Editing Navier–Stokes equations (section)

{{Short description|Equations describing the motion of viscous fluid substances}}

{{Continuum mechanics|fluid}}The '''Navier–Stokes equations''' ({{IPAc-en|n|æ|v|ˈ|j|eː|_|s|t|əʊ|k|s}} {{Respell|nav|YAY|_|STOHKS}}) are [[partial differential equation]]s which describe the motion of [[viscous fluid]] substances.  They were named after French engineer and physicist [[Claude-Louis Navier]] and the Irish physicist and mathematician [[Sir George Stokes, 1st Baronet|George Gabriel Stokes]].  They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

The Navier–Stokes equations mathematically express [[momentum]] balance for [[Newtonian fluid]]s and make use of [[conservation of mass]]. They are sometimes accompanied by an [[equation of state]] relating [[pressure]], [[temperature]]  and [[density]].<ref>{{cite book |title=Understanding Aerodynamics: Arguing from the Real Physics |first=Doug |last=McLean |publisher=John Wiley & Sons |year=2012 |chapter=Continuum Fluid Mechanics and the Navier-Stokes Equations |pages=13&ndash;78 |isbn=9781119967514 |quote=The main relationships comprising the NS equations are the basic conservation laws for mass, momentum, and energy. To have a complete equation set we also need an equation of state relating temperature, pressure, and density... |chapter-url=https://books.google.com/books?id=UE3sxu28R0wC&pg=PA13 }}</ref> They arise from applying [[Newton's second law|Isaac Newton's second law]] to [[Fluid dynamics|fluid motion]], together with the assumption that the [[stress (mechanics)|stress]] in the fluid is the sum of a [[diffusion|diffusing]] [[viscosity|viscous]] term (proportional to the [[gradient]] of velocity) and a [[pressure]] term—hence describing ''viscous flow''. The difference between them and the closely related [[Euler equations (fluid dynamics)|Euler equations]] is that Navier–Stokes equations take [[viscosity]] into account while the Euler equations model only [[inviscid flow]]. As a result, the Navier–Stokes are an [[Elliptic partial differential equation|elliptic equation]] and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never [[completely integrable]]).

The Navier–Stokes equations are useful because they describe the physics of many phenomena of [[scientific]] and [[engineering]] interest. They may be used to [[model (abstract)|model]] the weather, [[ocean current]]s, water [[flow conditioning|flow in a pipe]] and air flow around a [[airfoil|wing]]. The Navier–Stokes equations, in their full and simplified forms, help with the design of [[Aircraft design process#Preliminary design phase|aircraft]] and cars, the study of [[Hemodynamics|blood flow]], the design of [[power station]]s, the analysis of [[pollution]], and many other problems. Coupled with [[Maxwell's equations]], they can be used to model and study [[magnetohydrodynamics]].

The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always [[Existence theorem|exist]] in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in the [[Domain (mathematical analysis)|domain]]. This is called the [[Navier–Stokes existence and smoothness]] problem. The [[Clay Mathematics Institute]] has called this one of the [[Millennium Prize Problems|seven most important open problems in mathematics]] and has offered a [[US$]]1&nbsp;million prize for a solution or a counterexample.<ref>{{Citation
| url = http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation
| website = claymath.org
| title = Millennium Prize Problems—Navier–Stokes Equation
| publisher = Clay Mathematics Institute
| date = March 27, 2017
| access-date = 2017-04-02
| archive-date = 2015-12-22
| archive-url = https://web.archive.org/web/20151222054637/http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation
| url-status = dead
}}</ref><ref>{{cite web
 | url = http://www.claymath.org/sites/default/files/navierstokes.pdf
 | title = Existence and smoothness of the Navier–Stokes equation
 | last = Fefferman
 | first = Charles L.
 | website = claymath.org
 | publisher = Clay Mathematics Institute
 | access-date = 2017-04-02
 | archive-url = https://web.archive.org/web/20150415042320/http://www.claymath.org/sites/default/files/navierstokes.pdf
 | archive-date = 2015-04-15
 | url-status = dead
 }}</ref>