Editing Navier–Stokes equations (section)

===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>

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