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

$$\sigma_{rr} =-p + 2\,\mu\,\frac{\partial v_r}{\partial r},$$ (1.142)

$$\sigma_{\theta\theta} =-p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$$ (1.143)

$$\sigma_{zz} = -p + 2\,\mu\,\frac{\partial v_z}{\partial z},$$ (1.144)

$$\sigma_{r\theta}=\sigma_{\theta r} =\mu\left(\frac{1}{r}\,\frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$$ (1.145)

$$\sigma_{rz}=\sigma_{zr} =\mu\left(\frac{\partial v_r}{\partial z} + \frac{\partial v_z}{\partial r}\right),$$ (1.146)

$$\sigma_{\theta z} = \sigma_{z\theta} = \mu\left(\frac{1}{r}\,\frac{\partial v_z}{\partial\theta}+\frac{\partial v_\theta}{\partial z}\right),$$ (1.147)

whereas the equations of compressible fluid flow become

$$\frac{D\rho}{Dt} =-\rho\,{\mit\Delta},$$ (1.148)

$$\frac{Dv_r}{Dt}-\frac{v_\theta^{\,2}}{r} = - \frac{1}{\rho}\,\frac{\partial p}{\partial r} - \frac{\partial{\mit\Psi}}{\partial r}$$

$$\phantom{=}+ \frac{\mu}{\rho}\left(\nabla^{\,2} v_r -\frac{v_r}{r... ...}{\partial\theta}+ \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial r}\right),$$ (1.149)

$$\frac{Dv_\theta}{Dt}+\frac{v_r\,v_\theta}{r} = - \frac{1}{\rho\,r}\,\frac{\partial p}{\partial \theta} - \frac{1}{r}\frac{\partial{\mit\Psi}}{\partial \theta}$$

$$\phantom{=}+\frac{\mu}{\rho}\left(\nabla^{\,2} v_\theta +\frac{2}... ...{r^{\,2}}+ \frac{1}{3\,r}\,\frac{\partial{\mit\Delta}}{\partial \theta}\right),$$ (1.150)

$$\frac{Dv_z}{Dt} = - \frac{1}{\rho}\,\frac{\partial p}{\partial z} - \frac{\partia... ...\nabla^{\,2} v_z + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial z}\right),$$ (1.151)

$$\frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right) =\chi + \frac{\kappa\,M}R\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$$ (1.152)

where

$${\mit\Delta} =\frac{1}{r}\,\frac{\partial (r\,v_r)}{\partial r} +\frac{1}{r}\, \frac{\partial v_\theta}{\partial \theta} + \frac{\partial v_z}{\partial z},$$ (1.153)

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + v_r\,\frac{\partial }{\partial r}... ...theta}{r}\,\frac{\partial}{\partial \theta} + v_z\,\frac{\partial}{\partial z},$$ (1.154)

$$\nabla^{\,2} =\frac{1}{r}\frac{\partial}{\partial r}\!\left(r\,\frac{\partial}... ...\partial^{\,2}}{\partial \theta^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}},$$ (1.155)

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

$$\phantom{=}\left.+\frac{1}{2}\left(\frac{\partial v_r}{\partial z... ...{\partial z}+\frac{1}{r}\,\frac{\partial v_z}{\partial \theta}\right)^2\right].$$ (1.156)