Editing Navier–Stokes equations (section)

==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 |pages=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
| publisher = [[MathWorld]]
| date = 2005-10-26
| access-date = 2008-01-22
| author-link = Eric W. Weisstein
}}</ref> <math display="inline">0 \leq \theta \leq \pi</math>):
<math display="block">\begin{align}
 r:\ &\rho \left({\partial_t u_r} + u_r {\partial_r u_r} + \frac{u_\varphi}{r \sin\theta} {\partial_\varphi u_r} +
 \frac{u_\theta}{r} {\partial_\theta u_r} - \frac{u_\varphi^2 + u_\theta^2}{r}\right) \\ 
 &\quad = -{\partial_r p} \\
 &\qquad  + \mu \left(\frac{1}{r^2} \partial_r \left(r^2 {\partial_r u_r}\right) + \frac{1}{r^2 \sin^2\theta} {\partial_\varphi^2 u_r} + \frac{1}{r^2 \sin\theta} \partial_\theta \left(\sin\theta {\partial_\theta u_r}\right) - 2\frac{u_r + {\partial_\theta u_\theta} + u_\theta \cot\theta}{r^2} - \frac{2}{r^2 \sin\theta} {\partial_\varphi u_\varphi} \right) \\
 &\qquad  + \frac{1}{3}\mu \partial_r \left( \frac{1}{r^2} \partial_r\left(r^2 u_r\right) + \frac{1}{r \sin\theta} \partial_\theta \left( u_\theta\sin\theta \right) + \frac{1}{r\sin\theta} {\partial_\varphi u_\varphi} \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 \sin\theta} {\partial_\varphi u_\varphi} + \frac{u_\theta}{r} {\partial_\theta u_\varphi} +
 \frac{u_r u_\varphi + u_\varphi u_\theta \cot\theta}{r}\right) \\
 &\quad = -\frac{1}{r \sin\theta} {\partial_\varphi p} \\
 &\qquad  + \mu \left(\frac{1}{r^2} \partial_r \left(r^2 {\partial_r u_\varphi}\right) + \frac{1}{r^2 \sin^2\theta} {\partial_\varphi^2 u_\varphi} + \frac{1}{r^2 \sin\theta} \partial_\theta \left(\sin\theta {\partial_\theta u_\varphi}\right) + \frac{2 \sin\theta {\partial_\varphi u_r} + 2 \cos\theta {\partial_\varphi u_\theta} - u_\varphi}{r^2 \sin^2\theta} \right) \\
 &\qquad  + \frac{1}{3}\mu\frac{1}{r \sin\theta} \partial_\varphi \left( \frac{1}{r^2} \partial_r \left(r^2 u_r\right) + \frac{1}{r \sin\theta} \partial_\theta \left( u_\theta\sin\theta \right) + \frac{1}{r\sin\theta} {\partial_\varphi u_\varphi} \right) \\
 &\qquad  + \rho g_\varphi \\[8px]
\end{align}</math>
<math display="block">\begin{align}
 \theta:\ &\rho \left({\partial_t u_\theta} + u_r {\partial_r u_\theta} +
 \frac{u_\varphi}{r \sin\theta} {\partial_\varphi u_\theta} +
 \frac{u_\theta}{r} {\partial_\theta u_\theta} + \frac{u_r u_\theta - u_\varphi^2 \cot\theta}{r}\right) \\
 &\quad = -\frac{1}{r} {\partial_\theta p} \\
 &\qquad  + \mu \left(\frac{1}{r^2} \partial_r \left(r^2 {\partial_r u_\theta}\right) + \frac{1}{r^2 \sin^2\theta} {\partial_\varphi^2 u_\theta} + \frac{1}{r^2 \sin\theta} \partial_\theta \left(\sin\theta {\partial_\theta u_\theta}\right) + \frac{2}{r^2} {\partial_\theta u_r} - \frac{u_\theta + 2 \cos\theta {\partial_\varphi u_\varphi}}{r^2 \sin^2\theta} \right) \\
 &\qquad  + \frac{1}{3}\mu\frac{1}{r} \partial_\theta \left( \frac{1}{r^2} \partial_r \left(r^2 u_r\right) + \frac{1}{r \sin\theta} \partial_\theta  \left( u_\theta\sin\theta \right) + \frac{1}{r\sin\theta} {\partial_\varphi u_\varphi} \right) \\
 &\qquad  + \rho g_\theta.
\end{align}</math>

Mass continuity will read:
<math display="block">{\partial_t \rho} + \frac{1}{r^2} \partial_r \left(\rho r^2 u_r\right) + \frac{1}{r \sin\theta}{\partial_\varphi (\rho u_\varphi)} + \frac{1}{r \sin\theta} \partial_\theta \left(\sin\theta \rho u_\theta\right) = 0.</math>

These equations could be (slightly) compacted by, for example, factoring <math display="inline">\frac{1}{r^2}</math> from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities.