Editing Navier–Stokes equations (section)

===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.