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Fluid Equations in Cylindrical Coordinates

Let us adopt the cylindrical coordinate system, ($ r$ , $ \theta $ , $ z$ ). Making use of the results quoted in Section C.3, the components of the stress tensor are

$\displaystyle \sigma_{rr}$ $\displaystyle =-p + 2\,\mu\,\frac{\partial v_r}{\partial r},$ (1.142)
$\displaystyle \sigma_{\theta\theta}$ $\displaystyle =-p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$ (1.143)
$\displaystyle \sigma_{zz}$ $\displaystyle = -p + 2\,\mu\,\frac{\partial v_z}{\partial z},$ (1.144)
$\displaystyle \sigma_{r\theta}=\sigma_{\theta r}$ $\displaystyle =\mu\left(\frac{1}{r}\,\frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$ (1.145)
$\displaystyle \sigma_{rz}=\sigma_{zr}$ $\displaystyle =\mu\left(\frac{\partial v_r}{\partial z} + \frac{\partial v_z}{\partial r}\right),$ (1.146)
$\displaystyle \sigma_{\theta z} = \sigma_{z\theta}$ $\displaystyle = \mu\left(\frac{1}{r}\,\frac{\partial v_z}{\partial\theta}+\frac{\partial v_\theta}{\partial z}\right),$ (1.147)

whereas the equations of compressible fluid flow become

$\displaystyle \frac{D\rho}{Dt}$ $\displaystyle =-\rho\,{\mit\Delta},$ (1.148)
$\displaystyle \frac{Dv_r}{Dt}-\frac{v_\theta^{\,2}}{r}$ $\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial r} - \frac{\partial{\mit\Psi}}{\partial r}$    
  $\displaystyle \phantom{=}+ \frac{\mu}{\rho}\left(\nabla^{\,2} v_r -\frac{v_r}{r...
...}{\partial\theta}+ \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial r}\right),$ (1.149)
$\displaystyle \frac{Dv_\theta}{Dt}+\frac{v_r\,v_\theta}{r}$ $\displaystyle = - \frac{1}{\rho\,r}\,\frac{\partial p}{\partial \theta} - \frac{1}{r}\frac{\partial{\mit\Psi}}{\partial \theta}$    
  $\displaystyle \phantom{=}+\frac{\mu}{\rho}\left(\nabla^{\,2} v_\theta +\frac{2}...
...{r^{\,2}}+ \frac{1}{3\,r}\,\frac{\partial{\mit\Delta}}{\partial \theta}\right),$ (1.150)
$\displaystyle \frac{Dv_z}{Dt}$ $\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial z} - \frac{\partia...
...\nabla^{\,2} v_z + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial z}\right),$ (1.151)
$\displaystyle \frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right)$ $\displaystyle =\chi + \frac{\kappa\,M}R\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$ (1.152)

where

$\displaystyle {\mit\Delta}$ $\displaystyle =\frac{1}{r}\,\frac{\partial (r\,v_r)}{\partial r} +\frac{1}{r}\, \frac{\partial v_\theta}{\partial \theta} + \frac{\partial v_z}{\partial z},$ (1.153)
$\displaystyle \frac{D}{Dt}$ $\displaystyle = \frac{\partial}{\partial t} + v_r\,\frac{\partial }{\partial r}...
...theta}{r}\,\frac{\partial}{\partial \theta} + v_z\,\frac{\partial}{\partial z},$ (1.154)
$\displaystyle \nabla^{\,2}$ $\displaystyle =\frac{1}{r}\frac{\partial}{\partial r}\!\left(r\,\frac{\partial}...
...\partial^{\,2}}{\partial \theta^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}},$ (1.155)
$\displaystyle \chi$ $\displaystyle =2\,\mu\left[\left(\frac{\partial v_r}{\partial r}\right)^2+\left...
...\theta}+\frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right)^2\right.$    
  $\displaystyle \phantom{=}\left.+\frac{1}{2}\left(\frac{\partial v_r}{\partial z...
...{\partial z}+\frac{1}{r}\,\frac{\partial v_z}{\partial \theta}\right)^2\right].$ (1.156)


next up previous
Next: Fluid Equations in Spherical Up: Mathematical Models of Fluid Previous: Fluid Equations in Cartesian
Richard Fitzpatrick 2016-01-22