Editing Navier–Stokes equations (section)

===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>