Fluid Equations in Spherical Coordinates

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

$$\sigma_{rr} = -p + 2\,\mu\,\frac{\partial v_r}{\partial r},$$ (157)

$$\sigma_{\theta\theta} = -p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$$ (158)

$$\sigma_{\phi\phi} = -p + 2\,\mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_\phi}{\partial \phi}+\frac{v_r}{r}+\frac{\cot\theta\,v_\theta}{r}\right),$$ (159)

$$\sigma_{r\theta}=\sigma_{\theta r} = \mu\left(\frac{1}{r}\,\frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$$ (160)

$$\sigma_{r\phi}=\sigma_{\phi r} = \mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_r}{\partial \phi} + \frac{\partial v_\phi}{\partial r}-\frac{v_\phi}{r}\right),$$ (161)

$$\sigma_{\theta \phi} = \sigma_{\phi\theta} = \mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_\theta}{\parti... ...r}\,\frac{\partial v_\phi}{\partial\theta}-\frac{\cot\theta\,v_\phi}{r}\right),$$ (162)

whereas the equations of compressible fluid flow become

$$\frac{D\rho}{Dt} = -\rho\,\Delta,$$ (163)

$$\frac{Dv_r}{Dt}-\frac{v_\theta^{\,2}+v_\phi^{\,2}}{r} = - \frac{1}{\rho}\,\frac{\partial p}{\partial r} - \frac{\partial{... ...frac{2 v_r}{r^2}-\frac{2}{r^2}\,\frac{\partial v_\theta}{\partial\theta}\right.$$

$$\left.-\frac{2\cot\theta\,v_\theta}{r^2}-\frac{2}{r^2\,\sin\theta... ...l v_\phi}{\partial\phi}+ \frac{1}{3}\,\frac{\partial\Delta}{\partial r}\right),$$ (164)

$$\frac{Dv_\theta}{Dt}+\frac{v_r\,v_\theta-\cot\theta\,v_\phi^{\,2}}{r} = - \frac{1}{\rho\,r}\,\frac{\partial p}{\partial \theta} - \frac{1... ...t(\nabla^2 v_\theta +\frac{2}{r^2}\,\frac{\partial v_r}{\partial\theta} \right.$$ (165)

$$\left.-\frac{v_\theta}{r^2\,\sin^2\theta}-\frac{2\,\cot\theta}{r^... ...hi}{\partial\phi}+ \frac{1}{3r}\,\frac{\partial\Delta}{\partial \theta}\right),$$ (166)

$$\frac{Dv_\phi}{Dt}+\frac{v_r\,v_\phi + \cot\theta\,v_\theta\,v_\phi}{r} = - \frac{1}{\rho\,r\,\sin\theta}\,\frac{\partial p}{\partial \phi}... ... \frac{\mu}{\rho}\left(\nabla^2 v_\phi-\frac{v_\phi}{r^2\,\sin^2\theta} \right.$$

$$\left. + \frac{2}{r^2\,\sin^2\theta}\,\frac{\partial v_r}{\partia... ...al\phi}+ \frac{1}{3r\,\sin\theta}\,\frac{\partial\Delta}{\partial \phi}\right),$$ (167)

$$\frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right) = \chi + \frac{\kappa\,{\cal M}}{\cal R}\,\nabla^2\left(\frac{p}{\rho}\right),$$ (168)

where

$$\Delta = \frac{1}{r^2}\,\frac{\partial (r^2\,v_r)}{\partial r} +\frac{1}{r... ...rtial \theta} + \frac{1}{r\,\sin\theta}\,\frac{\partial v_\phi}{\partial \phi},$$ (169)

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + v_r\,\frac{\partial }{\partial r} +... ...partial \theta} + \frac{v_\phi}{r\,\sin\theta}\,\frac{\partial}{\partial \phi},$$ (170)

$$\nabla^2 = \frac{1}{r^2}\,\frac{\partial}{\partial r}\!\left(r^2\,\frac{\par... ...\theta}\right)+\frac{1}{r^2\,\sin^2\theta}\,\frac{\partial^2}{\partial \phi^2},$$ (171)

$$\chi = 2\mu\left[\left(\frac{\partial v_r}{\partial r}\right)^2+\left(\f... ...hi}{\partial \phi}+\frac{v_r}{r}+\frac{\cot\theta\,v_\theta}{r}\right)^2\right.$$

$$+\frac{1}{2}\left(\frac{1}{r}\,\frac{\partial v_r}{\partial \thet... ..._r}{\partial \phi}+\frac{\partial v_\phi}{\partial r}-\frac{v_\phi}{r}\right)^2$$

$$\left.+\frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v... ...\partial v_\phi}{\partial \theta}-\frac{\cot\theta\,v_\phi}{r}\right)^2\right].$$ (172)