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Fluid Equations in Spherical Coordinates

Let us, finally, adopt the spherical coordinate system, ($ r$ , $ \theta $ , $ \varphi$ ). Making use of the results quoted in Section C.4, the components of the stress tensor are

$\displaystyle \sigma_{rr}$ $\displaystyle =-p + 2\,\mu\,\frac{\partial v_r}{\partial r},$ (1.157)
$\displaystyle \sigma_{\theta\theta}$ $\displaystyle =-p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$ (1.158)
$\displaystyle \sigma_{\varphi\varphi}$ $\displaystyle =-p + 2\,\mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_\varphi}{\partial \varphi}+\frac{v_r}{r}+\frac{\cot\theta\,v_\theta}{r}\right),$ (1.159)
$\displaystyle \sigma_{r\theta}=\sigma_{\theta r}$ $\displaystyle =\mu\left(\frac{1}{r}\,\frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$ (1.160)
$\displaystyle \sigma_{r\varphi}=\sigma_{\varphi r}$ $\displaystyle =\mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_r}{\partial \varphi} + \frac{\partial v_\varphi}{\partial r}-\frac{v_\varphi}{r}\right),$ (1.161)
$\displaystyle \sigma_{\theta \varphi} = \sigma_{\varphi\theta}$ $\displaystyle = \mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_\theta}{\par...
...rac{\partial v_\varphi}{\partial\theta}-\frac{\cot\theta\,v_\varphi}{r}\right),$ (1.162)

whereas the equations of compressible fluid flow become

$\displaystyle \frac{D\rho}{Dt}$ $\displaystyle =-\rho\,{\mit\Delta},$ (1.163)
$\displaystyle \frac{Dv_r}{Dt}-\frac{v_\theta^{\,2}+v_\varphi^{\,2}}{r}$ $\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial r} - \frac{\partia...
..._r}{r^{\,2}}-\frac{2}{r^{\,2}}\,\frac{\partial v_\theta}{\partial\theta}\right.$    
  $\displaystyle \phantom{=}\left.-\frac{2\cot\theta\,v_\theta}{r^{\,2}}-\frac{2}{...
...{\partial\varphi}+ \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial r}\right),$ (1.164)
$\displaystyle \frac{Dv_\theta}{Dt}+\frac{v_r\,v_\theta-\cot\theta\,v_\varphi^{\,2}}{r}$ $\displaystyle = - \frac{1}{\rho\,r}\,\frac{\partial p}{\partial \theta} - \frac...
...^{\,2} v_\theta +\frac{2}{r^{\,2}}\,\frac{\partial v_r}{\partial\theta} \right.$    
  $\displaystyle \phantom{=}\left.-\frac{v_\theta}{r^{\,2}\,\sin^2\theta}-\frac{2\...
...l\varphi}+ \frac{1}{3\,r}\,\frac{\partial{\mit\Delta}}{\partial \theta}\right),$ (1.165)
$\displaystyle \frac{Dv_\varphi}{Dt}+\frac{v_r\,v_\varphi + \cot\theta\,v_\theta\,v_\varphi}{r}$ $\displaystyle = - \frac{1}{\rho\,r\,\sin\theta}\,\frac{\partial p}{\partial \va...
...ho}\left(\nabla^{\,2} v_\varphi-\frac{v_\varphi}{r^{\,2}\,\sin^2\theta} \right.$    
  $\displaystyle \phantom{=}\left. + \frac{2}{r^{\,2}\,\sin^2\theta}\,\frac{\parti...
...rac{1}{3\,r\,\sin\theta}\,\frac{\partial{\mit\Delta}}{\partial \varphi}\right),$ (1.166)
$\displaystyle \frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right)$ $\displaystyle =\chi + \frac{\kappa\,M}R\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$ (1.167)

where

$\displaystyle {\mit\Delta}$ $\displaystyle =\frac{1}{r^{\,2}}\,\frac{\partial (r^{\,2}\,v_r)}{\partial r} +\...
...\theta} + \frac{1}{r\,\sin\theta}\,\frac{\partial v_\varphi}{\partial \varphi},$ (1.168)
$\displaystyle \frac{D}{Dt}$ $\displaystyle = \frac{\partial}{\partial t} + v_r\,\frac{\partial }{\partial r}...
...l \theta} + \frac{v_\varphi}{r\,\sin\theta}\,\frac{\partial}{\partial \varphi},$ (1.169)
$\displaystyle \nabla^{\,2}$ $\displaystyle =\frac{1}{r^{\,2}}\,\frac{\partial}{\partial r}\!\left(r^{\,2}\,\...
...\frac{1}{r^{\,2}\,\sin^2\theta}\,\frac{\partial^{\,2}}{\partial \varphi^{\,2}},$ (1.170)
$\displaystyle \chi$ $\displaystyle =2\,\mu\left[\left(\frac{\partial v_r}{\partial r}\right)^2+\left...
...{\partial \varphi}+\frac{v_r}{r}+\frac{\cot\theta\,v_\theta}{r}\right)^2\right.$    
  $\displaystyle \phantom{=}+\frac{1}{2}\left(\frac{1}{r}\,\frac{\partial v_r}{\pa...
...ial \varphi}+\frac{\partial v_\varphi}{\partial r}-\frac{v_\varphi}{r}\right)^2$    
  $\displaystyle \phantom{=}\left.+\frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac...
...al v_\varphi}{\partial \theta}-\frac{\cot\theta\,v_\varphi}{r}\right)^2\right].$ (1.171)


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Next: Exercises Up: Mathematical Models of Fluid Previous: Fluid Equations in Cylindrical
Richard Fitzpatrick 2016-03-31