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Fluid Equations in Spherical Coordinates
Let us, finally, adopt the spherical coordinate system, (
,
,
). Making use of the results quoted
in Section C.4, the components of the stress tensor are
![$\displaystyle \sigma_{rr}$](img473.png) |
![$\displaystyle =-p + 2\,\mu\,\frac{\partial v_r}{\partial r},$](img474.png) |
(1.157) |
![$\displaystyle \sigma_{\theta\theta}$](img475.png) |
![$\displaystyle =-p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$](img476.png) |
(1.158) |
![$\displaystyle \sigma_{\varphi\varphi}$](img496.png) |
![$\displaystyle =-p + 2\,\mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_\varphi}{\partial \varphi}+\frac{v_r}{r}+\frac{\cot\theta\,v_\theta}{r}\right),$](img497.png) |
(1.159) |
![$\displaystyle \sigma_{r\theta}=\sigma_{\theta r}$](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) |
(1.160) |
![$\displaystyle \sigma_{r\varphi}=\sigma_{\varphi r}$](img498.png) |
![$\displaystyle =\mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_r}{\partial \varphi} + \frac{\partial v_\varphi}{\partial r}-\frac{v_\varphi}{r}\right),$](img499.png) |
(1.161) |
![$\displaystyle \sigma_{\theta \varphi} = \sigma_{\varphi\theta}$](img500.png) |
![$\displaystyle = \mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_\theta}{\par...
...rac{\partial v_\varphi}{\partial\theta}-\frac{\cot\theta\,v_\varphi}{r}\right),$](img501.png) |
(1.162) |
whereas the equations of compressible fluid flow become
![$\displaystyle \frac{D\rho}{Dt}$](img346.png) |
![$\displaystyle =-\rho\,{\mit\Delta},$](img454.png) |
(1.163) |
![$\displaystyle \frac{Dv_r}{Dt}-\frac{v_\theta^{\,2}+v_\varphi^{\,2}}{r}$](img502.png) |
![$\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial r} - \frac{\partia...
..._r}{r^{\,2}}-\frac{2}{r^{\,2}}\,\frac{\partial v_\theta}{\partial\theta}\right.$](img503.png) |
|
|
![$\displaystyle \phantom{=}\left.-\frac{2\cot\theta\,v_\theta}{r^{\,2}}-\frac{2}{...
...{\partial\varphi}+ \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial r}\right),$](img504.png) |
(1.164) |
![$\displaystyle \frac{Dv_\theta}{Dt}+\frac{v_r\,v_\theta-\cot\theta\,v_\varphi^{\,2}}{r}$](img505.png) |
![$\displaystyle = - \frac{1}{\rho\,r}\,\frac{\partial p}{\partial \theta} - \frac...
...^{\,2} v_\theta +\frac{2}{r^{\,2}}\,\frac{\partial v_r}{\partial\theta} \right.$](img506.png) |
|
|
![$\displaystyle \phantom{=}\left.-\frac{v_\theta}{r^{\,2}\,\sin^2\theta}-\frac{2\...
...l\varphi}+ \frac{1}{3\,r}\,\frac{\partial{\mit\Delta}}{\partial \theta}\right),$](img507.png) |
(1.165) |
![$\displaystyle \frac{Dv_\varphi}{Dt}+\frac{v_r\,v_\varphi + \cot\theta\,v_\theta\,v_\varphi}{r}$](img508.png) |
![$\displaystyle = - \frac{1}{\rho\,r\,\sin\theta}\,\frac{\partial p}{\partial \va...
...ho}\left(\nabla^{\,2} v_\varphi-\frac{v_\varphi}{r^{\,2}\,\sin^2\theta} \right.$](img509.png) |
|
|
![$\displaystyle \phantom{=}\left. + \frac{2}{r^{\,2}\,\sin^2\theta}\,\frac{\parti...
...rac{1}{3\,r\,\sin\theta}\,\frac{\partial{\mit\Delta}}{\partial \varphi}\right),$](img510.png) |
(1.166) |
![$\displaystyle \frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right)$](img461.png) |
![$\displaystyle =\chi + \frac{\kappa\,M}R\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$](img489.png) |
(1.167) |
where
Next: Exercises
Up: Mathematical Models of Fluid
Previous: Fluid Equations in Cylindrical
Richard Fitzpatrick
2016-03-31