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.

The continuity equation reads:

\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} .

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