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Fluid Equations in Cylindrical Coordinates
Let us adopt the cylindrical coordinate system, (
,
,
). Making use of the results quoted
in Section C.3, the components of the stress tensor are
![$\displaystyle \sigma_{rr}$](img473.png) |
![$\displaystyle =-p + 2\,\mu\,\frac{\partial v_r}{\partial r},$](img474.png) |
(1.142) |
![$\displaystyle \sigma_{\theta\theta}$](img475.png) |
![$\displaystyle =-p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$](img476.png) |
(1.143) |
![$\displaystyle \sigma_{zz}$](img444.png) |
![$\displaystyle = -p + 2\,\mu\,\frac{\partial v_z}{\partial z},$](img445.png) |
(1.144) |
![$\displaystyle \sigma_{r\theta}=\sigma_{\theta r}$](img477.png) |
![$\displaystyle =\mu\left(\frac{1}{r}\,\frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$](img478.png) |
(1.145) |
![$\displaystyle \sigma_{rz}=\sigma_{zr}$](img479.png) |
![$\displaystyle =\mu\left(\frac{\partial v_r}{\partial z} + \frac{\partial v_z}{\partial r}\right),$](img480.png) |
(1.146) |
![$\displaystyle \sigma_{\theta z} = \sigma_{z\theta}$](img481.png) |
![$\displaystyle = \mu\left(\frac{1}{r}\,\frac{\partial v_z}{\partial\theta}+\frac{\partial v_\theta}{\partial z}\right),$](img482.png) |
(1.147) |
whereas the equations of compressible fluid flow become
![$\displaystyle \frac{D\rho}{Dt}$](img346.png) |
![$\displaystyle =-\rho\,{\mit\Delta},$](img454.png) |
(1.148) |
![$\displaystyle \frac{Dv_r}{Dt}-\frac{v_\theta^{\,2}}{r}$](img483.png) |
![$\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial r} - \frac{\partial{\mit\Psi}}{\partial r}$](img484.png) |
|
|
![$\displaystyle \phantom{=}+ \frac{\mu}{\rho}\left(\nabla^{\,2} v_r -\frac{v_r}{r...
...}{\partial\theta}+ \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial r}\right),$](img485.png) |
(1.149) |
![$\displaystyle \frac{Dv_\theta}{Dt}+\frac{v_r\,v_\theta}{r}$](img486.png) |
![$\displaystyle = - \frac{1}{\rho\,r}\,\frac{\partial p}{\partial \theta} - \frac{1}{r}\frac{\partial{\mit\Psi}}{\partial \theta}$](img487.png) |
|
|
![$\displaystyle \phantom{=}+\frac{\mu}{\rho}\left(\nabla^{\,2} v_\theta +\frac{2}...
...{r^{\,2}}+ \frac{1}{3\,r}\,\frac{\partial{\mit\Delta}}{\partial \theta}\right),$](img488.png) |
(1.150) |
![$\displaystyle \frac{Dv_z}{Dt}$](img459.png) |
![$\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial z} - \frac{\partia...
...\nabla^{\,2} v_z + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial z}\right),$](img460.png) |
(1.151) |
![$\displaystyle \frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right)$](img461.png) |
![$\displaystyle =\chi + \frac{\kappa\,M}R\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$](img489.png) |
(1.152) |
where
Next: Fluid Equations in Spherical
Up: Mathematical Models of Fluid
Previous: Fluid Equations in Cartesian
Richard Fitzpatrick
2016-03-31