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

$$\sigma_{rr} = -p + 2\,\mu\,\frac{\partial v_r}{\partial r},$$ (142)

$$\sigma_{\theta\theta} = -p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$$ (143)

$$\sigma_{zz} = -p + 2\,\mu\,\frac{\partial v_z}{\partial z},$$ (144)

$$\sigma_{r\theta}=\sigma_{\theta r} = \mu\left(\frac{1}{r}\,\frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$$ (145)

$$\sigma_{rz}=\sigma_{zr} = \mu\left(\frac{\partial v_r}{\partial z} + \frac{\partial v_z}{\partial r}\right),$$ (146)

$$\sigma_{\theta z} = \sigma_{z\theta} = \mu\left(\frac{1}{r}\,\frac{\partial v_z}{\partial\theta}+\frac{\partial v_\theta}{\partial z}\right),$$ (147)

whereas the equations of compressible fluid flow become

$$\frac{D\rho}{Dt} = -\rho\,\Delta,$$ (148)

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

$$+ \frac{\mu}{\rho}\left(\nabla^2 v_r -\frac{v_r}{r^2}-\frac{2}{r^... ...\theta}{\partial\theta}+ \frac{1}{3}\,\frac{\partial\Delta}{\partial r}\right),$$ (149)

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

$$+\frac{\mu}{\rho}\left(\nabla^2 v_\theta +\frac{2}{r^2}\,\frac{\p... ...ac{v_\theta}{r^2}+ \frac{1}{3r}\,\frac{\partial\Delta}{\partial \theta}\right),$$ (150)

$$\frac{Dv_z}{Dt} = - \frac{1}{\rho}\,\frac{\partial p}{\partial z} - \frac{\partial{... ...rho}\left(\nabla^2 v_z + \frac{1}{3}\,\frac{\partial\Delta}{\partial z}\right),$$ (151)

$$\frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right) = \chi + \frac{\kappa\,{\cal M}}{\cal R}\,\nabla^2\left(\frac{p}{\rho}\right),$$ (152)

where

$$\Delta = \frac{1}{r}\,\frac{\partial (r\,v_r)}{\partial r} +\frac{1}{r}\, \frac{\partial v_\theta}{\partial \theta} + \frac{\partial v_z}{\partial z},$$ (153)

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + v_r\,\frac{\partial }{\partial r} +... ...theta}{r}\,\frac{\partial}{\partial \theta} + v_z\,\frac{\partial}{\partial z},$$ (154)

$$\nabla^2 = \frac{1}{r}\frac{\partial}{\partial r}\!\left(r\,\frac{\partial}{... ...1}{r^2}\, \frac{\partial^2}{\partial \theta^2}+\frac{\partial^2}{\partial z^2},$$ (155)

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

$$\left.+\frac{1}{2}\left(\frac{\partial v_r}{\partial z}+\frac{\pa... ...{\partial z}+\frac{1}{r}\,\frac{\partial v_z}{\partial \theta}\right)^2\right].$$ (156)