Editing Navier–Stokes equations (section)

==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>
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}}</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 |pages=055508 |arxiv=1309.0703 |doi=10.1088/0169-5983/46/5/055508}}</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 |url-status=dead |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
}}</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
| 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]].