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{{Continuum mechanics|fluid}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
In [[physics]], the '''Navier–Stokes equations''' [{{IPA|navˈjeː stəʊks}}], named after [[Claude-Louis Navier]] and [[Sir George Stokes, 1st Baronet|George Gabriel Stokes]], describe the motion of [[fluid]] substances. These equations arise from applying [[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 equations are useful because they describe the physics of many things 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 and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with [[Maxwell's equations]] they can be used to model and study [[magnetohydrodynamics]].
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The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, it has not yet been proven that in three dimensions solutions always exist ([[Existence theorem|existence]]), or that if they do exist, then they do not contain any [[Mathematical singularity|singularity]] (they are smooth). These are called the [[Navier–Stokes existence and smoothness]] problems. 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$1,000,000 prize for a solution or a counter-example.<ref>
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{{Citation
| url = http://www.claymath.org/millennium-problems
| title = Millennium Prize Problems
| publisher = Clay Mathematics Institute
| accessdate = 2014-01-14
}}</ref>


==Velocity field==
'''MathML'''
The solution of the Navier–Stokes equations is a [[flow velocity]] not [[position (vector)|position]]. It is called a flow [[velocity field]] or flow field, which is a description of the flow velocity at a given point in space and time. Once the velocity field is solved for, other quantities of interest, such as [[Pressure]] or [[Temperature]], may be found. 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; however for visualization purposes one can compute various [[Streamlines, streaklines, and pathlines|trajectories]].
:<math forcemathmode="mathml">E=mc^2</math>


==Properties==
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


===Nonlinearity===
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The Navier–Stokes equations are [[nonlinear]] [[partial differential equations]] in almost every real situation.<ref>Fluid Mechanics (Schaum's Series), M. Potter, D.C. Wiggert, Schaum's Outlines, McGraw-Hill (USA), 2008, ISBN 978-0-07-148781-8</ref><ref>Vectors, Tensors, and the basic Equations of Fluid Mechanics, R. Aris, Dover Publications, 1989, ISBN(10) 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.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


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>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3</ref>
==Demos==


===Turbulence===
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


[[Turbulence]] is the time-dependent [[Chaos theory|chaotic]] behavior 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), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</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|k-ω]] (k-omega), [[turbulence kinetic energy|k-ε]] (k-epsilon), and 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.
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** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


===Applicability===
==Test pages ==
{{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 [[Special relativity#Relativistic mechanics|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. Depending on the [[Knudsen number]] of the problem, the [[Boltzmann equation]] may be a suitable replacement; failing that, one may find the techniques of [[statistical mechanics]] sufficient or have to resort to [[molecular dynamics]].
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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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 rewritten 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, as of 2012, exist .{{citation needed|date=December 2013}}
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==Derivation and description==
==Bug reporting==
 
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{{Main|Derivation of the Navier–Stokes equations}}
 
The derivation of the Navier–Stokes equations begins with an application of [[Newton's second law]]: conservation of momentum (often alongside mass and energy conservation) being written for an arbitrary portion of the fluid. In an [[inertial frame of reference]], the general form of the equations of fluid motion is:<ref>Batchelor (1967) pp. 137 & 142.</ref>
 
{{Equation box 1
|indent=:
|title='''Navier–Stokes equations''' ''(general)''
|equation=<math> \rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \nabla \cdot\boldsymbol \tau + \mathbf{f}</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4
}}
where
: <math>\mathbf{u}</math> is the flow velocity,
: <math>\rho</math> is the fluid density,
: <math>p</math> is the pressure,
: <math>\boldsymbol \tau</math> is the ([[Cauchy stress tensor#Stress deviator tensor|deviatoric]]) component of the [[Cauchy stress tensor|total stress tensor]], which has order two,
: <math>\mathbf{f}</math> represents [[body force]]s (per unit volume) acting on the fluid,
: <math>\nabla</math> is the [[del]] operator.
 
This is a statement of the conservation of momentum in a fluid and it is an application of Newton's second law to a [[Continuum mechanics|continuum]]; in fact this equation is applicable to any non-relativistic continuum and is known as the [[Cauchy momentum equation]].
 
This equation is often written using the [[material derivative]] ''D'''''u'''/''Dt'', making it more apparent that this is a statement of [[Newton's second law]]:
 
:<math>\rho \frac{D \mathbf{u}}{D t} = -\nabla p + \nabla \cdot\boldsymbol \tau + \mathbf{f}.</math>
 
The left side of the equation describes acceleration, and may be composed of time-dependent or convective effects (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of body forces (such as gravity) and divergence of stress (pressure and shear stress).
 
===Convective acceleration===
 
[[Image:ConvectiveAcceleration.png|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 Navier–Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a fluid 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.
 
Regardless of what kind of fluid is being dealt with, convective acceleration is a [[nonlinear]] effect. Convective acceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamic effect is disregarded in [[creeping flow]] (also called Stokes flow). Convective acceleration is represented by the [[nonlinear]] quantity:
 
:<math>\mathbf{u} \cdot \nabla \mathbf{u}</math>
 
which may be interpreted either as  <math>(\mathbf{u}\cdot\nabla)\,\mathbf{u}</math> or as <math>\mathbf{u}\cdot(\nabla\mathbf{u}),</math> with <math>\nabla \mathbf{u}</math> the [[tensor derivative]] of the velocity vector <math>\mathbf{u}.</math> Both interpretations give the same result, independent of the coordinate system&nbsp;— provided <math>\nabla</math> is interpreted as the [[covariant derivative]].<ref name=Emanuel>{{Citation | last=Emanuel | first=G. | title=Analytical fluid dynamics | publisher=CRC Press | year=2001 | edition=second | isbn=0-8493-9114-8 }} pp. 6–7.</ref>
 
====Advection operator====
 
The convection term is often written as
 
:<math>(\mathbf{u} \cdot \nabla) \mathbf{u}</math>
 
where the [[advection|advection operator]] <math>\mathbf{u} \cdot \nabla</math> is used. Usually this representation is preferred as it is simpler than the one in terms of the tensor derivative <math>\nabla \mathbf{u}.</math><ref name=Emanuel/>
 
====Tensor derivative====
Here <math>\nabla \mathbf{u}</math> is the tensor derivative of the velocity vector, equal in Cartesian coordinates to the component-by-component gradient. Note that the gradient of a vector is being defined as <math>\left[\nabla \mathbf{u}\right]_{mi}=\partial_m v_i</math>, so that <math>\left[\mathbf{u}\cdot\left(\nabla \mathbf{u}\right)\right]_i=v_m \partial_m v_i=\left[(\mathbf{u}\cdot\nabla)\mathbf{u}\right]_i</math>.
 
====In irrotational flow====
The convection term may, by a [[vector calculus identities#Vector dot product|vector calculus identity]], be expressed without a tensor derivative:<ref>See Batchelor (1967), §3.5, p. 160.</ref><ref>{{Citation
| url = http://mathworld.wolfram.com/ConvectiveDerivative.html
| title = Convective Derivative
| author = [[Eric W. Weisstein]]
| publisher = [[MathWorld]]
| accessdate = 2008-05-20
}}</ref>
 
:<math>\mathbf{u} \cdot \nabla \mathbf{u} = \nabla \left( \frac{\|\mathbf{u}\|^2}{2} \right)  + \left( \nabla \times \mathbf{u} \right) \times \mathbf{u}.</math>
 
The form has use in irrotational flow, where the [[Curl (mathematics)|curl]] of the velocity (called [[vorticity]]) <math>\omega=\nabla \times \mathbf{u}</math> is equal to zero. Therefore, this reduces to only
 
:<math>\mathbf{u} \cdot \nabla \mathbf{u} = \nabla \left( \frac{\|\mathbf{u}\|^2}{2} \right).</math>
 
===Stresses===<!--Lift (force) has a wikilink to here-->
 
The effect of stress in the fluid is represented by the <math>\nabla p</math> and <math>\nabla \cdot\boldsymbol \tau</math> terms; these are [[gradient]]s of surface forces, analogous to stresses in a solid. Here <math>\nabla p</math> is called the pressure gradient and arises from the isotropic part of the [[Cauchy stress tensor]], which has order two. This part is given by [[normal stress]]es that turn up in almost all situations, dynamic or not. The anisotropic part of the stress tensor gives rise to <math>\nabla \cdot\boldsymbol \tau</math>, which conventionally describes viscous forces; for incompressible flow, this is only a shear effect. Thus, <math>\boldsymbol \tau</math> is the [[deviatoric stress tensor]], and the stress tensor is equal to:<ref>Batchelor (1967) p. 142.</ref>
 
:<math>\boldsymbol \sigma = - p \mathbf 1+ \boldsymbol \tau</math>
 
where <math>\mathbf 1</math> is the 3×3 [[identity matrix]]. In the Navier–Stokes equations, only the ''gradient'' of pressure matters, not the pressure itself. The effect of the pressure gradient on the flow is to accelerate the fluid in the direction from high pressure to low pressure.
 
The stress terms ''p'' and <math>\boldsymbol \tau</math> are yet unknown, so the general form of the equations of motion is not usable to solve problems. Besides the equations of motion—Newton's second law—a force model is needed relating the stresses to the fluid motion.<ref>
{{citation
| first1=Richard P.
| last=Feynman
| authorlink1=R. P. Feynman
| first2=Robert B.
| last2=Leighton
| authorlink2=R. B. Leighton
| first3=Matthew
| last3=Sands
| authorlink3=M. Sands
| year=1963
| title=[[The Feynman Lectures on Physics]]
| isbn=0-201-02116-1
| publisher=Addison-Wesley
| location=Reading, Massachusetts
}}, Vol. 1, §9–4 and §12–1.</ref>  For this reason, assumptions based on natural observations are often applied to specify the stresses in terms of the other flow variables, such as velocity and density.
 
The Navier–Stokes equations result from the following assumptions on the deviatoric stress tensor <math>\boldsymbol \tau</math>:<ref name=Batchelor_142_148>Batchelor (1967) pp. 142–148.</ref>
 
*the deviatoric stress vanishes for a fluid at rest, and, by [[Galilean invariance]], also does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity
 
*the deviatoric stress is expressed as the [[Dyadics#Double-dot product|double-dot product]] of the tensor gradient <math>\nabla\mathbf{u}</math> of the flow velocity with, a fourth-order viscosity tensor <math>\mathbf V</math>, i.e., <math>\boldsymbol \tau = \mathbf V :\left( \nabla\mathbf{u} \right)</math>
 
*the fluid is assumed to be [[isotropic]], as with gases and simple liquids, and consequently <math>\mathbf  V</math> is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, it can be expressed in terms of two scalar [[dynamic viscosity|dynamic viscosities]] ''μ'' and ''μ''": <math>\boldsymbol \tau = 2 \mu \boldsymbol \varepsilon + \mu'' \mathbf 1 \, \nabla\cdot\mathbf{u} ,</math> where <math>\boldsymbol \varepsilon=\tfrac12 \left( \nabla\mathbf{u} \right) + \tfrac12 \left( \nabla\mathbf{u} \right)^\text{T}</math> is the rate-of-[[strain tensor]] and <math>\nabla\cdot\mathbf{u}</math> is the rate of expansion of the flow
 
*the deviatoric stress tensor has zero [[trace (linear algebra)|trace]], so for a three-dimensional flow 2''μ''&nbsp;+&nbsp;3''μ''"&nbsp;=&nbsp;0
 
As a result, the deviatoric stress tensor has the following form:<ref name=Batchelor_142_148/>
 
:<math>\boldsymbol \tau = 2 \mu \left( \boldsymbol \varepsilon - \tfrac13 \mathbf 1 \, \nabla\cdot\mathbf{u}\right)</math>
 
with the quantity between brackets being the non-isotropic part of the rate-of-strain tensor <math>\boldsymbol \varepsilon.</math> The dynamic viscosity ''μ'' need not be constant – in general, it depends on conditions like temperature and pressure, and, in [[turbulence]] modeling, [[eddy viscosity]] approximates the average deviatoric stress.
 
The pressure ''p'' is modeled by an [[equation of state]].<ref>Batchelor (1967) p. 165.</ref> 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]] velocity field with <math>\nabla\cdot\mathbf{u}=0.</math><ref>Batchelor (1967) p. 75.</ref>
 
===Other forces===
The vector field <math>\mathbf{f}</math> represents [[body force]]s. Typically, these consist of only [[gravity]] forces, but may include others, such as electromagnetic forces. In non-inertial coordinate frames, other "forces" associated with [[Fictitious force|rotating coordinates]] may arise.
 
Often, these forces may be represented as the gradient of some scalar quantity <math>\chi \,</math>, with <math> \mathbf{f} = \nabla \chi ,</math> in which case they are called [[conservative force]]s. Gravity in the ''z'' direction, for example, is the gradient of <math>-\rho g z</math>. Because pressure from such gravitation arises only as a gradient, we may include it in the pressure term as a body force <math> h = p - \chi. </math>  The pressure and force terms on the right-hand side of the Navier–Stokes equation become
 
:<math>-\nabla p + \mathbf{f} = -\nabla p + \nabla \chi = -\nabla \left( p - \chi \right) = -\nabla h.</math>
 
===Other equations===
The Navier–Stokes equations are strictly a statement of the conservation 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, conservation of energy, and/or an [[equation of state]].
 
====Continuity equation====
{{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]], given in its most general form as:
 
:<math>\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0</math>
 
or, using the [[substantive derivative]]:
 
:<math>\frac{D\rho}{Dt} + \rho (\nabla \cdot \mathbf{u}) = 0.</math>
 
==Incompressible flow of Newtonian fluids==
 
The equations are simplified in the case of [[incompressible flow]] of a [[Newtonian fluid]]. Incompressibility rules out sound propagation or [[shock wave]]s, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well even with a "compressible" fluid — such as air at room temperature — at low [[Mach number]]s up to about Mach 0.3. With incompressible flow and constant viscosity, the Navier–Stokes equations read<ref name="Ach">See Acheson (1990).</ref>
 
{{Equation box 1
|indent=:
|title='''[[Incompressible flow|Incompressible]] Navier–Stokes equations'''
|equation=<math>\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}.</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
 
in tensor notation:
 
{{Equation box 1
|indent=:
|title='''[[Incompressible flow|Incompressible]] Navier–Stokes equations'''
|equation=<math>\rho\left(\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j} \right)=-\frac{\partial p}{\partial x_i}+\mu\frac{\partial^2 u_i}{\partial x_j\partial x_j}+f_i </math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
 
Here '''f''' represents "other" [[body force]]s (per unit volume), such as [[gravity]] or [[centrifugal force]].  The [[shear stress]] term <math>\nabla \cdot \boldsymbol{\mathsf{T}}</math> becomes <math>\mu \nabla^2 \mathbf{u}</math>, where <math>\nabla^2</math> is the [[vector Laplacian]].<ref>Batchelor (1967) pp. 21 & 147.</ref>
 
It's well worth observing the meaning of each term (compare to the [[Cauchy momentum equation]]):
 
:<math>
\overbrace{\rho \Big(
\underbrace{\frac{\partial \mathbf{u}}{\partial t}}_{
\begin{smallmatrix}
  \text{Unsteady}\\
  \text{acceleration}
\end{smallmatrix}} +
\underbrace{\mathbf{u} \cdot \nabla \mathbf{u}}_{
\begin{smallmatrix}
  \text{Convective} \\
  \text{acceleration}
\end{smallmatrix}}\Big)}^{\text{Inertia (per volume)}} =
\overbrace{\underbrace{-\nabla p}_{
\begin{smallmatrix}
  \text{Pressure} \\
  \text{gradient}
\end{smallmatrix}} +
\underbrace{\mu \nabla^2 \mathbf{u}}_{\text{Viscosity}}}^{\text{Divergence of stress}} +
\underbrace{\mathbf{f}.}_{
\begin{smallmatrix}
  \text{Other} \\
  \text{body} \\
  \text{forces}
\end{smallmatrix}}
</math>
 
Only the convective terms are nonlinear for incompressible Newtonian flow. Convective acceleration is caused by a (possibly steady) change in velocity over ''position'', for example, by the speeding up of fluid entering a converging [[nozzle]]. Though individual fluid particles are accelerated and thus under unsteady motion, the flow field, a velocity distribution, will not necessarily be time-dependent.
 
Another important observation is that viscosity is represented by the [[vector Laplacian]] of the velocity field, interpreted here 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 [[diffusion]] of heat in the [[heat equation]],  also expressed with the Laplacian.
 
If temperature effects are also neglected, the only "other" equation (apart from initial/boundary conditions) needed is the mass continuity equation. Under the assumption of incompressibility, the density of a [[fluid particle]] is constant, and when using the substantive derivative [[Incompressible_flow#Derivation|it follows easily]] that the [[Navier–Stokes equations#Continuity equation|continuity equation]] simplifies to:
 
:<math>\nabla \cdot \mathbf{u} = 0.</math>
 
This is more specifically a statement of the conservation of volume (see [[divergence]] and [[isochoric process]]).
 
These equations are commonly used in 3 coordinates systems: [[Cartesian coordinate system|Cartesian]], [[Cylindrical coordinate system|cylindrical]], and [[Spherical coordinate system|spherical]]. While the Cartesian equations seem to follow directly from the vector equation above, the vector form of the  Navier–Stokes equation involves some [[tensor calculus]] which means that writing it in other coordinate systems is not as simple as doing so for scalar equations (such as the [[heat equation]]).
 
==3D representations==
===Cartesian coordinates===
From the general form of the Navier-Stokes, with the velocity vector expanded as <math>\mathbf{u} = (u_x,u_y,u_z)</math> (sometimes respectively named ''u'', ''v'', ''w''), we may write the vector equation explicitly,
 
:<math>\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} + u_z \frac{\partial u_x}{\partial z}\right)
    &= -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2} + \frac{\partial^2 u_x}{\partial z^2}\right) - \mu \frac{\partial}{\partial x} \left( \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} \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}+ u_z \frac{\partial u_y}{\partial z}\right)
    &= -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2} + \frac{\partial^2 u_y}{\partial z^2}\right) - \mu \frac{\partial}{\partial y} \left( \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} \right) + \rho g_y \\
  \rho \left(\frac{\partial u_z}{\partial t} + u_x \frac{\partial u_z}{\partial x} + u_y \frac{\partial u_z}{\partial y}+ u_z \frac{\partial u_z}{\partial z}\right)
    &= -\frac{\partial p}{\partial z} + \mu \left(\frac{\partial^2 u_z}{\partial x^2} + \frac{\partial^2 u_z}{\partial y^2} + \frac{\partial^2 u_z}{\partial z^2}\right) - \mu \frac{\partial}{\partial z} \left( \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} \right) + \rho g_z.
\end{align}</math>
 
Note that gravity has been accounted for as a body force, and the values of ''g<sub>x</sub>'', ''g<sub>y</sub>'', ''g<sub>z</sub>'' will depend on the orientation of gravity with respect to the chosen set of coordinates.
 
The continuity equation reads:
:<math>{\partial \rho \over \partial t} + {\partial (\rho u_x ) \over \partial x} + {\partial (\rho u_y) \over \partial y} + {\partial (\rho u_z) \over \partial z} = 0.</math>
 
When the flow is incompressible, <math>\rho</math> does not change for any fluid particle, and its [[material derivative]] vanishes: <math>{D\rho \over Dt} = 0</math>. The continuity equation is reduced to:
 
:<math>{\partial u_x \over \partial x} + {\partial u_y \over \partial y} + {\partial u_z \over \partial 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 [[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 ''r'', <math>\phi</math>, and ''z'':
 
 
:<math>\begin{align}
  r:\ &\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} +
                  \frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\phi}^2}{r}\right) = {}\\
      &-\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) +
        \frac{1}{r^2}\frac{\partial^2 u_r}{\partial \phi^2} + \frac{\partial^2 u_r}{\partial z^2} - \frac{u_r}{r^2} -
        \frac{2}{r^2}\frac{\partial u_\phi}{\partial \phi} \right] + \rho g_r \\
  \phi:\ &\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} +
                      \frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + u_z \frac{\partial u_{\phi}}{\partial z} + \frac{u_r u_{\phi}}{r}\right) = {}\\
        &-\frac{1}{r}\frac{\partial p}{\partial \phi} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) +
          \frac{1}{r^2}\frac{\partial^2 u_{\phi}}{\partial \phi^2} + \frac{\partial^2 u_{\phi}}{\partial z^2} + \frac{2}{r^2}\frac{\partial u_r}{\partial \phi}-\frac{u_{\phi}}{r^2}\right] + \rho g_{\phi} \\
  z:\ &\rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_z}{\partial \phi} +
              u_z \frac{\partial u_z}{\partial z}\right) = {}\\
      &-\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) +
        \frac{1}{r^2}\frac{\partial^2 u_z}{\partial \phi^2} + \frac{\partial^2 u_z}{\partial z^2}\right] + \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>
  \frac{\partial\rho}{\partial t} + \frac{1}{r}\frac{\partial}{\partial r}\left(\rho r u_r\right) +
  \frac{1}{r}\frac{\partial (\rho u_\phi)}{\partial \phi} + \frac{\partial (\rho u_z)}{\partial z}
    = 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>u_{\phi} = 0</math>), and the remaining quantities are independent of <math>\phi</math>:
 
:<math>\begin{align}
  \rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + u_z \frac{\partial u_r}{\partial z}\right)
    &= -\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) +
      \frac{\partial^2 u_r}{\partial z^2} - \frac{u_r}{r^2}\right] + \rho g_r \\
  \rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + u_z \frac{\partial u_z}{\partial z}\right)
    &= -\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) +
      \frac{\partial^2 u_z}{\partial z^2}\right] + \rho g_z \\
  \frac{1}{r}\frac{\partial}{\partial r}\left(r u_r\right) + \frac{\partial u_z}{\partial z} &= 0.
\end{align}</math>
 
===Spherical coordinates===
In [[spherical coordinates]], the ''r'', ''ϕ'', and ''θ'' momentum equations are<ref name="Ach"/> (note the convention used: ''θ'' is polar angle, or [[colatitude]],<ref>{{Citation
| url = http://mathworld.wolfram.com/SphericalCoordinates.html
| title = Spherical Coordinates
| author = [[Eric W. Weisstein]]
| publisher = [[MathWorld]]
| date = 2005-10-26
| accessdate = 2008-01-22
}}</ref> 0 ≤ ''θ'' ≤ π):
:<math>\begin{align}
  r:\  &\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_r}{\partial \phi} +
                  \frac{u_{\theta}}{r} \frac{\partial u_r}{\partial \theta} - \frac{u_{\phi}^2 + u_{\theta}^2}{r}\right) =
          -\frac{\partial p}{\partial r} + \rho g_r + \\
      &\mu \left[\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_r}{\partial r}\right) +
                  \frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_r}{\partial \phi^2} +
                  \frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_r}{\partial \theta}\right) - 2\frac{u_r +
                  \frac{\partial u_{\theta}}{\partial \theta} + u_{\theta} \cot(\theta)}{r^2} - \frac{2}{r^2 \sin(\theta)} \frac{\partial u_{\phi}}{\partial \phi}
            \right] \\
 
  \phi:\  &\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} +
                      \frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_{\phi}}{\partial \phi} + \frac{u_{\theta}}{r} \frac{\partial u_{\phi}}{\partial \theta} +
                      \frac{u_r u_{\phi} + u_{\phi} u_{\theta} \cot(\theta)}{r}\right) =
              -\frac{1}{r \sin(\theta)} \frac{\partial p}{\partial \phi} + \rho g_{\phi} + \\
          &\mu \left[\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_{\phi}}{\partial r}\right) +
                    \frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_{\phi}}{\partial \phi^2} +
                    \frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_{\phi}}{\partial \theta}\right) +
                    \frac{2 \sin(\theta) \frac{\partial u_r}{\partial \phi} + 2 \cos(\theta) \frac{\partial u_{\theta}}{\partial \phi} -
                    u_{\phi}}{r^2 \sin(\theta)^2}
              \right] \\
 
  \theta:\  &\rho \left(\frac{\partial u_{\theta}}{\partial t} + u_r \frac{\partial u_{\theta}}{\partial r} +
                        \frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_{\theta}}{\partial \phi} +
                        \frac{u_{\theta}}{r} \frac{\partial u_{\theta}}{\partial \theta} + \frac{u_r u_{\theta} - u_{\phi}^2 \cot(\theta)}{r}\right) =
                -\frac{1}{r} \frac{\partial p}{\partial \theta} + \rho g_{\theta} + \\
            &\mu \left[\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_{\theta}}{\partial r}\right) +
                      \frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_{\theta}}{\partial \phi^2} +
                      \frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_{\theta}}{\partial \theta}\right) +
                      \frac{2}{r^2} \frac{\partial u_r}{\partial \theta} - \frac{u_{\theta} +
                      2 \cos(\theta) \frac{\partial u_{\phi}}{\partial \phi}}{r^2 \sin(\theta)^2}
                \right].
\end{align}</math>
 
Mass continuity will read:
 
:<math>
  \frac{\partial \rho}{\partial t} + \frac{1}{r^2}\frac{\partial}{\partial r}\left(\rho r^2 u_r\right) +
  \frac{1}{r \sin(\theta)}\frac{\partial \rho u_\phi}{\partial \phi} +
  \frac{1}{r \sin(\theta)}\frac{\partial}{\partial \theta}\left(\sin(\theta) \rho u_\theta\right)
    = 0.
</math>
 
These equations could be (slightly) compacted by, for example, factoring <math>1/r^2</math> from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities.
 
==Stream function formulation==
Taking the [[Curl (mathematics)|curl]] of the 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>u_z = 0</math> and no dependence of anything on ''z''), where the equations reduce to:
 
:<math>\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 ''y'', the second with respect to ''x'' and subtracting the resulting equations will eliminate pressure and any [[conservative force]]. Defining the [[stream function]] <math>\psi</math> through
 
:<math>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>\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>\nabla^4</math> is the (2D) [[biharmonic operator]] and <math>\nu</math> is the [[kinematic viscosity]], <math>\nu = \frac{\mu}{\rho}</math>. We can also express this compactly using the [[Jacobian matrix and determinant|Jacobian determinant]]:
 
:<math>\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial\left(\psi, \nabla^2\psi \right)}{\partial\left(y,x\right)} = \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.
 
===Pressure-free velocity formulation===
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 above eliminates the pressure (in 2D) at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest.
 
The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations,
:<math>\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>\Pi^S</math> and <math>\Pi^I</math> are solenoidal and irrotational projection operators satisfying <math>\Pi^S+\Pi^I=1</math> and
<math>\mathbf{f}^S</math> and <math>\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>\Pi^S\,\mathbf{F}(\mathbf{r}) = \frac{1}{4\pi}\nabla\times\int \frac{\nabla^\prime\times\mathbf{F}(\mathbf{r}^\prime)}{|\mathbf{r}-\mathbf{r}^\prime|} d V^\prime, \quad \Pi^I = 1-\Pi^S</math>
 
with a similar structure in 2D. Thus the governing equation is an [[integro-differential equation]] and 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>\left(\mathbf{w},\frac{\partial\mathbf{u}}{\partial t}\right) = -(\mathbf{w},\mathbf{u}\cdot\nabla\mathbf{u})-\nu(\nabla\mathbf{w}: \nabla\mathbf{u})+(\mathbf{w},\mathbf{f}^S)</math>
 
for divergence-free test functions <math>\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 imminently suited to finite element computation of divergence-free flow, as we shall see in the next section. There we 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 law. This all would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition.
 
====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>\left(\mathbf{w}_i, \frac{\partial\mathbf{u}_j}{\partial t}\right) = -(\mathbf{w}_i, \mathbf{u}\cdot\nabla\mathbf{u}_j) - \nu(\nabla\mathbf{w}_i: \nabla\mathbf{u}_j) + \left(\mathbf{w}_i, \mathbf{f}^S\right).</math>
 
It is desirable to choose basis functions which 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>\nabla\phi = \left[\frac{\partial \phi}{\partial x},\,\frac{\partial \phi}{\partial y}\right]^\mathrm{T}, \quad
\nabla\times\phi = \left[\frac{\partial \phi}{\partial y},\,-\frac{\partial \phi}{\partial x}\right]^\mathrm{T}.</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.
  | author-link = J.T. Holdeman
  | title = A Hermite finite element method for incompressible fluid flow
  | journal = Int. J. Numer. Meth. FLuids
  | volume = 64  | pages = 376–408  | year = 2010
  | doi = 10.1002/fld.2154
  | issue = 4 |bibcode = 2010IJNMF..64..376H
}}</ref><ref>
{{Citation
  | last = Holdeman  | first = J.T.
  | author-link = J.T. Holdeman
  | last2 = Kim  | first2 = J.W.
  | title = Computation of incompressible thermal flows using Hermite finite elements
  | journal = Comput. Methods Appl. Mech. Engrg.
  | volume = 199  | pages = 3297–3304  | year = 2010
  | authorlink2 = Jin-Whan Kim
  | 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>(\mathbf{g}_i, \nabla p) = -(\mathbf{g}_i, \mathbf{u}\cdot\nabla\mathbf{u}_j) - \nu(\nabla\mathbf{g}_i: \nabla\mathbf{u}_j) + (\mathbf{g}_i, \mathbf{f}^I)</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>\mathbf{g}_i</math> one would choose the irrotational vector elements obtained from the gradient of the pressure element.
 
==Compressible flow of Newtonian fluids==
 
There are some phenomena that are closely linked with fluid [[compressibility]]. One of the obvious examples is sound. Description of such phenomena requires more general presentation of the Navier–Stokes equation that takes into account fluid compressibility. If viscosity is assumed a constant, one additional term appears, as shown here:<ref>Landau & Lifshitz (1987) pp. 44–45.</ref><ref>Batchelor (1967) pp. 147 & 154.</ref>
 
:<math>\rho \left(\frac{\partial  \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right)  -  \mu \nabla^2 \mathbf{u} - \left(\zeta + \frac{\mu}{3}\right) \nabla (\nabla \cdot \mathbf{u})= -\nabla p + \mathbf{f}, </math>
 
where <math>\zeta</math> is the [[volume viscosity| second viscosity]].
 
==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.png|thumb|400px|Visualization of a) parallel flow and b) radial flow.]]
 
a)
Assume steady, parallel, one dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) [[boundary value problem]] is:
:<math>\frac{d^2 u}{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>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.
 
b)
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)</math> that must satisfy:
:<math>\frac{d^2 f}{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 [[nonlinear]] term makes this a very difficult problem to solve analytically (a lengthy [[Implicit function|implicit]] solution may be found which involves [[elliptic integrals]] and [[Cubic formula|roots of cubic polynomials]]). Issues with the actual existence of solutions arise for R > 1.41 (approximately; this is not the [[square root of 2]]), the parameter R being the Reynolds number with appropriately chosen scales.<ref name="TM Shah">{{cite web|last=Shah |first=Tasneem Mohammad|title=Analysis of the multigrid method|url=http://adsabs.harvard.edu/abs/1989STIN...9123418S|publisher=Published by Dr. TM Shah; republished by The Smithsonian/NASA Astrophysics Data System|accessdate=8 January 2013|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"/>
 
==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; for example the [[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>
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>
Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.
 
{{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,\phi),</math> the velocity components <math>(u_r,u_\phi)</math> and pressure ''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>
    u_r = \frac{A}{r}, \qquad
  u_\phi = B\left(\frac{1}{r} - r^{\frac{A}{\nu} + 1}\right), \qquad
      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)}{2\frac{A}{\nu} + 2}
</math>
 
where A and B  are arbitrary constants. This solution is valid in the domain ''r''&nbsp;≥&nbsp;1 and for <math> A < -2\nu</math>.
 
In Cartesian coordinates, when the viscosity is zero (<math>\nu = 0</math>), this is:
:<math>
  \mathbf{v}(x,y) = \frac{1}{x^2 + y^2}\begin{pmatrix} Ax + By \\ Ay - Bx \end{pmatrix}, \qquad
          p(x,y) = -\frac{A^2 + B^2}{2(x^2 + y^2)}
</math>
 
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|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>\nu=0</math>) — radial flow in [[Cartesian coordinates]] <math>(x,y,z),</math> the velocity vector <math>\mathbf{v}</math> and pressure ''p'' are:{{citation needed|date=January 2014}}
 
:<math>
  \mathbf{v}(x, y, z) = \frac{A}{x^2 + y^2 + z^2}\begin{pmatrix} x \\ y\\ z \end{pmatrix}, \qquad
          p(x, y, z) = -\frac{A^2}{2(x^2 + y^2 + z^2)}.
</math>
 
There is a singularity at <math>x = y = z = 0</math>.
 
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|headerstyle = text-align:left
}}
 
===A three-dimensional steady-state vortex solution===
[[Image:Hopfkeyrings.jpg|right|250px|thumb|Some of the flow lines along a [[Hopf fibration]].]]
A nice steady-state example with no singularities comes from considering the flow along the lines of a [[Hopf fibration]]. Let r be a constant radius to 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
| year=1982
| title=Topological solitons in magnetohydrodynamics
| first=A. M
| last= Kamchatno
}}</ref>
:<math>\begin{align}
        \rho(x, y, z) &= \frac{3B}{r^2 + x^2 + y^2 + z^2} \\
          p(x, y, z) &= \frac{-A^2B}{(r^2 + x^2 + y^2 + z^2)^3} \\
  \mathbf{u}(x, y, z) &= \frac{A}{(r^2 + x^2 + y^2 + z^2)^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 A and B. 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>\rho</math> is a constant, 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>z = 0, x^2 + y^2 = r^2</math>:
:<math>\begin{align}
  \rho(x, y, z) &= \frac{20B(x^2 + y^2)}{(r^2 + x^2 + y^2 + z^2)^3} \\
    p(x, y, z) &= \frac{-A^2B}{(r^2 + x^2 + y^2 + z^2)^4} + \frac{-4A^2B(x^2 + y^2)}{(r^2 + x^2 + y^2 + z^2)^5}.
\end{align}</math>
 
In fact in general there are simple solutions for any polynomial function f where the density is:
:<math>\rho(x, y, z) = \frac{1}{r^2 + x^2 + y^2 + z^2} f\left(\frac{x^2 + y^2}{(r^2 + x^2 + y^2 + z^2)^2}\right).</math>
 
|style = border: 1px solid lightgray; width: 90%;
|headerstyle = text-align:left
}}
 
==Wyld diagrams==<!-- [[Wyld diagrams]] redirects here-->
'''Wyld diagrams''' are bookkeeping [[graph (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|Keldysh]]'s technique for nonequilibrium processes in fluid dynamics.  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=0-19-923652-6 }} pp. 121–128.</ref>
 
==Navier–Stokes equations use in games==<!-- [[Wyld diagrams]] redirects here-->
The Navier–Stokes equations are used extensively in video games in order to model a wide variety of natural phenomena. Simulations of small-scale gaseous fluids, such as fire and smoke are often based on the seminal paper "Real-Time Fluid Dynamics for Games"<ref>{{citation
| url= http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf
| year=2003
| title=Real-Time Fluid Dynamics for Games
| first=Jos
| last=Stam
}}</ref> by [[Jos Stam]], which elaborates one of the methods proposed in Stam's earlier, more famous paper "Stable Fluids"<ref>{{citation
| url= http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/ns.pdf
| year=1999
| title=Stable Fluids
| first=Jos
| last=Stam
}}</ref> from 1999. Stam proposes stable fluid simulation using a Navier–Stokes solution method from 1968, coupled with an unconditionally stable semi-Lagrangian advection scheme, as first proposed in 1992.
 
More recent implementations based upon this work run on the GPU as opposed to the CPU and achieve a much higher degree of performance.<ref>{{citation
| year=2004
| title=GPUGems - Fast Fluid Dynamics Simulation on the GPU
| chapter=38
| first=Mark J.
| last=Harris
}}</ref><ref>{{citation
| year=2007
| title=ShaderX5 - Explicit Early-Z Culling for Efficient Fluid Flow Simulation
| chapter=9.6
| pages=553–564
| first=P.
| last=Sander
| first2=N.
| last2=Tatarchuck
| first3=J.L.
| last3=Mitchell
}}</ref>
Many improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass.
 
An introduction to interactive fluid simulation can be found in the 2007 [[ACM SIGGRAPH]] course, Fluid Simulation for Computer Animation.<ref>{{cite web|author1=Robert Bridson|author2=Matthias Müller-Fischer|title=Fluid Simulation for Computer Animation|url=http://www.cs.ubc.ca/~rbridson/fluidsimulation/|website=www.cs.ubc.ca}}</ref>
 
==See also==
{{columns-list|colwidth=25em|
*[[Adhémar Jean Claude Barré de Saint-Venant]]
*[[Boltzmann equation]]
*[[Churchill–Bernstein equation]]
*[[Coandă effect]]
*[[Computational fluid dynamics]]
*[[Euler equations (fluid dynamics)|Euler equations]]
*[[Fokker–Planck equation]]
*[[Hagen–Poiseuille flow from the Navier–Stokes equations]]
*[[Derivation of the Navier–Stokes equations]]
*[[Non-dimensionalization and scaling of the Navier–Stokes equations]]
*[[Pressure-correction method]]
*[[Reynolds transport theorem]]
*[[Vlasov equation]]
}}
 
==Notes==
{{reflist|colwidth=30em}}
 
==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 = 0-19-859679-0 }}
*{{citation | first=G. K. | last=Batchelor | authorlink=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0-521-66396-2 }}
*{{citation | title=Fluid mechanics | first1=L. D. | last1=Landau | author1-link=Lev Landau | author2-link=Evgeny Lifshitz | last2=Lifshitz | first2= E. M. | year=1987 | publisher=Pergamon Press | series=[[Course of Theoretical Physics]] | volume=6 | edition=2nd revised | isbn=0-08-033932-8 | oclc=15017127 }}
*{{citation | first=Inge L. | last=Rhyming | title=Dynamique des fluides | year=1991 | publisher=[[Presses polytechniques et universitaires romandes]]}}
*{{citation | first1=A. D. | last1=Polyanin | first2=A. M. | last2=Kutepov | first3=A. V. | last3=Vyazmin | first4=D. A. | last4=Kazenin | title=Hydrodynamics, Mass and Heat Transfer in Chemical Engineering | publisher=Taylor & Francis, London | year=2002 | isbn=0-415-27237-8 }}
*{{Citation
  | last = Currie
  | first = I. G.
  | title = Fundamental Mechanics of Fluids
  | publisher = [[McGraw-Hill]]
  | year = 1974
  | isbn = 0-07-015000-1 }}
* V. Girault and P.A. Raviart.  ''Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms.''  Springer Series in Computational Mathematics.  Springer-Verlag, 1986.
 
==External links==
* [http://www.allstar.fiu.edu/aero/Flow2.htm Simplified derivation of the Navier–Stokes equations]
* [http://www.claymath.org/sites/default/files/navierstokes.pdf Millennium Prize problem description.]
* [http://www.cfd-online.com/Wiki/Codes CFD online software list] A compilation of codes, including Navier–Stokes solvers.
* [http://nerget.com/fluidSim/ Online fluid dynamics simulator] Solves the Navier–Stokes equation numerically and visualizes it using Javascript.
* [https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html Three-dimensional unsteady form of the Navier-Stokes equations] Glenn Research Center, NASA
 
{{DEFAULTSORT:Navier-Stokes Equations}}
[[Category:Concepts in physics]]
[[Category:Equations of fluid dynamics]]
[[Category:Aerodynamics]]
[[Category:Partial differential equations]]
[[Category:Conservation equations]]

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