# Navier Stokes Equation in cylindrical coordinates

for incompressible flow, without external forces:

$\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\phi}^2}{r} = -\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_r}{\partial \phi^2} + \frac{\partial^2 u_r}{\partial z^2}-\frac{u_r}{r^2}-\frac{2}{r^2}\frac{\partial u_\phi}{\partial \phi} \right]$ $\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + u_z \frac{\partial u_{\phi}}{\partial z} + \frac{u_r u_{\phi}}{r} = -\frac{1}{r}\frac{\partial p}{\partial \phi} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_{\phi}}{\partial \phi^2} + \frac{\partial^2 u_{\phi}}{\partial z^2} + \frac{2}{r^2}\frac{\partial u_r}{\partial \phi} - \frac{u_{\phi}}{r^2}\right]$ $\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_z}{\partial \phi} + u_z \frac{\partial u_z}{\partial z} = -\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_z}{\partial \phi^2} + \frac{\partial^2 u_z}{\partial z^2}\right]$

Here $u$ is the velocity field written in the cylindrical coordinate system $(r,\phi,z)$, where $r$ is the radius and $\phi$ is the angle in the $(x,y)$ plane, $p$ is the pressure, and $\mu$ is the viscosity coefficient.

$\frac{1}{r}\frac{\partial}{\partial r}\left(r u_r\right) +\frac{1}{r}\frac{\partial (u_\phi)}{\partial \phi} + \frac{\partial u_z}{\partial z} = 0$

NB: $u_\phi$ is given by the formula $u_\phi = (-y u_x + x u_y)/r$, and the velocity field is:

$u = \frac{u_\phi}{r}\frac{\partial}{\partial \phi} + u_r \frac{\partial}{\partial r} + u_z \frac{\partial}{\partial z}$

Cylindrical coordinates are probably most convenient for the study of axisymmetric solutions, i.e. when the velocity field $u$ does not depend on the angle coordinate $\phi$. Then we get the following equations:

$\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\phi}^2}{r} = -\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{\partial^2 u_r}{\partial z^2}-\frac{u_r}{r^2} \right]$ $\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + u_z \frac{\partial u_{\phi}}{\partial z} + \frac{u_r u_{\phi}}{r} = \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) + \frac{\partial^2 u_{\phi}}{\partial z^2} - \frac{u_{\phi}}{r^2}\right]$ $\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + u_z \frac{\partial u_z}{\partial z} = -\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) + \frac{\partial^2 u_z}{\partial z^2}\right]$

The above equations still look too complicated and unintuitive. Instead of the variables $u_r, u_\theta, u_z$, we will use the new triple of variables $(u_\phi, \psi_\phi,\omega_\phi)$, where

$u_r = - \partial_z \psi_\theta, u_z = (1/r) \partial_r (r\psi_\phi)$

(such a $\psi_\phi$ exists because of the continuity equation), and $\omega_\phi$ is the the phi-component of the vorticity vector:

$\omega = - (\partial u_\phi / \partial z) \frac{\partial}{\partial r} + (\omega_\phi/r) \frac{\partial}{\partial \phi} + (1/r) (\partial (r u_\phi) /\partial r)) \frac{\partial}{\partial z}$.