Editing Navier–Stokes equations (section)

===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]].