View source for Navier–Stokes equations

{{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|_|ˈ|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 British physicist and mathematician [[Sir George Stokes, 1st Baronet|Sir George Gabriel Stokes, Bt]].  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=978-1-119-96751-4 |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
}}</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
 }}</ref>

==Flow velocity==
The solution of the equations is a [[flow velocity]]. It is a [[vector field]]—to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is studied in three spatial dimensions and one time dimension, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest such as [[pressure]] or [[temperature]] may be found using dynamical equations and relations. This is different from what one normally sees in [[classical mechanics]], where solutions are typically trajectories of position of a [[particle]] or deflection of a [[Continuum (theory)|continuum]]. Studying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various [[Streamlines, streaklines, and pathlines|trajectories]]. In particular, the [[Streamlines, streaklines, and pathlines|streamlines]] of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel. These paths are the [[integral curve]]s whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time.

==General continuum equations==
{{Main|Derivation of the Navier–Stokes equations}}
{{See also|Cauchy momentum equation#Conservation form}}

The Navier–Stokes momentum equation can be derived as a particular form of the [[Cauchy momentum equation]], whose general convective form is:
<math display="block"> \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \frac 1 \rho \nabla \cdot \boldsymbol{\sigma} + \mathbf{f}.</math>
By setting the [[Cauchy stress tensor]] <math display="inline">\boldsymbol{\sigma}</math> to be the sum of a viscosity term <math display="inline">\boldsymbol{\tau}</math> (the [[deviatoric stress]]) and a pressure term <math display="inline">-p \mathbf{I}</math> (volumetric stress), we arrive at:

{{Equation box 1
|indent=:
|title='''Cauchy momentum equation'''&nbsp;&nbsp;''(convective form)''
|equation=<math>
\rho\frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{a}
</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #DCDCDC
}}

where
* <math display="inline">\frac{\mathrm{D}}{\mathrm{D}t}</math> is the [[material derivative]], defined as <math display="inline">\frac{\partial}{\partial t} + \mathbf{u} \cdot \nabla</math>,
* <math display="inline">\rho</math> is the (mass) density,
* <math display="inline">\mathbf{u}</math> is the flow velocity,
* <math display="inline">\nabla \cdot \,</math> is the [[divergence]],
* <math display="inline">p</math> is the [[pressure]],
* <math display="inline">t</math> is [[time]],
* <math display="inline">\boldsymbol{\tau}</math> is the [[Deviatoric stress|deviatoric stress tensor]], which has order 2,
* <math display="inline">\mathbf{a}</math> represents [[body force#acceleration|body acceleration]]s acting on the continuum, for example [[gravity]], [[Fictitious force|inertial accelerations]], [[Coulomb's law|electrostatic accelerations]], and so on.

In this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the [[Euler equations (fluid dynamics)|Euler equations]].

Assuming [[conservation of mass]], with the known properties of [[divergence]] and [[gradient]] we can use the mass [[continuity equation]], which represents the mass per unit volume of a [[homogenous]] fluid with respect to space and time (i.e., [[material derivative]] <math>\frac{\mathbf{D}}{\mathbf{Dt}}</math>) of any finite volume ('''V''') to represent the change of velocity in fluid media:
<math display="block">
\begin{align}
& \frac{\mathbf{D}m}{\mathbf{Dt}} = \iiint\limits_V \left(\frac{ \mathbf{D}\rho}{\mathbf{Dt}} + \rho (\nabla \cdot \mathbf{u})\right) \, dV \\[5pt]
& \frac{\mathbf{D}\rho}{\mathbf{Dt}} + \rho (\nabla \cdot \mathbf{u} )=\frac{\partial\rho}{\partial t} + (\nabla \rho) \cdot \mathbf{u} + \rho(\nabla \cdot \mathbf{u})= \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho \mathbf{u})= 0 
\end{align}</math>where
* <math display="inline">\frac{\mathrm{D}m}{\mathrm{D}t}</math> is the [[material derivative]] of [[mass]] per unit volume ([[density]], <math>\rho</math>),
* <math display="inline"> \iiint \limits_V (F(x_1, x_2, x_3 ,t)) \, dV</math> is the mathematical operation for the [[Volume integral|integration throughout the volume]] (''V''),
* <math display="inline">\frac{\partial }{\partial t}</math> is the [[partial derivative]] mathematical operator,
* <math display="inline">\nabla \cdot \mathbf{u}\,</math> is the [[divergence]] of the flow velocity (<math>\mathbf{u}</math>), which is a [[scalar field]], [[Del|<sup>Note 1</sup>]]
* <math display="inline"> \nabla \rho \,</math> is the [[gradient]] of [[density]] (<math>\rho</math>), which is the vector derivative of a [[scalar field]], [[Del|<sup>Note 1</sup>]]
<sup>[[Del|Note 1 – Refer to the mathematical operator del represented by the nabla]] (<math>\nabla</math>) [[Del|symbol.]]</sup>

to arrive at the conservation form of the equations of motion. This is often written:<ref>{{harvp|Batchelor|1967|pp=137 & 142}}</ref>

{{Equation box 1
|indent=:
|title='''Cauchy momentum equation''' ''(conservation form)''
|equation=<math>
\frac {\partial}{\partial t} (\rho\,\mathbf{u})
   + \nabla \cdot (\rho\,\mathbf{u} \otimes \mathbf{u})
 = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{a}
</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #DCDCDC
}}

where <math display="inline">\otimes</math> is the [[outer product]] of the flow velocity (<math>\mathbf{u}</math>):<math display="block">\mathbf u \otimes \mathbf u = \mathbf u \mathbf u^{\mathrm T}</math>

The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity).

All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a [[constitutive relation]]. By expressing the deviatoric (shear) stress tensor in terms of [[viscosity]] and the fluid [[Shear velocity|velocity]] gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below.

===Convective acceleration===
{{See also|Cauchy momentum equation#Convective acceleration}}
[[Image:ConvectiveAcceleration vectorized.svg|thumb|An example of convection. Though the flow may be steady (time-independent), the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow), hence there is an acceleration happening over position.]]

A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of a flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.

==Compressible flow==
Remark: here, the deviatoric stress tensor is denoted <math display="inline">\boldsymbol{\tau}</math> as it was in the [[#General continuum equations|general continuum equations]] and in the [[#Incompressible flow|incompressible flow section]].

The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:<ref name="Batchelor_142_148"/>

<ul>
<li>the stress is '''[[Galilean invariance|Galilean invariant]]''': it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient <math display="inline">\nabla \mathbf{u}</math>, or more simply the [[strain-rate tensor|rate-of-strain tensor]]: <math display="inline">\boldsymbol{\varepsilon}\left(\nabla \mathbf{u}\right) \equiv \frac{1}{2}\nabla \mathbf{u} + \frac{1}{2} \left(\nabla \mathbf{u}\right)^T</math></li>
<li>the deviatoric stress is '''linear''' in this variable: <math display="inline">\boldsymbol{\sigma}(\boldsymbol \varepsilon) = -p \mathbf I + \mathbf{C} : \boldsymbol \varepsilon</math>, where <math display="inline">p</math> is independent on the strain rate tensor, <math display="inline">\mathbf{C}</math> is the fourth-order tensor representing the constant of proportionality, called the viscosity or [[elasticity tensor]], and : is the [[Dyadics#Double-dot product|double-dot product]].</li>
<li>the fluid is assumed to be [[isotropic]], as with gases and simple liquids, and consequently <math display="inline">\mathbf{C}</math> is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, by [[Helmholtz decomposition]] it can be expressed in terms of two scalar [[Lamé parameters]], the [[second viscosity]] <math display="inline">\lambda</math> and the [[dynamic viscosity]] <math display="inline">\mu</math>, as it is usual in [[linear elasticity]]:
{{Equation box 1
|indent=:
|title='''Linear stress [[constitutive equation]]''' ''(expression similar to the one for elastic solid)''
|equation=<math>\boldsymbol \sigma(\boldsymbol \varepsilon) =  - p \mathbf I + \lambda \operatorname{tr} (\boldsymbol \varepsilon) \mathbf I + 2 \mu \boldsymbol \varepsilon</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}
where <math display="inline">\mathbf{I}</math> is the [[Identity matrix|identity tensor]], and <math display="inline">\operatorname{tr} (\boldsymbol \varepsilon)</math> is the [[trace (linear algebra)|trace]] of the rate-of-strain tensor. So this decomposition can be explicitly defined as:
<math display="block">\boldsymbol \sigma = -p \mathbf I + \lambda (\nabla\cdot\mathbf{u}) \mathbf I + \mu \left(\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T}\right).</math>
</li>
</ul>

Since the [[trace (linear algebra)|trace]] of the rate-of-strain tensor in three dimensions is the [[divergence]] (i.e. rate of expansion) of the flow:
<math display="block">\operatorname{tr} (\boldsymbol \varepsilon) = \nabla\cdot\mathbf{u}.</math>

Given this relation, and since the trace of the identity tensor in three dimensions is three:
<math display="block">\operatorname{tr} (\boldsymbol I) = 3.</math>

the trace of the stress tensor in three dimensions becomes:
<math display="block">\operatorname{tr} (\boldsymbol \sigma ) = -3p + (3 \lambda + 2 \mu )\nabla\cdot\mathbf{u}.</math>

So by alternatively decomposing the stress tensor into '''isotropic''' and '''deviatoric''' parts, as usual in fluid dynamics:<ref>{{cite book |last1=Chorin |first1=Alexandre E. |last2=Marsden |first2=Jerrold E. |date=1993 |title=A Mathematical Introduction to Fluid Mechanics |page=33}}</ref>
<math display="block">\boldsymbol \sigma = - \left[ p - \left(\lambda + \tfrac23 \mu\right) \left(\nabla\cdot\mathbf{u}\right) \right] \mathbf I + \mu \left(\nabla\mathbf{u} + \left( \nabla\mathbf{u} \right)^\mathrm{T} - \tfrac23 \left(\nabla\cdot\mathbf{u}\right)\mathbf I\right)</math>

Introducing the [[volume viscosity|bulk viscosity]] <math display="inline">\zeta</math>,
<math display="block"> \zeta \equiv \lambda + \tfrac23 \mu ,</math>

we arrive to the linear [[constitutive equation]] in the form usually employed in [[thermal hydraulics]]:<ref name=Batchelor_142_148/>

{{Equation box 1
|indent=:
|title='''Linear stress constitutive equation''' ''(expression used for fluids)''
|equation=<math>\boldsymbol \sigma = -[ p - \zeta (\nabla\cdot\mathbf{u})] \mathbf I + \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right]</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}

which can also be arranged in the other usual form:<ref>Bird, Stewart, Lightfoot, Transport Phenomena, 1st ed., 1960, eq. (3.2-11a)</ref>
<math display="block">\boldsymbol \sigma = -p \mathbf I + \mu \left(\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T}\right) + \left(\zeta - \frac 2 3 \mu \right) (\nabla\cdot\mathbf{u}) \mathbf I.</math>

Note that in the compressible case the pressure is no more proportional to the [[hydrostatic stress|isotropic stress]] term, since there is the additional bulk viscosity term:
<math>p = - \frac 1 3 \operatorname{tr} (\boldsymbol \sigma)  + \zeta (\nabla\cdot\mathbf{u})</math>

and the [[deviatoric stress tensor]] <math>\boldsymbol \sigma'</math> is still coincident with the shear stress tensor <math>\boldsymbol \tau</math> (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which is proportional to the shear viscosity:

<math>\boldsymbol \sigma' = \boldsymbol \tau = \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right]</math>

Both bulk viscosity <math display="inline">\zeta</math> and dynamic viscosity <math display="inline">\mu</math> need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these [[transport coefficient]] in the [[conservation variable]]s is called an [[equation of state]].<ref name="Batchelor 1967 p. 165"/>

The most general of the Navier–Stokes equations become
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''convective form'')
|equation=<math> \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = - \nabla p  + \nabla \cdot \left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\} + \nabla[\zeta (\nabla\cdot\mathbf{u})] + \rho\mathbf{a} .</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}
in index notation, the equation can be written as<ref name="landau">Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6. Vol. 6. Elsevier, 2013.</ref>
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''index notation'')
|equation=<math> \rho \left(\frac{\partial u_i}{\partial t} + u_k \frac{\partial u_i}{\partial x_k}\right) = - \frac{\partial p}{\partial x_i} + \frac{\partial }{\partial x_k}\left[\mu \left(\frac{\partial u_i}{\partial x_k}+ \frac{\partial u_k}{\partial x_i}-\frac{2}{3}\delta_{ik} \frac{\partial u_l}{\partial x_l}\right)\right] + \frac{\partial }{\partial x_i}\left(\zeta \frac{\partial u_\ell}{\partial x_\ell}\right)+\rho a_i.</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}

The corresponding equation in conservation form can be obtained by considering that, given the mass [[continuity equation]], the left side is equivalent to:

: <math> \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \frac {\partial}{\partial t} (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u)</math>

to give finally:
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''conservative form'')
|equation=<math> \frac {\partial}{\partial t} (\rho \mathbf u) + \nabla \cdot \left(\rho \mathbf u \otimes \mathbf u + [p - \zeta (\nabla\cdot\mathbf{u})] \mathbf I - \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u}) \mathbf I\right] \right) = \rho\mathbf{a} .</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}

Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the ''dispersion''. In some cases, the [[volume viscosity|second viscosity]] <math display="inline">\zeta</math> can be assumed to be constant in which case, the effect of the volume viscosity <math display="inline">\zeta</math> is that the mechanical pressure is not equivalent to the thermodynamic [[pressure]]:<ref>{{harvp|Landau|Lifshitz|1987|pp=44–45, 196}}</ref> as demonstrated below.<math display="block">\nabla\cdot(\nabla\cdot \mathbf u)\mathbf I=\nabla (\nabla \cdot \mathbf u),</math><math display="block"> \bar{p} \equiv p - \zeta \, \nabla \cdot \mathbf{u} ,</math>{{Anchor|Stokes hypothesis}}However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves,<ref>{{harvp|White|2006|p=67}}</ref> where second viscosity coefficient becomes important) by explicitly assuming <math display="inline">\zeta = 0</math>. The assumption of setting <math display="inline">\zeta = 0</math> is called as the '''Stokes hypothesis'''.<ref>Stokes, G. G. (1845). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids.</ref> The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory;<ref>Vincenti, W. G., Kruger Jr., C. H. (1975). Introduction to physical gas dynamic. Introduction to physical gas dynamics/Huntington.</ref> for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' (''convective form, Stokes hypothesis'')
|equation=<math> \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = - \nabla  p + \nabla \cdot \left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\} + \rho\mathbf{a} .</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}

If the dynamic {{mvar|μ}} and bulk <math>\zeta</math> viscosities are  assumed to be uniform in space, the equations in convective form can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor <math display="inline">\nabla \mathbf{u}</math> is <math display="inline">\nabla^2 \mathbf{u}</math> and the divergence of tensor <math display="inline">\left(\nabla \mathbf{u}\right)^{\mathrm T}</math> is <math display="inline">\nabla \left(\nabla \cdot \mathbf{u}\right) </math>, one finally arrives to the compressible Navier–Stokes momentum equation:<ref>{{harvp|Batchelor|1967|pp=147 & 154}}</ref>
{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' with uniform shear and bulk viscosities (''convective form'')
|equation=<math> \frac {D \mathbf{u}}{D t} = - \frac 1 \rho \nabla p + \nu \, \nabla^2 \mathbf u + (\tfrac13 \nu + \xi) \, \nabla (\nabla\cdot\mathbf{u}) + \mathbf{a} .</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}
where <math display="inline">\frac{\mathrm{D}}{\mathrm{D}t} </math> is the [[material derivative]]. <math>\nu=\frac \mu \rho</math> is the shear [[kinematic viscosity]] and <math>\xi=\frac \zeta \rho</math> is the bulk kinematic viscosity. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation.
By bringing the operator on the flow velocity on the left side, one also has:

{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation''' with uniform shear and bulk viscosities (''convective form'')
|equation=<math> \left(\frac {\partial}{\partial t} + \mathbf u \cdot \nabla - \nu \, \nabla^2 - (\tfrac13 \nu + \xi) \, \nabla (\nabla\cdot) \right)\mathbf{u} = - \frac 1\rho \nabla p + \mathbf{a} .</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}

The convective acceleration term can also be written as
<math display="block"> \mathbf u\cdot\nabla\mathbf u = (\nabla\times\mathbf u)\times\mathbf u + \tfrac12\nabla\mathbf u^2,</math>
where the vector <math display="inline"> (\nabla \times \mathbf{u}) \times \mathbf{u}</math> is known as the [[Lamb vector]].

For the special case of an [[incompressible flow]], the pressure constrains the flow so that the volume of [[fluid element]]s is constant: [[isochoric process|isochoric flow]] resulting in a [[Solenoidal vector field|solenoidal]] velocity field with <math display="inline"> \nabla \cdot \mathbf{u} = 0</math>.<ref>{{harvp|Batchelor|1967|p=75}}</ref>

==Incompressible flow==
The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:<ref name="Batchelor_142_148">{{harvp|Batchelor|1967|pp=142–148}}</ref>

<ul>
<li>the stress is '''[[Galilean invariance|Galilean invariant]]''': it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient <math display="inline">\nabla \mathbf{u}</math>.</li>
<li>the fluid is assumed to be [[isotropic]], as with gases and simple liquids, and consequently <math display="inline">\boldsymbol{\tau}</math> is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the [[dynamic viscosity]] <math display="inline">\mu</math>:
{{Equation box 1
|indent=:
|title='''Stokes' stress [[constitutive equation]]''' ''(expression used for incompressible elastic solids)''
|equation=:<math>\boldsymbol \tau = 2 \mu \boldsymbol \varepsilon</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#DCDCDC
}}
where
<math display="block">\boldsymbol{\varepsilon} = \tfrac{1}{2} \left( \mathbf{\nabla u} + \mathbf{\nabla u}^\mathrm{T} \right)</math>
is the rate-of-[[strain tensor]]. So this decomposition can be made explicit as:<ref name=Batchelor_142_148/>
{{Equation box 1
|indent=:
|title='''Stokes's stress constitutive equation''' ''(expression used for incompressible viscous fluids)''
|equation=:<math>\boldsymbol \tau = \mu \left[\nabla\mathbf{u} +  (\nabla\mathbf{u}) ^\mathrm{T}\right]</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#DCDCDC
}}
</li>
</ul>

This is constitutive equation is also called the '''[[Newtonian fluid#Newtonian law of viscosity|Newtonian law of viscosity]]'''.
Dynamic viscosity {{mvar|μ}} need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these [[transport coefficient]] in the [[conservative variable]]s is called an [[equation of state]].<ref name="Batchelor 1967 p. 165">{{harvp|Batchelor|1967|p=165}}</ref>

The divergence of the deviatoric stress in case of uniform viscosity is given by:
<math display="block">\nabla \cdot \boldsymbol \tau = 2 \mu \nabla \cdot \boldsymbol \varepsilon = \mu \nabla \cdot \left( \nabla\mathbf{u} + \nabla\mathbf{u} ^\mathrm{T} \right) = \mu \, \nabla^2 \mathbf{u}</math>
because <math display="inline"> \nabla \cdot \mathbf{u} = 0</math> for an incompressible fluid.

Incompressibility rules out density and pressure waves like sound or [[shock wave]]s, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well with all fluids at low [[Mach number]]s (say up to about Mach 0.3), such as for modelling air winds at normal temperatures.<ref name="Ach">{{harvp|Acheson|1990}}</ref> the incompressible Navier–Stokes equations are best visualized by dividing for the density:<ref>{{Cite journal |last1=Abdulkadirov |first1=Ruslan |last2=Lyakhov |first2=Pavel |date=2022-02-22 |title=Estimates of Mild Solutions of Navier–Stokes Equations in Weak Herz-Type Besov–Morrey Spaces |journal=Mathematics |language=en |volume=10 |issue=5 |page=680 |doi=10.3390/math10050680 |issn=2227-7390|doi-access=free }}</ref>

{{Equation box 1
|indent=:
|title='''[[Incompressible flow|Incompressible]] Navier–Stokes equations''' with uniform viscosity (''convective form'')
|equation=<math>\frac{D \mathbf{u}}{D t} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = \nu \,\nabla^2 \mathbf{u} - \frac{1}{\rho}\nabla p + \frac{1}{\rho} \mathbf{f} </math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#DCDCDC
}}

where <math display="inline">\nu = \frac{\mu}{\rho}</math> is called the [[kinematic viscosity]]. 
By isolating the fluid velocity, one can also state:
{{Equation box 1
|indent=:
|title='''[[Incompressible flow|Incompressible]] Navier–Stokes equations''' with constant viscosity (''alternative convective form'')
|equation=<math> \left(\frac{\partial}{\partial t} + \mathbf{u} \cdot \nabla - \nu \,\nabla^2 \right) \mathbf{u} = - \frac{1}{\rho}\nabla p + \frac{1}{\rho} \mathbf{f}.</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#DCDCDC
}}

If the density is constant throughout the fluid domain, or, in other words, if all fluid elements have the same density, <math display="inline">\rho</math>, then we have

{{Equation box 1
|indent=:
|title='''[[Incompressible flow|Incompressible]] Navier–Stokes equations with constant density and viscosity''' (''convective form'')
|equation=<math>\frac{D \mathbf{u}}{D t} = \nu \,\nabla^2 \mathbf{u} - \nabla \frac{p}{\rho} + \frac{1}{\rho} \mathbf{f},</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#DCDCDC
}}
where <math display="inline">p/\rho</math> is called the unit [[pressure head]].

In incompressible flows, the pressure field satisfies the [[Poisson equation]],{{r|landau}}

:<math>\nabla^2 p = - \rho \frac{\partial u_i}{\partial x_k}\frac{\partial u_k}{\partial x_i} = - \rho \frac{\partial^2 u_iu_k}{\partial x_kx_i},</math>

which is obtained by taking the divergence of the momentum equations.

{{hidden
| A laminar flow example |
Velocity profile (laminar flow):
<math display="block">u_x = u(y), \quad u_y = 0, \quad u_{z} = 0</math>
for the {{mvar|x}}-direction, simplify the Navier–Stokes equation:
<math display="block">0 = -\frac{\mathrm{d} P}{\mathrm{d} x} + \mu\left(\frac{\mathrm{d}^2 u}{\mathrm{d} y^2}\right)</math>

Integrate twice to find the velocity profile with boundary conditions {{math|1=''y'' = ''h''}}, {{math|1=''u'' = 0}}, {{math|1=''y'' = −''h''}}, {{math|1=''u'' = 0}}:
<math display="block">u = \frac{1}{2\mu}\frac{\mathrm{d} P}{\mathrm{d} x} y^2 + Ay + B</math>

From this equation, substitute in the two boundary conditions to get two equations:
<math display="block">\begin{align}
0 &= \frac{1}{2 \mu}\frac{\mathrm{d} P}{\mathrm{d} x} h^2 + Ah + B \\
0 &= \frac{1}{2 \mu}\frac{\mathrm{d} P}{\mathrm{d} x} h^2 - Ah + B
\end{align}</math>

Add and solve for {{mvar|B}}:
<math display="block">B = -\frac{1}{2 \mu}\frac{\mathrm{d} P}{\mathrm{d} x} h^2</math>

Substitute and solve for {{mvar|A}}:
<math display="block">A = 0</math>

Finally this gives the velocity profile:
<math display="block">u = \frac{1}{2 \mu}\frac{\mathrm{d} P}{\mathrm{d} x} \left(y^2 - h^2\right)</math>
|style = border: 1px solid lightgray;
|headerstyle = text-align:left
}}

It is well worth observing the meaning of each term (compare to the [[Cauchy momentum equation]]):

<math display="block">\overbrace{ 
    \vphantom{\frac{}{}} 
    \underbrace{
        \frac{\partial \mathbf{u}}{\partial t}
    }_{\text{Variation}} 
    + 
    \underbrace{ 
        \vphantom{\frac{}{}} 
        (\mathbf{u} \cdot \nabla) \mathbf{u}
    }_{\begin{smallmatrix}
        \text{Convective}\\
        \text{acceleration}
    \end{smallmatrix}} 
}^{\text{Inertia (per volume)}} 
= 
\overbrace{ 
    \vphantom{\frac{\partial}{\partial}} 
    \underbrace{ 
        \vphantom{\frac{}{}} 
        -\nabla w
    }_{\begin{smallmatrix}
            \text{Internal}\\
            \text{source}
        \end{smallmatrix}
    } 
    + 
    \underbrace{ 
        \vphantom{\frac{}{}} 
        \nu \nabla^2 \mathbf{u}
    }_{\text{Diffusion}} 
}^{\text{Divergence of stress}} 
+ 
\underbrace{ 
    \vphantom{\frac{}{}} 
    \mathbf{g}
}_{\begin{smallmatrix}
        \text{External}\\
        \text{source}
    \end{smallmatrix}}.</math>

The higher-order term, namely the [[shear stress]] divergence <math display="inline">\nabla \cdot \boldsymbol{\tau}</math>, has simply reduced to the [[vector Laplacian]] term <math display="inline">\mu \nabla^2 \mathbf{u}</math>.<ref>{{harvp|Batchelor|1967|pp=21 & 147}}</ref> This Laplacian term can be interpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume. This implies that – for a Newtonian fluid – viscosity operates as a ''diffusion of momentum'', in much the same way as the [[heat conduction]]. In fact neglecting the convection term, incompressible Navier–Stokes equations lead to a vector [[diffusion equation]] (namely [[Stokes flow|Stokes equations]]), but in general the convection term is present, so incompressible Navier–Stokes equations belong to the class of [[convection–diffusion equation]]s.

In the usual case of an external field being a [[conservative field]]:
<math display="block"> \mathbf g = - \nabla \varphi </math>
by defining the [[hydraulic head]]:
<math display="block">h \equiv w + \varphi </math>

one can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field:
<math display="block">\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} - \nu \, \nabla^2 \mathbf{u} = - \nabla h.</math>

The incompressible Navier–Stokes equations with uniform density and viscosity and conservative external field is the '''fundamental equation of [[hydraulics]]'''. The domain for these equations is commonly a 3 or fewer dimensional [[Euclidean space]], for which an [[orthogonal coordinate]] reference frame is usually set to explicit the system of scalar partial differential equations to be solved. In 3-dimensional orthogonal coordinate systems are 3: [[Cartesian coordinate system|Cartesian]], [[Cylindrical coordinate system|cylindrical]], and [[Spherical coordinate system|spherical]]. Expressing the Navier–Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate systems. But for the higher order terms (the two coming from the divergence of the deviatoric stress that distinguish Navier–Stokes equations from Euler equations) some [[tensor calculus]] is required for deducing an expression in non-cartesian orthogonal coordinate systems.
A special case of the fundamental equation of hydraulics is the [[Bernoulli's equation]].

The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations,
<math display="block">\begin{align}
\frac{\partial\mathbf{u}}{\partial t} &= \Pi^S\left(-(\mathbf{u}\cdot\nabla)\mathbf{u} + \nu\,\nabla^2\mathbf{u}\right) + \mathbf{f}^S \\
\rho^{-1}\,\nabla p &= \Pi^I\left(-(\mathbf{u}\cdot\nabla)\mathbf{u} + \nu\,\nabla^2\mathbf{u}\right) + \mathbf{f}^I
\end{align}</math>
where <math display="inline">\Pi^S</math> and <math display="inline">\Pi^I</math> are solenoidal and [[Conservative vector field|irrotational]] projection operators satisfying <math display="inline">\Pi^S + \Pi^I = 1</math>, and <math display="inline">\mathbf{f}^S</math> and <math display="inline">\mathbf{f}^I</math> are the non-conservative and conservative parts of the body force. This result follows from the [[Helmholtz decomposition|Helmholtz theorem]] (also known as the fundamental theorem of vector calculus). The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to the pressure Poisson equation.

The explicit functional form of the projection operator in 3D is found from the Helmholtz Theorem:
<math display="block">\Pi^S\,\mathbf{F}(\mathbf{r}) = \frac{1}{4\pi}\nabla\times\int \frac{\nabla^\prime\times\mathbf{F}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d} V', \quad \Pi^I = 1-\Pi^S</math>
with a similar structure in 2D. Thus the governing equation is an [[integro-differential equation]] similar to [[Coulomb's law|Coulomb's]] and [[Biot–Savart law|Biot–Savart's law]], not convenient for numerical computation.

An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation,<ref>
{{citation
 | last = Temam | first = Roger
 | author-link = Roger Temam
 | title = Navier–Stokes Equations, Theory and Numerical Analysis
 | pages = 107–112
 | year = 2001 | publisher = AMS Chelsea
}}</ref> is given by,
<math display="block">\left(\mathbf{w},\frac{\partial\mathbf{u}}{\partial t}\right) = -\bigl(\mathbf{w}, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}\bigr) - \nu \left(\nabla\mathbf{w}: \nabla\mathbf{u}\right) + \left(\mathbf{w}, \mathbf{f}^S\right)</math>

for divergence-free test functions <math display="inline">\mathbf{w}</math> satisfying appropriate boundary conditions. Here, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces. The discrete form of this is eminently suited to finite element computation of divergence-free flow, as we shall see in the next section. There, one will be able to address the question, "How does one specify pressure-driven (Poiseuille) problems with a pressureless governing equation?".

The absence of pressure forces from the governing velocity equation demonstrates that the equation is not a dynamic one, but rather a kinematic equation where the divergence-free condition serves the role of a conservation equation. This would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition.

===Weak form of the incompressible Navier–Stokes equations===
====Strong form====
Consider the incompressible Navier–Stokes equations for a [[Newtonian fluid]] of constant density <math display="inline">\rho</math> in a domain
<math display="block"> \Omega \subset \mathbb R^d \quad (d=2, 3)</math>
with boundary
<math display="block"> \partial \Omega = \Gamma_D \cup \Gamma_N ,</math>
being <math display="inline">\Gamma_D</math> and <math display="inline">\Gamma_N</math> portions of the boundary where respectively a [[Dirichlet boundary condition|Dirichlet]] and a [[Neumann boundary condition]]  is applied (<math display="inline">\Gamma_D \cap \Gamma_N = \emptyset</math>):<ref name="Quarteroni">{{cite book |author-link1= Alfio Quarteroni |last1=Quarteroni |first1=Alfio |title=Numerical models for differential problems |publisher=Springer |isbn=978-88-470-5522-3 |edition=Second|date=2014-04-25 }}</ref>
<math display="block">
\begin{cases}
\rho \dfrac{\partial \mathbf{u}}{\partial t} + \rho (\mathbf{u} \cdot \nabla) \mathbf{u} - \nabla \cdot \boldsymbol{\sigma} (\mathbf{u}, p) = \mathbf{f} & \text{ in } \Omega \times (0, T) \\
\nabla \cdot \mathbf{u} = 0  & \text{ in } \Omega \times (0, T) \\
\mathbf{u} = \mathbf{g} & \text{ on } \Gamma_D \times (0, T) \\
 \boldsymbol{\sigma} (\mathbf{u}, p) \hat{\mathbf{n}} = \mathbf{h} & \text{ on } \Gamma_N \times (0, T) \\
\mathbf{u}(0)= \mathbf{u}_0 & \text{ in } \Omega \times \{ 0\}
\end{cases}
</math>
<math display="inline">\mathbf{u}</math> is the fluid velocity, <math display="inline">p</math> the fluid pressure, <math display="inline">\mathbf{f}</math> a given forcing term, <math>\hat{\mathbf{n}}</math> the outward directed unit normal vector to <math display="inline">\Gamma_N</math>, and <math display="inline">\boldsymbol{\sigma}(\mathbf{u}, p)</math> the [[viscous stress tensor]] defined as:<ref name="Quarteroni" /> 
<math display="block">\boldsymbol{\sigma} (\mathbf{u}, p) = -p \mathbf{I} + 2 \mu \boldsymbol{\varepsilon}(\mathbf{u}).</math>
Let <math display="inline">\mu</math> be the dynamic viscosity of the fluid, <math display="inline">\mathbf{I}</math> the second-order [[Identity matrix|identity tensor]] and <math display="inline">\boldsymbol{\varepsilon}(\mathbf{u})</math> the [[strain-rate tensor]] defined as:<ref name="Quarteroni" /> 
<math display="block">
\boldsymbol{\varepsilon} (\mathbf{u}) = \frac{1}{2} \left(\left( \nabla \mathbf{u} \right) + \left( \nabla \mathbf{u} \right)^\mathrm{T}\right).
</math>
The functions <math display="inline">
\mathbf{g}
</math> and <math display="inline">
\mathbf{h}
</math> are given Dirichlet and Neumann boundary data, while <math display="inline">
\mathbf{u}_0
</math> is the [[initial condition]]. The first equation is the momentum balance equation, while the second represents the [[conservation of mass|mass conservation]], namely the [[continuity equation]]. 
Assuming constant dynamic viscosity, using the vectorial identity
<math display="block"> \nabla \cdot \left( \nabla \mathbf{f} \right)^\mathrm{T} = \nabla ( \nabla \cdot \mathbf{f} )</math>
and exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as:<ref name="Quarteroni" /> 
<math display="block">
\begin{align}
\nabla \cdot \boldsymbol{\sigma} (\mathbf{u}, p)
& = \nabla \cdot \left(-p \mathbf{I} + 2 \mu \boldsymbol{\varepsilon}(\mathbf{u}) \right) \\
& = - \nabla p + 2 \mu \nabla \cdot \boldsymbol{\varepsilon}(\mathbf{u}) \\
& = - \nabla p + 2 \mu \nabla \cdot \left [ \tfrac{1}{2} \left(\left(\nabla \mathbf{u} \right) + \left(\nabla \mathbf{u} \right)^\mathrm{T}\right) \right] \\
& = -\nabla p + \mu \left(\Delta \mathbf{u} + \nabla \cdot \left(\nabla \mathbf{u} \right)^\mathrm{T} \right) \\
& =  -\nabla p + \mu \bigl( \Delta \mathbf{u} + \nabla  \underbrace{(\nabla \cdot \mathbf{u})}_{=0} \bigr)
= -\nabla p + \mu \, \Delta \mathbf{u}.
\end{align}
</math>
Moreover, note that the Neumann boundary conditions can be rearranged as:<ref name="Quarteroni" /> 
<math display="block">
 \boldsymbol{\sigma}(\mathbf{u}, p) \hat{\mathbf{n}} = \left(-p \mathbf{I} + 2 \mu \boldsymbol{\varepsilon}(\mathbf{u})\right)\hat{\mathbf{n}} = -p \hat{\mathbf{n}} + \mu \frac{\partial \boldsymbol u}{\partial \hat{\mathbf{n}}}.
</math>

====Weak form====
In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation<ref name="Quarteroni" /> 
<math display="block">
\rho \frac{\partial \mathbf{u}}{\partial t}  - \mu \Delta \mathbf{u} + \rho (\mathbf{u} \cdot \nabla) \mathbf{u} + \nabla p  = \mathbf{f} 
</math>
multiply it for a test function <math display="inline">
\mathbf{v}
</math>, defined in a suitable space <math display="inline">
V
</math>, and integrate both members with respect to the domain <math display="inline">
\Omega
</math>:<ref name="Quarteroni" />
<math display="block">
\int \limits_\Omega \rho \frac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v}  - \int \limits_\Omega \mu \Delta \mathbf{u} \cdot \mathbf{v} + \int \limits_\Omega \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v} + \int \limits_\Omega \nabla p \cdot \mathbf{v} = \int \limits_\Omega \mathbf{f} \cdot \mathbf{v}
</math>
Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem:<ref name="Quarteroni" /> 
<math display="block">\begin{align}
-\int \limits_\Omega \mu \Delta \mathbf{u} \cdot \mathbf{v} &= \int_\Omega \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} - \int \limits_{\partial \Omega} \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} \cdot \mathbf{v} \\
\int \limits_\Omega \nabla p \cdot \mathbf{v} &= -\int \limits_\Omega p \nabla \cdot \mathbf{v} + \int \limits_{\partial \Omega} p \mathbf{v} \cdot {\hat{\mathbf{n}}}
\end{align}</math>

Using these relations, one gets:<ref name="Quarteroni" /> 
<math display="block">
\int \limits_\Omega \rho \dfrac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v}
 + \int \limits_\Omega \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v}
 + \int \limits_\Omega \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v}
 - \int \limits_\Omega p \nabla \cdot \mathbf{v}
= \int \limits_\Omega \mathbf{f} \cdot \mathbf{v}
 + \int \limits_{\partial \Omega} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}}\right) \cdot \mathbf{v}
 \quad \forall \mathbf{v} \in V.
</math>
In the same fashion, the continuity equation is multiplied for a test function {{mvar|q}} belonging to a space <math display="inline">
Q
</math> and integrated in the domain <math display="inline">
\Omega
</math>:<ref name="Quarteroni" />
<math display="block">\int \limits_\Omega q \nabla \cdot \mathbf{u} = 0. \quad \forall q \in Q. </math>
The space functions are chosen as follows: 
<math display="block">\begin{align}
V = \left[H_0^1(\Omega) \right]^d &= \left\{ \mathbf{v} \in \left[H^1(\Omega)\right]^d: \quad \mathbf{v} = \mathbf{0} \text{ on } \Gamma_D \right\}, \\
Q &= L^2(\Omega)
\end{align}</math>
Considering that the test function {{math|'''''v'''''}} vanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as:<ref name="Quarteroni" /> 
<math display="block">
\int \limits_{\partial \Omega} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right) \cdot \mathbf{v} 
= 
\underbrace{
    \int \limits_{\Gamma_D} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right) \cdot \mathbf{v}
}_{ \mathbf{v} = \mathbf{0} \text{ on } \Gamma_D \ } 
+ 
\int \limits_{\Gamma_N} \underbrace{ \vphantom{\int \limits_{\Gamma_N} }
    \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right)
}_{= \mathbf{h} \text{ on } \Gamma_N} \cdot \mathbf{v}
= 
\int \limits_{\Gamma_N} \mathbf{h} \cdot \mathbf{v}.
</math>
Having this in mind, the weak formulation of the Navier–Stokes equations is expressed as:<ref name="Quarteroni" /> 
<math display="block">\begin{align}
&\text{find } \mathbf{u} \in L^2 \left(\mathbb R^+\; \left[H^1(\Omega)\right]^d\right) \cap C^0\left(\mathbb R^+ \; \left[L^2(\Omega)\right]^d\right) \text{ such that: } \\[5pt]
&\quad\begin{cases}
\displaystyle \int \limits_{\Omega}\rho \dfrac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v} + \int  \limits_{\Omega} \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} + \int \limits_{\Omega} \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v} - \int \limits_{\Omega} p \nabla \cdot \mathbf{v} = \int \limits_{\Omega}\mathbf{f} \cdot \mathbf{v} +  \int \limits_{\Gamma_N} \mathbf{h} \cdot \mathbf{v} \quad \forall \mathbf{v} \in V, \\
\displaystyle  \int \limits_{\Omega} q \nabla \cdot \mathbf{u} = 0 \quad \forall q \in Q. 
\end{cases}\end{align}
</math>

===Discrete velocity===
With partitioning of the problem domain and defining [[basis function]]s on the partitioned domain, the discrete form of the governing equation is
<math display="block">\left(\mathbf{w}_i, \frac{\partial\mathbf{u}_j}{\partial t}\right) = -\bigl(\mathbf{w}_i, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}_j\bigr) - \nu\left(\nabla\mathbf{w}_i: \nabla\mathbf{u}_j\right) + \left(\mathbf{w}_i, \mathbf{f}^S\right).</math>

It is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by [[Stokes' theorem]]. Discussion will be restricted to 2D in the following.

We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from the [[Bending of plates|plate-bending]] literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,
<math display="block">\begin{align}
\nabla\varphi &= \left(\frac{\partial \varphi}{\partial x},\,\frac{\partial \varphi}{\partial y}\right)^\mathrm{T}, \\[5pt]
\nabla\times\varphi &= \left(\frac{\partial \varphi}{\partial y},\,-\frac{\partial \varphi}{\partial x}\right)^\mathrm{T}.
\end{align}</math>

Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements.

Taking the curl of the scalar stream function elements gives divergence-free velocity elements.<ref>
{{Citation
 | last = Holdeman | first = J. T.
 | title = A Hermite finite element method for incompressible fluid flow
 | journal = Int. J. Numer. Methods Fluids
 | volume = 64 | pages = 376–408 | year = 2010
 | doi = 10.1002/fld.2154
 | issue = 4 |bibcode = 2010IJNMF..64..376H
| s2cid = 119882803
 }}</ref><ref>
{{Citation
 | last1 = Holdeman | first1 = J. T.
 | last2 = Kim | first2 = J. W.
 | title = Computation of incompressible thermal flows using Hermite finite elements
 | journal = Comput. Meth. Appl. Mech. Eng.
 | volume = 199 | pages = 3297–3304 | year = 2010
 | doi = 10.1016/j.cma.2010.06.036
 | issue = 49–52 |bibcode = 2010CMAME.199.3297H
}}</ref> The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces.

Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces.
Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may be used with some problems. These are all Dirichlet conditions.

The algebraic equations to be solved are simple to set up, but of course are [[#Nonlinearity|non-linear]], requiring iteration of the linearized equations.

Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists no simple relation between the gradient and the curl as was the case in 2D.

===Pressure recovery===
Recovering pressure from the velocity field is easy. The discrete weak equation for the pressure gradient is,
<math display="block">(\mathbf{g}_i, \nabla p) = -\left(\mathbf{g}_i, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}_j\right) - \nu\left(\nabla\mathbf{g}_i: \nabla\mathbf{u}_j\right) + \left(\mathbf{g}_i, \mathbf{f}^I\right)</math>

where the test/weight functions are irrotational. Any conforming scalar finite element may be used. However, the pressure gradient field may also be of interest. In this case, one can use scalar Hermite elements for the pressure. For the test/weight functions <math display="inline">\mathbf{g}_i
</math> one would choose the irrotational vector elements obtained from the gradient of the pressure element.

==Non-inertial frame of reference==

The rotating frame of reference introduces some interesting pseudo-forces into the equations through the [[material derivative]] term. Consider a stationary inertial frame of reference <math display="inline">K
</math>&nbsp;, and a non-inertial frame of reference <math display="inline">K'
</math>, which is translating with velocity <math display="inline">\mathbf{U}(t)
</math> and rotating with angular velocity <math display="inline">\Omega(t)
</math> with respect to the stationary frame. The Navier–Stokes equation observed from the non-inertial frame then becomes

{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation in non-inertial frame''' 
|equation=<math> \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = - \nabla p  + \nabla \cdot \left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\} + \nabla[\zeta (\nabla\cdot\mathbf{u})] + \rho\mathbf{f} - \rho \left[2\mathbf\Omega\times\mathbf u + \mathbf\Omega\times(\mathbf\Omega\times\mathbf x)+ \frac{\mathrm{d} \mathbf U}{\mathrm{d} t} + \frac{\mathrm{d} \mathbf \Omega}{\mathrm{d} t}\times\mathbf x\right].</math>
|cellpadding
|border
|border colour = #FF0000
|background colour = #DCDCDC
}}

Here <math display="inline">\mathbf{x}
</math> and <math display="inline">\mathbf{u}
</math> are measured in the non-inertial frame. The first term in the parenthesis represents [[Coriolis acceleration]], the second term is due to [[centrifugal force|centrifugal acceleration]], the third is due to the linear acceleration of <math display="inline">K'
</math> with respect to <math display="inline">K
</math> and the fourth term is due to the angular acceleration of <math display="inline">K'
</math> with respect to <math display="inline">K
</math>.

==Other equations==
The Navier–Stokes equations are strictly a statement of the balance of momentum. To fully describe fluid flow, more information is needed, how much depending on the assumptions made. This additional information may include boundary data ([[no-slip condition|no-slip]], [[capillary surface]], etc.), conservation of mass, [[First law of thermodynamics (fluid mechanics)|balance of energy]], and/or an [[equation of state]].

===Continuity equation for incompressible fluid===
{{Main|Continuity equation}}

Regardless of the flow assumptions, a statement of the [[conservation of mass]] is generally necessary. This is achieved through the mass [[continuity equation]], as discussed above in the "General continuum equations" within this article, as follows:
<math display="block">
\begin{align}
\frac{\mathbf{D}m}{{\mathbf{Dt}}}&={\iiint\limits_V}({\frac{\mathbf{D}\rho}{{\mathbf{Dt}}} + \rho (\nabla \cdot \mathbf{u})})dV \\
\frac{\mathbf{D}\rho}{{\mathbf{Dt}}} + \rho (\nabla \cdot{\mathbf{u}})&=\frac{\partial\rho}{\partial t} + ({\nabla \rho}) \cdot{\mathbf{u}} + {\rho}(\nabla \cdot \mathbf{u})= \frac{\partial\rho}{\partial t} + \nabla\cdot({\rho \mathbf{u}})= 0 
\end{align}</math>
A fluid media for which the [[density]] (<math>\rho</math>) is constant is called [[Incompressible flow|''incompressible'']]. Therefore, the rate of change of [[density]] (<math>\rho</math>) with respect to time <math>(\frac{\partial\rho}{\partial t})</math> and the [[gradient]] of density <math>(\nabla \rho)</math> are equal to zero <math>(0)</math>. In this case the general equation of continuity, <math>\frac{\partial\rho}{\partial t} + \nabla\cdot({\rho \mathbf{u}})= 0</math>, reduces to:  <math>\rho(\nabla{\cdot}{\mathbf{u}}) = 0</math>. Furthermore, assuming that [[density]] (<math>\rho</math>) is a non-zero constant <math>(\rho \neq 0)</math> means that the right-hand side of the equation <math>(0)</math> is divisible by [[density]] (<math>\rho</math>). Therefore, the continuity equation for an [[Incompressible flow|incompressible fluid]] reduces further to:<big><math display="block">(\nabla{\cdot{\mathbf{u}}}) = 0</math></big>This relationship,  <small><math display="inline">(\nabla{\cdot{\mathbf{u}}}) = 0</math></small>,  identifies that the [[divergence]] of the flow velocity [[Vector field|vector]] (<math>\mathbf{u}</math>) is equal to zero <math>(0)</math>, which means that for an [[Incompressible flow|incompressible fluid]] the [[Flow velocity|flow velocity field]] is a [[solenoidal vector field]] or a [[Divergence-free|divergence-free vector field]]. Note that this relationship can be expanded upon due to its uniqueness with the [[vector Laplace operator]] <math>(\nabla ^{2} \mathbf{u} =\nabla (\nabla \cdot \mathbf{u} )-\nabla \times (\nabla \times \mathbf{u} ))</math>, and [[vorticity]] <math>(\vec \omega = \nabla \times \mathbf{u})</math> which is now expressed like so, for an   [[Incompressible flow|incompressible fluid]]:<math display="block">\nabla ^{2}\mathbf {u} = - (\nabla \times (\nabla \times \mathbf {u} )) = - (\nabla \times \vec \omega)</math>

==Stream function for incompressible 2D fluid==
Taking the [[Curl (mathematics)|curl]] of the incompressible Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like in the degenerate 3D case with <math display="inline">u_z = 0
</math> and no dependence of anything on <math display="inline">z
</math>), where the equations reduce to:
<math display="block">\begin{align}
 \rho \left(\frac{\partial u_x}{\partial t} + u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y}\right)
 &= -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2}\right) + \rho g_x \\
 \rho \left(\frac{\partial u_y}{\partial t} + u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y}\right)
 &= -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2}\right) + \rho g_y.
\end{align}</math>

Differentiating the first with respect to <math display="inline">y
</math>, the second with respect to <math display="inline">x
</math> and subtracting the resulting equations will eliminate pressure and any [[conservative force]]. 
For incompressible flow, defining the [[stream function]] <math display="inline">\psi
</math> through
<math display="block">u_x = \frac{\partial \psi}{\partial y}; \quad u_y = -\frac{\partial \psi}{\partial x}</math>
results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation condense into one equation:
<math display="block">\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial \psi}{\partial y} \frac{\partial}{\partial x}\left(\nabla^2 \psi\right) - \frac{\partial \psi}{\partial x} \frac{\partial}{\partial y}\left(\nabla^2 \psi\right) = \nu \nabla^4 \psi</math>

where <math display="inline">\nabla^4
</math> is the 2D [[biharmonic operator]] and <math display="inline">\nu
</math> is the [[kinematic viscosity]], <math display="inline">\nu = \frac{\mu}{\rho}
</math>. We can also express this compactly using the [[Jacobian matrix and determinant|Jacobian determinant]]:
<math display="block">\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial\left(\psi, \nabla^2\psi \right)}{\partial(y,x)} = \nu \nabla^4 \psi.</math>

This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for [[creeping flow]] results when the left side is assumed zero.

In [[axisymmetric]] flow another stream function formulation, called the [[Stokes stream function]], can be used to describe the velocity components of an incompressible flow with one [[scalar (mathematics)|scalar]] function.

The incompressible Navier–Stokes equation is a [[differential algebraic equation]], having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. The stream function formulation eliminates the pressure but only in two dimensions and at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest.

==Properties==
===Nonlinearity===

The Navier–Stokes equations are [[Nonlinearity|nonlinear]] [[partial differential equations]] in the general case and so remain in almost every real situation.<ref>{{cite book |title=Fluid Mechanics |first1=M. |last1=Potter |first2=D. C. |last2=Wiggert |series=Schaum's Outlines |publisher=McGraw-Hill |year=2008 |isbn=978-0-07-148781-8}}</ref><ref>{{cite book |title=Vectors, Tensors, and the basic Equations of Fluid Mechanics |first=R. |last=Aris |publisher=Dover Publications |year=1989 |isbn=0-486-66110-5 }}</ref> In some cases, such as one-dimensional flow and [[Stokes flow]] (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the [[turbulence]] that the equations model.

The nonlinearity is due to [[convective]] acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but [[laminar flow|laminar]] (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging [[nozzle]]. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.<ref>{{cite book |title=McGraw Hill Encyclopaedia of Physics |edition=2nd |first=C. B. |last=Parker |year=1994 |publisher=McGraw-Hill |isbn=0-07-051400-3 }}</ref>

===Turbulence===

[[Turbulence]] is the time-dependent [[Chaos theory|chaotic]] behaviour seen in many fluid flows. It is generally believed that it is due to the [[inertia]] of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the [[Reynolds number]] quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.<ref>Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, {{ISBN|3-527-26954-1}} (Verlagsgesellschaft), {{ISBN|0-89573-752-3}} (VHC Inc.)</ref>

The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or [[direct numerical simulation]]. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the [[Reynolds-averaged Navier–Stokes equations]] (RANS), supplemented with turbulence models, are used in practical [[computational fluid dynamics]] (CFD) applications when modeling turbulent flows. Some models include the [[Spalart–Allmaras turbulence model|Spalart–Allmaras]], [[k-omega turbulence model|{{mvar|k}}–{{mvar|ω}}]], [[turbulence kinetic energy|{{mvar|k}}–{{mvar|ε}}]], and [[SST (Menter's Shear Stress Transport)|SST]] models, which add a variety of additional equations to bring closure to the RANS equations. [[Large eddy simulation]] (LES) can also be used to solve these equations numerically. This approach is computationally more expensive—in time and in computer memory—than RANS, but produces better results because it explicitly resolves the larger turbulent scales.

===Applicability===
{{Further|Discretization of Navier–Stokes equations}}
Together with supplemental equations (for example, conservation of mass) and well-formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations.

The Navier–Stokes equations assume that the fluid being studied is a [[Continuum mechanics|continuum]] (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at [[Relativistic velocity|relativistic velocities]]. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. For example, [[capillarity]] of internal layers in fluids appears for flow with high gradients.<ref>{{Citation
| last1    = Gorban
| first1   = A.N.
| last2   = Karlin
| first2  = I. V.
| year     = 2016
| title    = Beyond Navier–Stokes equations: capillarity of ideal gas
| type    = Review article
| journal = Contemporary Physics
| volume   = 58
| issue   = 1
| pages   = 70–90
| doi = 10.1080/00107514.2016.1256123 
| url = https://www.researchgate.net/publication/310825466 
| arxiv= 1702.00831
| bibcode= 2017ConPh..58...70G| s2cid = 55317543
}}</ref>  For large [[Knudsen number]] of the problem, the [[Boltzmann equation]] may be a suitable replacement.<ref>{{Citation
| last    = Cercignani
| first   = C.
| year     = 2002
| title    = Handbook of mathematical fluid dynamics 
| chapter  = The Boltzmann equation and fluid dynamics 
| volume  = 1
| publisher = North-Holland
| place    = Amsterdam
| pages = 1–70
| editor-last   = Friedlander
| editor-first  = S.
| editor2-last  = Serre
| editor2-first = D.
| isbn = 978-0-444-50330-5
}}</ref> 
Failing that, one may have to resort to [[molecular dynamics]] or various hybrid methods.<ref>{{Citation
| last1    = Nie
| first1   = X.B.
| last2   = Chen
| first2  = S.Y.
| last3   = Robbins
| first3  = M.O.
| year     = 2004
| title    =  A continuum and molecular dynamics hybrid method for micro-and nano-fluid flow
| type    = Research article
| journal = Journal of Fluid Mechanics
| volume  = 500
| pages = 55–64 
| doi = 10.1017/S0022112003007225
| url = https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/a-continuum-and-molecular-dynamics-hybrid-method-for-micro-and-nano-fluid-flow/BE0D4513A0F90F844CD21D64F6D3F9EF
| bibcode= 2004JFM...500...55N
| s2cid = 122867563
| url-access= subscription
}}</ref>

Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this reason, these equations are usually written for [[Newtonian fluid]]s where the viscosity model is [[linear]]; truly general models for the flow of other kinds of fluids (such as blood) do not exist.<ref>{{Citation
| last    = Öttinger 
| first   = H.C.
| year     = 2012
| title    = Stochastic processes in polymeric fluids
| publisher = Springer Science & Business Media
| place    = Berlin, Heidelberg
| isbn = 978-3-540-58353-0
| doi = 10.1007/978-3-642-58290-5
}}</ref>

==Application to specific problems==
The Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as [[multiphase flow]] driven by [[surface tension]].

Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by [[Scale analysis (mathematics)|scale analysis]] to further simplify the problem.

[[Image:NSConvection vectorial.svg|thumb|400px|Visualization of '''(a)''' parallel flow and '''(b)''' radial flow]]

===Parallel flow===
Assume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) [[boundary value problem]] is:
<math display="block">\frac{\mathrm{d}^2 u}{\mathrm{d} y^2} = -1; \quad u(0) = u(1) = 0.</math>

The boundary condition is the [[no slip condition]]. This problem is easily solved for the flow field:
<math display="block">u(y) = \frac{y - y^2}{2}.</math>

From this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.

===Radial flow===
Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the ''radial'' flow between parallel plates; this involves convection and thus non-linearity. The velocity field may be represented by a function {{math|''f''(''z'')}} that must satisfy:
<math display="block">\frac{\mathrm{d}^2 f}{\mathrm{d} z^2} + R f^2 = -1; \quad f(-1) = f(1) = 0.</math>

This [[ordinary differential equation]] is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The [[Nonlinearity|nonlinear]] term makes this a very difficult problem to solve analytically (a lengthy [[Implicit function|implicit]] solution may be found which involves [[elliptic integral]]s and [[Cubic formula|roots of cubic polynomials]]). Issues with the actual existence of solutions arise for <math display="inline">R > 1.41
</math> (approximately; this is not [[square root of 2|{{sqrt|2}}]]), the parameter <math display="inline">R
</math> being the Reynolds number with appropriately chosen scales.<ref name="TM Shah">{{Cite journal|last=Shah |first=Tasneem Mohammad|title=Analysis of the multigrid method|journal=NASA Sti/Recon Technical Report N|volume=91|page=23418|bibcode=1989STIN...9123418S|year=1972}}</ref> This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.<ref name="TM Shah"/>

===Convection===
A type of natural convection that can be described by the Navier–Stokes equation is the [[Rayleigh–Bénard convection]]. It is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility.

==Exact solutions of the Navier–Stokes equations==
Some exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases—with the non-linear terms in the Navier–Stokes equations equal to zero—are [[Hagen-Poiseuille equation|Poiseuille flow]], [[Couette flow]] and the oscillatory [[Stokes boundary layer]]. But also, more interesting examples, solutions to the full non-linear equations, exist, such as [[Jeffery–Hamel flow]], [[Von Kármán swirling flow]], [[stagnation point flow]], [[Landau–Squire jet]], and [[Taylor–Green vortex]].<ref>
{{citation
| journal=Annual Review of Fluid Mechanics
| volume=23
| pages=159–177
| year=1991
| doi=10.1146/annurev.fl.23.010191.001111
| title=Exact solutions of the steady-state Navier–Stokes equations
| first=C. Y.
| last=Wang
|bibcode = 1991AnRFM..23..159W
}}</ref><ref>{{harvp|Landau|Lifshitz|1987|pp=75–88}}</ref><ref>
{{citation
 | last1=Ethier
 | first1=C. R.
 | last2=Steinman
 | first2=D. A.
 | title = Exact fully 3D Navier–Stokes solutions for benchmarking
 | journal=International Journal for Numerical Methods in Fluids
 | year=1994
 | volume=19
 | issue=5
 | pages=369–375
 | doi=10.1002/fld.1650190502
 |bibcode = 1994IJNMF..19..369E
}}</ref> Time-dependent [[Self-similar solution|self-similar]] solutions of the three-dimensional non-compressible Navier–Stokes equations in Cartesian coordinate can be given with the help of the [[Kummer's function]]s with quadratic arguments.<ref>{{Cite journal |last=Barna |first=I.F. |date=2011 |title=Self-Similar Solutions of Three-Dimensional Navier–Stokes Equation |url=https://iopscience.iop.org/article/10.1088/0253-6102/56/4/25 |journal=Communications in Theoretical Physics |volume=56 |issue=4 |pages=745–750 |arxiv=1102.5504 |doi=10.1088/0253-6102/56/4/25|bibcode=2011CoTPh..56..745I }}</ref> For the compressible Navier–Stokes equations the time-dependent self-similar solutions are however the [[Whittaker function]]s again with quadratic arguments when the [[Polytrope|polytropic]] [[equation of state]] is used as a closing condition.<ref>{{Cite journal |last1=Barna |first1=I.F. |last2=Mátyás |first2=L. |date=2014 |title=Analytic solutions for the three-dimensional compressible Navier-Stokes equation |url=https://iopscience.iop.org/article/10.1088/0169-5983/46/5/055508 |journal=Fluid Dynamics Research |volume=46 |issue=5 |article-number=055508 |arxiv=1309.0703 |doi=10.1088/0169-5983/46/5/055508|bibcode=2014FlDyR..46e5508B }}</ref> Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.

Under additional assumptions, the component parts can be separated.<ref>{{Cite web |title=Navier Stokes Equations |url=http://www.claudino.webs.com/Navier%20Stokes%20Equations.pps |archive-url=https://web.archive.org/web/20150619215817/http://www.claudino.webs.com/Navier%20Stokes%20Equations.pps |archive-date=2015-06-19 |access-date=2023-03-11 |website=www.claudino.webs.com}}</ref>

{{hidden
|A two-dimensional example
|For example, in the case of an unbounded planar domain with '''two-dimensional''' — incompressible and stationary — flow in [[polar coordinates]] {{math|(''r'',''φ'')}}, the velocity components {{math|(''u<sub>r</sub>'',''u<sub>φ</sub>'')}} and pressure {{mvar|p}} are:<ref>{{citation
| first=O. A.
| last= Ladyzhenskaya
| year=1969
| title=The Mathematical Theory of viscous Incompressible Flow
| edition=2nd
| page=preface, xi
}}</ref>
<math display="block">\begin{align}
u_r &= \frac{A}{r}, \\
u_\varphi &= B\left(\frac{1}{r} - r^{\frac{A}{\nu} + 1}\right), \\
p &= -\frac{A^2 + B^2}{2r^2} - \frac{2B^2 \nu r^\frac{A}{\nu}}{A} + \frac{B^2 r^\left(\frac{2A}{\nu} + 2\right)}{\frac{2A}{\nu} + 2}
\end{align}</math>

where {{mvar|A}} and {{mvar|B}} are arbitrary constants. This solution is valid in the domain {{math|''r'' ≥ 1}} and for {{math|''A'' < −2''ν''}}.

In Cartesian coordinates, when the viscosity is zero ({{math|1=''ν'' = 0}}), this is:
<math display="block">\begin{align}
\mathbf{v}(x,y) &= \frac{1}{x^2 + y^2}\begin{pmatrix} Ax + By \\ Ay - Bx \end{pmatrix}, \\
p(x,y) &= -\frac{A^2 + B^2}{2\left(x^2 + y^2\right)} \end{align}</math>

|style = border: 1px solid lightgray; width: 90%;
|headerstyle = text-align:left
}}

{{hidden
|A three-dimensional example
|For example, in the case of an unbounded Euclidean domain with '''three-dimensional''' — incompressible, stationary and with zero viscosity ({{math|1=''ν'' = 0}}) — radial flow in [[Cartesian coordinates]] {{math|(''x'',''y'',''z'')}}, the velocity vector {{math|'''v'''}} and pressure {{mvar|p}} are:{{citation needed|date=January 2014}}
<math display="block">\begin{align}
\mathbf{v}(x, y, z) &= \frac{A}{x^2 + y^2 + z^2}\begin{pmatrix} x \\ y\\ z \end{pmatrix}, \\
p(x, y, z) &= -\frac{A^2}{2\left(x^2 + y^2 + z^2\right)}. \end{align}</math>

There is a singularity at {{math|1=''x'' = ''y'' = ''z'' = 0}}.

|style = border: 1px solid lightgray; width: 90%;
|headerstyle = text-align:left
}}

===A three-dimensional steady-state vortex solution===
[[Image:Hopfkeyrings.jpg|right|250px|thumb|Wire model of flow lines along a [[Hopf fibration]]]]
A steady-state example with no singularities comes from considering the flow along the lines of a [[Hopf fibration]]. Let <math display="inline">r
</math> be a constant radius of the inner coil. One set of solutions is given by:<ref>{{citation
| url= http://www.jetp.ac.ru/cgi-bin/dn/e_055_01_0069.pdf |archive-url=https://web.archive.org/web/20160128200456/http://www.jetp.ac.ru/cgi-bin/dn/e_055_01_0069.pdf |archive-date=2016-01-28 |url-status=live
| year=1982
| title=Topological solitons in magnetohydrodynamics
| first=A. M.
| last= Kamchatno
|journal=Soviet Journal of Experimental and Theoretical Physics |volume=55 |issue=1 |page=69 |bibcode=1982JETP...55...69K }}</ref>
<math display="block">\begin{align}
\rho(x, y, z) &= \frac{3B}{r^2 + x^2 + y^2 + z^2} \\
p(x, y, z) &= \frac{-A^2B}{\left(r^2 + x^2 + y^2 + z^2\right)^3} \\
\mathbf{u}(x, y, z) &= \frac{A}{\left(r^2 + x^2 + y^2 + z^2\right)^2}\begin{pmatrix} 2(-ry + xz) \\ 2(rx + yz) \\ r^2 - x^2 - y^2 + z^2 \end{pmatrix} \\
g &= 0 \\
\mu &= 0
\end{align}</math>

for arbitrary constants <math display="inline">A
</math> and <math display="inline">B
</math>. This is a solution in a non-viscous gas (compressible fluid) whose density, velocities and pressure goes to zero far from the origin. (Note this is not a solution to the Clay Millennium problem because that refers to incompressible fluids where <math display="inline">\rho
</math> is a constant, and neither does it deal with the uniqueness of the Navier–Stokes equations with respect to any [[turbulence]] properties.) It is also worth pointing out that the components of the velocity vector are exactly those from the [[Pythagorean quadruple]] parametrization. Other choices of density and pressure are possible with the same velocity field:

{{hidden
|Other choices of density and pressure
|Another choice of pressure and density with the same velocity vector above is one where the pressure and density fall to zero at the origin and are highest in the central loop at {{math|1=''z'' = 0}}, {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = ''r''<sup>2</sup>}}:
<math display="block">\begin{align}
\rho(x, y, z) &= \frac{20B\left(x^2 + y^2\right)}{\left(r^2 + x^2 + y^2 + z^2\right)^3} \\
p(x, y, z) &= \frac{-A^2B}{\left(r^2 + x^2 + y^2 + z^2\right)^4} + \frac{-4A^2B\left(x^2 + y^2\right)}{\left(r^2 + x^2 + y^2 + z^2\right)^5}.
\end{align}</math>

In fact in general there are simple solutions for any polynomial function {{mvar|f}} where the density is:
<math display="block">\rho(x, y, z) = \frac{1}{r^2 + x^2 + y^2 + z^2} f\left(\frac{x^2 + y^2}{\left(r^2 + x^2 + y^2 + z^2\right)^2}\right).</math>

|style = border: 1px solid lightgray; width: 90%;
|headerstyle = text-align:left
}}

===Viscous three-dimensional periodic solutions===
Two examples of periodic fully-three-dimensional viscous solutions are described in.<ref>{{citation
| title=Tri-periodic fully three-dimensional analytic solutions for the Navier–Stokes equations
| first=M.
| last= Antuono
| journal=Journal of Fluid Mechanics
| year=2020
| volume=890
| article-number=A23
| doi=10.1017/jfm.2020.126
| bibcode=2020JFM...890A..23A
| s2cid=216463266
}}</ref>
These solutions are defined on a three-dimensional [[torus]] <math> \mathbb{T}^3 = [0, L]^3 </math> and are characterized by positive and negative [[hydrodynamical helicity|helicity]] respectively.
The solution with positive helicity is given by:
<math display="block">\begin{align}
u_x &= \frac{4 \sqrt{2}}{3 \sqrt{3}} \, U_0 \left[\, \sin(k x - \pi/3) \cos(k y + \pi/3) \sin(k z + \pi/2) - \cos(k z - \pi/3) \sin(k x + \pi/3) \sin(k y + \pi/2) \,\right]  e^{-3 \nu k^2 t} \\
u_y &= \frac{4 \sqrt{2}}{3 \sqrt{3}} \, U_0 \left[\, \sin(k y - \pi/3) \cos(k z + \pi/3) \sin(k x + \pi/2) - \cos(k x - \pi/3) \sin(k y + \pi/3) \sin(k z + \pi/2) \,\right]  e^{-3 \nu k^2 t} \\
u_z &= \frac{4 \sqrt{2}}{3 \sqrt{3}} \, U_0 \left[\, \sin(k z - \pi/3) \cos(k x + \pi/3) \sin(k y + \pi/2) - \cos(k y - \pi/3) \sin(k z + \pi/3) \sin(k x + \pi/2) \,\right]  e^{-3 \nu k^2 t}
\end{align}</math>
where <math>k = 2 \pi/L</math> is the wave number and the velocity components are normalized so that the average kinetic energy per unit of mass is <math>U_0^2/2</math> at <math> t = 0 </math>.
The pressure field is obtained from the velocity field as  <math> p = p_0 - \rho_0 \| \boldsymbol{u} \|^2/2</math> (where <math>p_0</math> and <math>\rho_0</math> are reference values for the pressure and density fields respectively).
Since both the solutions belong to the class of [[Beltrami flow]], the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by <math>\omega =\sqrt{3} \, k \, \boldsymbol{u}</math>. 
These solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor-Green [[Taylor–Green vortex]].

==Wyld diagrams==<!-- [[Wyld diagrams]] redirects here-->
'''Wyld diagrams''' are bookkeeping [[Graph (discrete mathematics)|graphs]] that correspond to the Navier–Stokes equations via a [[perturbation theory|perturbation expansion]] of the fundamental [[continuum mechanics]]. Similar to the [[Feynman diagram]]s in [[quantum field theory]], these diagrams are an extension of [[Mstislav Keldysh]]'s technique for nonequilibrium processes in fluid dynamics.{{cn|date=March 2025}} In other words, these diagrams assign [[graph theory|graphs]] to the (often) [[turbulence|turbulent]] phenomena in turbulent fluids by allowing [[correlation function|correlated]] and interacting fluid particles to obey [[stochastic processes]] associated to [[pseudo-random]] [[function (mathematics)|functions]] in [[probability distribution]]s.<ref>{{citation | title=Renormalization methods: A guide for beginners | first=W. D. | last=McComb | publisher=Oxford University Press | year=2008 | isbn=978-0-19-923652-7 | pages = 121–128}}</ref>

==Representations in 3D==

Note that the formulas in this section make use of the single-line notation for partial derivatives, where, e.g. <math display="inline">\partial_x u</math> means the partial derivative of <math display="inline">u</math> with respect to <math display="inline">x</math>, and <math display="inline">\partial_y^2 f_\theta</math> means the second-order partial derivative of <math display="inline">f_\theta</math> with respect to <math display="inline">y</math>.

A 2022 paper provides a less costly, dynamical and recurrent solution of the Navier-Stokes equation for 3D turbulent fluid flows. On suitably short time scales, the dynamics of turbulence is deterministic.<ref>{{cite journal|url=https://phys.org/news/2022-08-physicists-uncover-dynamical-framework-turbulence.amp|title=Physicists uncover new dynamical framework for turbulence|date=August 29, 2022|author=[[Georgia Institute of Technology]]|journal=Proceedings of the National Academy of Sciences of the United States of America |volume=119 |issue=34 |article-number=e2120665119 |publisher=[[Phys.org]]|doi=10.1073/pnas.2120665119|doi-access=free |pmid=35984901 |pmc=9407532 |s2cid=251693676 }}</ref>

===Cartesian coordinates===
From the general form of the Navier–Stokes, with the velocity vector expanded as <math display="inline">\mathbf{u} = (u_x, u_y, u_z)
</math>, sometimes respectively named <math display="inline">u</math>, <math display="inline">v</math>, <math display="inline">w</math>, we may write the vector equation explicitly,
<math display="block">\begin{align}
 x:\ &\rho \left({\partial_t u_x} + u_x \, {\partial_x u_x} + u_y \, {\partial_y u_x} + u_z \, {\partial_z u_x}\right) \\
 &\quad= -\partial_x p + \mu \left({\partial_x^2 u_x} + {\partial_y^2 u_x} + {\partial_z^2 u_x}\right) + \frac{1}{3} \mu \  \partial_x \left( {\partial_x u_x} + {\partial_y u_y} + {\partial_z u_z} \right) + \rho g_x \\
\end{align}</math>
<math display="block">\begin{align}
 y:\ &\rho \left({\partial_t u_y} + u_x {\partial_x u_y} + u_y {\partial_y u_y} + u_z {\partial_z u_y}\right) \\
 &\quad= -{\partial_y p} + \mu \left({\partial_x^2 u_y} + {\partial_y^2 u_y} + {\partial_z^2 u_y}\right) + \frac{1}{3} \mu \  \partial_y \left( {\partial_x u_x} + {\partial_y u_y} + {\partial_z u_z} \right) + \rho g_y \\
\end{align}</math>
<math display="block">\begin{align}
 z:\ &\rho \left({\partial_t u_z} + u_x {\partial_x u_z} + u_y {\partial_y u_z} + u_z {\partial_z u_z}\right) \\
 &\quad= -{\partial_z p} + \mu \left({\partial_x^2 u_z} + {\partial_y^2 u_z} + {\partial_z^2 u_z}\right) + \frac{1}{3} \mu \  \partial_z \left( {\partial_x u_x} + {\partial_y u_y} + {\partial_z u_z} \right) + \rho g_z.
\end{align}</math>

Note that gravity has been accounted for as a body force, and the values of <math display="inline">g_x</math>, <math display="inline">g_y</math>, <math display="inline">g_z</math> will depend on the orientation of gravity with respect to the chosen set of coordinates.

The continuity equation reads:
<math display="block">\partial_t \rho + \partial_x (\rho u_x) + \partial_y (\rho u_y) + \partial_z (\rho u_z) = 0.</math>

When the flow is incompressible, <math display="inline">\rho</math> does not change for any fluid particle, and its [[material derivative]] vanishes: <math display="inline">\frac{\mathrm{D} \rho}{\mathrm{D}t} = 0</math>. The continuity equation is reduced to:
<math display="block">\partial_x u_x + \partial_y u_y + \partial_z u_z = 0.</math>

Thus, for the incompressible version of the Navier–Stokes equation the second part of the viscous terms fall away (see [[Incompressible flow]]).

This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other representations, this is still a [[Nonlinearity|nonlinear]] system of [[partial differential equations]] for which solutions are difficult to obtain.

===Cylindrical coordinates===
A change of variables on the Cartesian equations will yield<ref name="Ach"/> the following momentum equations for <math display="inline">r</math>, <math display="inline">\phi</math>, and <math display="inline">z</math><ref>{{Citation| first=Mattia| last=de' Michieli Vitturi| title = Navier–Stokes equations in cylindrical coordinates| url = https://demichie.github.io/NS_cylindrical| access-date = 2016-12-26}}</ref>
<math display="block">\begin{align}
 r:\ & \rho \left({\partial_t u_r} + u_r {\partial_r u_r} + \frac{u_\varphi}{r} {\partial_\varphi u_r} + u_z {\partial_z u_r} - \frac{u_\varphi^2}{r}\right) \\
 &\quad = -{\partial_r p} \\
 &\qquad  + \mu \left(\frac{1}{r} \partial_r \left(r {\partial_r u_r}\right) +
                      \frac{1}{r^2} {\partial_\varphi^2 u_r} + {\partial_z^2 u_r} - \frac{u_r}{r^2} -
                      \frac{2}{r^2} {\partial_\varphi u_\varphi} \right) \\
 &\qquad  + \frac{1}{3}\mu \partial_r \left( \frac{1}{r} {\partial_r\left(r u_r\right)} + \frac{1}{r} {\partial_\varphi u_\varphi} + {\partial_z u_z} \right) \\
 &\qquad  + \rho g_r \\[8px]
\end{align}</math>
<math display="block">\begin{align}
 \varphi:\ & \rho \left({\partial_t u_\varphi} + u_r {\partial_r u_\varphi} +
 \frac{u_\varphi}{r} {\partial_\varphi u_\varphi} + u_z {\partial_z u_\varphi} + \frac{u_r u_\varphi}{r} \right) \\
 &\quad = -\frac{1}{r} {\partial_\varphi p} \\
 &\qquad  + \mu \left(\frac{1}{r} \  \partial_r \left(r {\partial_r u_\varphi}\right)
                    + \frac{1}{r^2} {\partial_\varphi^2 u_{\varphi}}
                    + {\partial_z^2 u_{\varphi}} - \frac{u_\varphi}{r^2} + \frac{2}{r^2} {\partial_\varphi u_r}\right) \\
 &\qquad  + \frac{1}{3}\mu \frac{1}{r} \partial_\varphi \left( \frac{1}{r} {\partial_r\left(r u_r\right)} + \frac{1}{r} {\partial_\varphi u_\varphi} + {\partial_z u_z} \right) \\
 &\qquad  + \rho g_\varphi \\[8px]
\end{align}</math>
<math display="block">\begin{align}
 z:\ & \rho \left({\partial_t u_z} + u_r {\partial_r u_z} + \frac{u_\varphi}{r} {\partial_\varphi u_z} +
 u_z {\partial_z u_z}\right) \\
 &\quad = -{\partial_z p} \\
 &\qquad  + \mu \left(\frac{1}{r} \partial_r \left(r {\partial_r u_z}\right)
                    + \frac{1}{r^2} {\partial_\varphi^2 u_z} + {\partial_z^2 u_z}\right) \\
 &\qquad  + \frac{1}{3}\mu \partial_z \left( \frac{1}{r} {\partial_r \left(r u_r\right)}
                                           + \frac{1}{r} {\partial_\varphi u_\varphi} + {\partial_z u_z} \right) \\
 &\qquad  + \rho g_z.
\end{align}</math>

The gravity components will generally not be constants, however for most applications either the coordinates are chosen so that the gravity components are constant or else it is assumed that gravity is counteracted by a pressure field (for example, flow in horizontal pipe is treated normally without gravity and without a vertical pressure gradient). The continuity equation is:
<math display="block">{\partial_t\rho} + \frac{1}{r} \partial_r \left(\rho r u_r\right) + \frac{1}{r} {\partial_\varphi \left(\rho u_\varphi\right)} + {\partial_z \left(\rho u_z\right)} = 0.
</math>

This cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above). Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. A very common case is axisymmetric flow with the assumption of no tangential velocity (<math display="inline">u_\phi = 0</math>), and the remaining quantities are independent of <math display="inline">\phi</math>:
<math display="block">\begin{align}
 \rho \left({\partial_t u_r} + u_r {\partial_r u_r} + u_z {\partial_z u_r}\right)
 &= -{\partial_r p} + \mu \left(\frac{1}{r} \partial_r \left(r {\partial_r u_r}\right) +
 {\partial_z^2 u_r} - \frac{u_r}{r^2}\right) + \rho g_r \\
 \rho \left({\partial_t u_z} + u_r {\partial_r u_z} + u_z {\partial_z u_z}\right)
 &= -{\partial_z p} + \mu \left(\frac{1}{r} \partial_r \left(r {\partial_r u_z}\right) +
 {\partial_z^2 u_z}\right) + \rho g_z \\
 \frac{1}{r} \partial_r\left(r u_r\right) + {\partial_z u_z} &= 0.
\end{align}</math>

===Spherical coordinates===
In [[spherical coordinates]], the  <math display="inline">r</math>, <math display="inline">\phi</math>, and <math display="inline">\theta</math> momentum equations are<ref name="Ach"/> (note the convention used: <math display="inline">\theta</math> is polar angle, or [[colatitude]].)<ref>{{Citation
| url = http://mathworld.wolfram.com/SphericalCoordinates.html
| title = Spherical Coordinates
| author = Eric W. Weisstein
}}</ref>

==See also==
{{columns-list|colwidth=25em|
*[[Adhémar Jean Claude Barré de Saint-Venant]]
*[[Boltzmann equation]]
*[[Cauchy momentum equation]]
*[[Cauchy stress tensor#Balance laws – Cauchy's equations of motion|Cauchy stress tensor]]
*[[Chapman–Enskog theory]]
*[[Churchill–Bernstein equation]]
*[[Coandă effect]]
*[[Computational fluid dynamics]]
*[[Cauchy's laws of motion|Continuum mechanics]]
*[[Convection–diffusion equation]]
*[[Derivation of the Navier–Stokes equations]]
*[[Einstein–Stokes equation]]
*[[Euler equations (fluid dynamics)|Euler equations]]
*[[Hagen–Poiseuille flow from the Navier–Stokes equations]]
*[[Millennium Prize Problems]]
*[[Newtonian fluid]]
*[[Non-dimensionalization and scaling of the Navier–Stokes equations]]
*[[Pressure-correction method]]
*[[Primitive equations]]
*[[Rayleigh–Bénard convection]]
*[[Reynolds transport theorem]]
*[[Stokes flow|Stokes equations]]
*[[Supersonic flow over a flat plate]]
*[[Vlasov equation]]
}}

==Citations==
{{Reflist|30em}}

==General references==

* {{Citation
 | last = Acheson
 | first = D. J.
 | title = Elementary Fluid Dynamics
 | publisher = [[Oxford University Press]]
 | series = Oxford Applied Mathematics and Computing Science Series
 | year = 1990
 | isbn = 978-0-19-859679-0 
 | url=https://books.google.com/books?id=IGfDBAAAQBAJ}}

*{{citation | last=Batchelor | first=G. K. | author-link=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=978-0-521-66396-0 }}
*{{Citation
 | last = Currie
 | first = I. G.
 | title = Fundamental Mechanics of Fluids
 | publisher = [[McGraw-Hill]]
 | year = 1974
 | isbn = 978-0-07-015000-3 }}
* [[Vivette Girault|V. Girault]] and P. A. Raviart. ''Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms''. Springer Series in Computational Mathematics. Springer-Verlag, 1986.
*{{citation | last1=Landau | first1=L. D. | author1-link=Lev Landau | author2-link=Evgeny Lifshitz | last2=Lifshitz | first2= E. M. | year=1987 | title=Fluid mechanics | publisher=Pergamon Press | volume=''[[Course of Theoretical Physics]]'' Volume 6 | edition=2nd revised | isbn=978-0-08-033932-0 | oclc=15017127 }}
*{{citation | last1=Polyanin | first1=A. D. | last2=Kutepov | first2=A. M. | last3=Vyazmin | first3=A. V. | last4=Kazenin | first4=D. A. | title=Hydrodynamics, Mass and Heat Transfer in Chemical Engineering | publisher=Taylor & Francis, London | year=2002 | isbn=978-0-415-27237-7 }}
*{{citation | last=Rhyming | first=Inge L. | title=Dynamique des fluides | year=1991 | publisher=[[Presses polytechniques et universitaires romandes]]}}
* Smits, Alexander J. (2014), ''A Physical Introduction to Fluid Mechanics'', Wiley, {{isbn|0-47-1253499}}
* Temam, Roger (1984): ''Navier–Stokes Equations: Theory and Numerical Analysis'', ACM Chelsea Publishing, {{ISBN|978-0-8218-2737-6}}
*[[L. M. Milne-Thomson|Milne-Thomson, L.M.]]  [[Order of the British Empire|C.B.E]] (1962), ''Theoretical Hydrodynamics'', Macmillan & Co Ltd.
*Tartar, L (2006), ''An Introduction to Navier Stokes Equation and Oceanography,'' Springer {{ISBN|3-540-35743-2}}
*[[Garrett Birkhoff|Birkhoff, Garrett]] (1960)'', Hydrodynamics,''  Princeton University Press 
*Campos, D.(Editor) (2017) ''Handbook on Navier-Stokes Equations Theory and Applied Analysis'', Nova Science Publisher  {{ISBN|978-1-53610-292-5}} 
*[[Charles R. Doering|Döring, C.E.]]  and J.D. Gibbon, J.D. (1995)  ''Applied analysis of the Navier-Stokes equations,'' Cambridge University Press, {{ISBN|0-521-44557-1}} 
*[[Alfred Barnard Basset|Basset, A.B.]] (1888) ''Hydrodynamics Volume I and  II,''  Cambridge: Delighton, Bell and Co
*Fox, R. W. McDonald, A.T. and Pritchard, P.J. (2004) ''Introduction to Fluid Mechanics'', John Wiley and Sons, {{ISBN|0-471-2023-2}}  
*Foias, C.  Mainley, O. Rosa, R. and Temam, R.  (2004) ''Navier–Stokes Equations and Turbulence'', Cambridge University Press, {{ISBN}0-521-36032-3}} 
*[[Pierre-Louis Lions|Lions, P-L]]. (1998) ''Mathematical Topics in Fluid Mechanics'' Volume 1 and 2, Clarendon Press, {{ISBN|0-19-851488-3}}  
*Deville, M.O. and Gatski, T. B. (2012)  ''Mathematical Modeling for Complex Fluids and Flows,'' Springer,  {{ISBN|978-3-642-25294-5}} 
*Kochin, N.E. Kibel, I.A. and Roze, N.V. (1964) ''Theoretical Hydromechanics,'' John Wiley & Sons, Ltd. 
*[[Horace Lamb|Lamb, H.]] (1879) ''Hydrodynamics,'' Cambridge University Press, 
*{{Citation
 | last = White
 | first = Frank M.
 | author-link = Frank M. White
 | title = Viscous Fluid Flow
 | publisher = [[McGraw-Hill]]
 | year = 2006
 | isbn = 978-0-07-124493-0 }}

==External links==
* [https://web.archive.org/web/20171129091339/http://www.allstar.fiu.edu/aero/Flow2.htm Simplified derivation of the Navier–Stokes equations]
* [https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html Three-dimensional unsteady form of the Navier–Stokes equations] Glenn Research Center, NASA

{{Topics in continuum mechanics}}
{{Authority control}}

{{DEFAULTSORT:Navier-Stokes equations}}
[[Category:Aerodynamics]]
[[Category:Computational fluid dynamics]]
[[Category:Concepts in physics]]
[[Category:Equations of fluid dynamics]]
[[Category:Functions of space and time]]
[[Category:Partial differential equations]]
[[Category:Transport phenomena]]