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

$$\sigma_{rr} =-p + 2\,\mu\,\frac{\partial v_r}{\partial r},$$ (1.157)

$$\sigma_{\theta\theta} =-p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$$ (1.158)

$$\sigma_{\varphi\varphi} =-p + 2\,\mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_\varphi}{\partial \varphi}+\frac{v_r}{r}+\frac{\cot\theta\,v_\theta}{r}\right),$$ (1.159)

$$\sigma_{r\theta}=\sigma_{\theta r} =\mu\left(\frac{1}{r}\,\frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$$ (1.160)

$$\sigma_{r\varphi}=\sigma_{\varphi r} =\mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_r}{\partial \varphi} + \frac{\partial v_\varphi}{\partial r}-\frac{v_\varphi}{r}\right),$$ (1.161)

$$\sigma_{\theta \varphi} = \sigma_{\varphi\theta} = \mu\left(\frac{1}{r\,\sin\theta}\,\frac{\partial v_\theta}{\par... ...rac{\partial v_\varphi}{\partial\theta}-\frac{\cot\theta\,v_\varphi}{r}\right),$$ (1.162)

whereas the equations of compressible fluid flow become

$$\frac{D\rho}{Dt} =-\rho\,{\mit\Delta},$$ (1.163)

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

$$\phantom{=}\left.-\frac{2\cot\theta\,v_\theta}{r^{\,2}}-\frac{2}{... ...{\partial\varphi}+ \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial r}\right),$$ (1.164)

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

$$\phantom{=}\left.-\frac{v_\theta}{r^{\,2}\,\sin^2\theta}-\frac{2\... ...l\varphi}+ \frac{1}{3\,r}\,\frac{\partial{\mit\Delta}}{\partial \theta}\right),$$ (1.165)

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

$$\phantom{=}\left. + \frac{2}{r^{\,2}\,\sin^2\theta}\,\frac{\parti... ...rac{1}{3\,r\,\sin\theta}\,\frac{\partial{\mit\Delta}}{\partial \varphi}\right),$$ (1.166)

$$\frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right) =\chi + \frac{\kappa\,M}R\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$$ (1.167)

where

$${\mit\Delta} =\frac{1}{r^{\,2}}\,\frac{\partial (r^{\,2}\,v_r)}{\partial r} +\... ...\theta} + \frac{1}{r\,\sin\theta}\,\frac{\partial v_\varphi}{\partial \varphi},$$ (1.168)

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + v_r\,\frac{\partial }{\partial r}... ...l \theta} + \frac{v_\varphi}{r\,\sin\theta}\,\frac{\partial}{\partial \varphi},$$ (1.169)

$$\nabla^{\,2} =\frac{1}{r^{\,2}}\,\frac{\partial}{\partial r}\!\left(r^{\,2}\,\... ...\frac{1}{r^{\,2}\,\sin^2\theta}\,\frac{\partial^{\,2}}{\partial \varphi^{\,2}},$$ (1.170)

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

$$\phantom{=}+\frac{1}{2}\left(\frac{1}{r}\,\frac{\partial v_r}{\pa... ...ial \varphi}+\frac{\partial v_\varphi}{\partial r}-\frac{v_\varphi}{r}\right)^2$$

$$\phantom{=}\left.+\frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac... ...al v_\varphi}{\partial \theta}-\frac{\cot\theta\,v_\varphi}{r}\right)^2\right].$$ (1.171)