are used in [[Engineering:Traffic engineering (transportation)|traffic engineering]] and crowd dynamics.

==Relationship to continuum mechanics==
Fluid mechanics is a subdiscipline of [[Physics:Continuum mechanics|continuum mechanics]], as illustrated in the following table.
{{Continuum mechanics context}}
In a mechanical view, a fluid is a substance that does not support [[Physics:Shear stress|shear stress]]; that is why a fluid at rest has the shape of its containing vessel.  A fluid at rest has no shear stress.

==Assumptions==

[[File:Reynolds.svg|thumb|right|Balance for some integrated fluid quantity in a control volume enclosed by a control surface.]]
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:
* [[Physics:Conservation of mass|Conservation of mass]]
* [[Physics:Conservation of energy|Conservation of energy]]
* [[Physics:Conservation of momentum|Conservation of momentum]]
* [[Knudsen number|The continuum assumption]]
For example, the assumption that mass is conserved means that for any fixed [[Physics:Control volume|control volume]] (for example, a spherical volume)—enclosed by a control surface—the [[Derivative|rate of change]] 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 [[Continuity equation#Integral form|equation in integral form]] over the control volume.{{r|Batchelor1967|p=74}}

The '''{{vanchor|continuum assumption|Continuum assumption}}''' is an idealization of [[Physics:Continuum mechanics|continuum mechanics]] under which fluids can be treated as [[Continuous function|continuous]], even though, on a microscopic scale, they are composed of molecules. 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.<ref name="Greenkorn2018">{{cite book |first=Robert |last=Greenkorn |title=Momentum, Heat, and Mass Transfer Fundamentals |url=https://books.google.com/books?id=pjFRDwAAQBAJ&q=%22Breakdown+of+continuum+assumption%22&pg=PA18 |date=3 October 2018 |publisher=CRC Press |isbn=978-1-4822-9297-8 |page=18}}</ref> Those problems for which the continuum hypothesis fails can be solved using [[Physics:Statistical mechanics|statistical mechanics]].  To determine whether or not the continuum hypothesis applies, the [[Knudsen number]], defined as the ratio of the molecular [[Physics:Mean free path|mean free path]] to the characteristic length scale, 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.
{{anchor|mathematical_fluid_mechanics}}

==Navier–Stokes equations==
{{main|Physics:Navier–Stokes equations}}
The '''Navier–Stokes equations''' (named after [[Biography:Claude-Louis Navier|Claude-Louis Navier]] and George Gabriel Stokes) are differential equations that describe the force balance at a given point within a fluid. For an incompressible fluid with vector velocity field <math>\mathbf{u}</math>, the Navier–Stokes equations are<ref>Constantin, P., & Foias, C. (1988). Navier-stokes equations. University of Chicago Press.</ref><ref>Temam, R. (2001). Navier-Stokes equations: theory and numerical analysis (Vol. 343). American Mathematical Society.</ref><ref>Foias, C., Manley, O., Rosa, R., & Temam, R. (2001). Navier-Stokes equations and turbulence (Vol. 83). Cambridge University Press.</ref><ref>Girault, V., & Raviart, P. A. (2012). Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5). Springer Science & Business Media.</ref> 

: <math>\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}  = - \frac{1}{\rho}\nabla p +   \nu \nabla^2 \mathbf{u}</math>.

These differential equations are the analogues for deformable materials to Newton's equations of motion for particles – the Navier–Stokes equations describe changes in [[Physics:Momentum|momentum]] ([[Physics:Force|force]]) in response to [[Physics:Pressure|pressure]] <math>p </math> and viscosity, parameterized by the [[Physics:Kinematic viscosity|kinematic viscosity]] <math>\nu </math>. Occasionally, [[Physics:Body force|body force]]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]]. 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]] is small. For more complex cases, especially those involving [[Physics:Turbulence|turbulence]], 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|computational fluid dynamics]].<ref>Anderson, J. D., & Wendt, J. (1995). Computational fluid dynamics (Vol. 206). New York: McGraw-Hill.</ref><ref>Chung, T. J. (2010). Computational fluid dynamics. Cambridge University Press.</ref><ref>Blazek, J. (2015). Computational fluid dynamics: principles and applications. Butterworth-Heinemann.</ref><ref>Wesseling, P. (2009). Principles of computational fluid dynamics (Vol. 29). Springer Science & Business Media.</ref><ref>Anderson, D., Tannehill, J. C., & Pletcher, R. H. (2016). Computational fluid mechanics and heat transfer. Taylor & Francis.</ref>

==Inviscid and viscous fluids==

An '''inviscid fluid''' has no [[Physics:Viscosity|viscosity]], <math>\nu=0 </math>. 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 [[Physics:Superfluidity|superfluidity]]. Otherwise, fluids are generally '''viscous''', a property that is often most important within a [[Physics:Boundary layer|boundary layer]] near a solid surface,<ref>{{cite book |last1=Kundu |first1=Pijush K. |last2=Cohen |first2=Ira M. |last3=Dowling |first3=David R. |title=Fluid Mechanics |publisher=Academic Press |isbn=978-0124059351 |edition=6th |chapter=10|date=27 March 2015 }}</ref> where the flow must match onto the [[No-slip condition|no-slip condition]] 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 [[Method of matched asymptotic expansions|matching]] its solution onto that for a thin [[Physics:Laminar flow|laminar]] 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 [[Physics:Speed of sound|subsonic]] speeds to assume that gas is incompressible—that is, the density of the gas does not change even though the speed and [[Physics:Static pressure|static pressure]] change.

==Newtonian versus non-Newtonian fluids==
A '''Newtonian fluid''' (named after [[Biography:Isaac Newton|Isaac Newton]]) is defined to be a [[Physics:Fluid|fluid]] whose [[Physics:Shear stress|shear stress]] is linearly proportional to the [[Physics:Velocity|velocity]] [[Gradient|gradient]] in the direction [[Perpendicular|perpendicular]] 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 [[Physics:Drag|drag]] of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare [[Physics:Friction|friction]]).  Important fluids, like water as well as most gasses, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth.{{r|Batchelor1967|p=145}}

By contrast, stirring a [[Physics:Non-Newtonian fluid|non-Newtonian fluid]] can leave a "hole" behind. This will gradually fill up over time—this behavior is seen in materials such as pudding, oobleck, or sand (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 [[Engineering:Paint|paint]]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.{{r|Batchelor1967|p=145}}

===Equations for a Newtonian fluid===
{{main|Physics:Newtonian fluid}}
The constant of proportionality between the viscous stress tensor and the velocity gradient is known as the [[Physics:Viscosity|viscosity]]. A simple equation to describe incompressible Newtonian fluid behavior is
:<math>\tau = -\mu\frac{\mathrm{d} u}{\mathrm{d} n}</math>

where
:<math>\tau</math> is the shear stress exerted by the fluid ("[[Physics:Drag|drag]]"),
:<math>\mu</math> is the fluid viscosity—a constant of proportionality, and
:<math>\frac{\mathrm{d} u}{\mathrm{d} n}</math> is the velocity gradient perpendicular to the direction of shear.

For a Newtonian fluid, the viscosity, by definition, depends only on [[Physics:Temperature|temperature]], not on the forces acting upon it. If the fluid is incompressible the equation governing the viscous stress (in [[Cartesian coordinate system|Cartesian coordinates]]) is

:<math>\tau_{ij} = \mu\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \right)</math>

where
:<math>\tau_{ij}</math> is the shear stress on the <math>i^{th}</math> face of a fluid element in the <math>j^{th}</math> direction
:<math>v_i</math> is the velocity in the <math>i^{th}</math> direction
:<math>x_j</math> is the <math>j^{th}</math> direction coordinate.

If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is

:<math>\tau_{ij} = \mu \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \nabla \cdot \mathbf{v} \right) + \kappa \delta_{ij} \nabla \cdot \mathbf{v} </math>
where <math> \kappa </math> is the second viscosity coefficient (or bulk viscosity). If a fluid does not obey this relation, it is termed a [[Physics:Non-Newtonian fluid|non-Newtonian fluid]], 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 <math> \mathbf{\tau} </math> in the Navier–Stokes equation vanishes. The equation reduced in this form is called the [[Euler equations (fluid dynamics)|Euler equation]].

==See also==
*[[Physics:Transport phenomena|Transport phenomena]]
*[[Physics:Aerodynamics|Aerodynamics]]
*[[Physics:Applied mechanics|Applied mechanics]]
*[[Bernoulli's principle]]
*[[Physics:Communicating vessels|Communicating vessels]]
*[[Computational fluid dynamics]]
*[[Compressor map]]
*[[Physics:Secondary flow|Secondary flow]]
*Different types of boundary conditions in fluid dynamics

== References ==
{{reflist|
refs=
<ref name="White2011">{{cite book |last1=White |first1=Frank M. |title=Fluid Mechanics |date=2011 |publisher=McGraw-Hill |isbn=978-0-07-352934-9 |edition=7th}}</ref>
<ref name="Batchelor1967">{{cite book | first=George K. | last=Batchelor |  title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0-521-66396-2 |page=74 }}</ref>
}}

==Further reading==
* {{citation | last=Falkovich | first=Gregory | year=2011 | title=Fluid Mechanics (A short course for physicists) | publisher=Cambridge University Press | isbn=978-1-107-00575-4 | doi=10.1017/CBO9780511794353 }}
* {{citation | last2=Cohen | first2=Ira M. | last1=Kundu | first1=Pijush K. | year=2008 | title=Fluid Mechanics | edition=4th revised | publisher=Academic Press | isbn=978-0-12-373735-9 }}
*{{Citation
  | last = Currie
  | first = I. G.
  | title = Fundamental Mechanics of Fluids
  | publisher = McGraw-Hill, Inc.
  | year = 1974
  | isbn = 0-07-015000-1 }}
* {{citation | last1=Massey | first1=B. | last2=Ward-Smith | first2=J. | year=2005 | title=Mechanics of Fluids | edition=8th | publisher=Taylor & Francis | isbn=978-0-415-36206-1 }}
* {{citation | last1=Nazarenko | first1=Sergey | year=2014 | title=Fluid Dynamics via Examples and Solutions | publisher=CRC Press (Taylor & Francis group) | isbn=978-1-43-988882-7 }}

==External links==
*[http://www.freebookcentre.net/Physics/Fluid-Mechanics-Books.html Free Fluid Mechanics books]
*[http://arjournals.annualreviews.org/loi/fluid Annual Review of Fluid Mechanics]. {{Webarchive|url=https://web.archive.org/web/20090119222743/http://arjournals.annualreviews.org/loi/fluid |date=2009-01-19 }}.
*[http://www.cfd-online.com/Wiki/Main_Page CFDWiki] – the Computational Fluid Dynamics reference wiki.
*[http://www.interactiveflows.com/downloads/ Educational Particle Image Velocimetry] – resources and demonstrations

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