Fluid Equations in Cylindrical Coordinates

$\displaystyle \sigma_{rr}$	$\displaystyle =-p + 2\,\mu\,\frac{\partial v_r}{\partial r},$	(1.142)
$\displaystyle \sigma_{\theta\theta}$	$\displaystyle =-p + 2\,\mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$	(1.143)
$\displaystyle \sigma_{zz}$	$\displaystyle = -p + 2\,\mu\,\frac{\partial v_z}{\partial z},$	(1.144)
$\displaystyle \sigma_{r\theta}=\sigma_{\theta r}$	$\displaystyle =\mu\left(\frac{1}{r}\,\frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$	(1.145)
$\displaystyle \sigma_{rz}=\sigma_{zr}$	$\displaystyle =\mu\left(\frac{\partial v_r}{\partial z} + \frac{\partial v_z}{\partial r}\right),$	(1.146)
$\displaystyle \sigma_{\theta z} = \sigma_{z\theta}$	$\displaystyle = \mu\left(\frac{1}{r}\,\frac{\partial v_z}{\partial\theta}+\frac{\partial v_\theta}{\partial z}\right),$	(1.147)

$\displaystyle \frac{D\rho}{Dt}$	$\displaystyle =-\rho\,{\mit\Delta},$	(1.148)
$\displaystyle \frac{Dv_r}{Dt}-\frac{v_\theta^{\,2}}{r}$	$\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial r} - \frac{\partial{\mit\Psi}}{\partial r}$
	$\displaystyle \phantom{=}+ \frac{\mu}{\rho}\left(\nabla^{\,2} v_r -\frac{v_r}{r... ...}{\partial\theta}+ \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial r}\right),$	(1.149)
$\displaystyle \frac{Dv_\theta}{Dt}+\frac{v_r\,v_\theta}{r}$	$\displaystyle = - \frac{1}{\rho\,r}\,\frac{\partial p}{\partial \theta} - \frac{1}{r}\frac{\partial{\mit\Psi}}{\partial \theta}$
	$\displaystyle \phantom{=}+\frac{\mu}{\rho}\left(\nabla^{\,2} v_\theta +\frac{2}... ...{r^{\,2}}+ \frac{1}{3\,r}\,\frac{\partial{\mit\Delta}}{\partial \theta}\right),$	(1.150)
$\displaystyle \frac{Dv_z}{Dt}$	$\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial z} - \frac{\partia... ...\nabla^{\,2} v_z + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial z}\right),$	(1.151)
$\displaystyle \frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right)$	$\displaystyle =\chi + \frac{\kappa\,M}R\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$	(1.152)

$\displaystyle {\mit\Delta}$	$\displaystyle =\frac{1}{r}\,\frac{\partial (r\,v_r)}{\partial r} +\frac{1}{r}\, \frac{\partial v_\theta}{\partial \theta} + \frac{\partial v_z}{\partial z},$	(1.153)
$\displaystyle \frac{D}{Dt}$	$\displaystyle = \frac{\partial}{\partial t} + v_r\,\frac{\partial }{\partial r}... ...theta}{r}\,\frac{\partial}{\partial \theta} + v_z\,\frac{\partial}{\partial z},$	(1.154)
$\displaystyle \nabla^{\,2}$	$\displaystyle =\frac{1}{r}\frac{\partial}{\partial r}\!\left(r\,\frac{\partial}... ...\partial^{\,2}}{\partial \theta^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}},$	(1.155)
$\displaystyle \chi$	$\displaystyle =2\,\mu\left[\left(\frac{\partial v_r}{\partial r}\right)^2+\left... ...\theta}+\frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right)^2\right.$
	$\displaystyle \phantom{=}\left.+\frac{1}{2}\left(\frac{\partial v_r}{\partial z... ...{\partial z}+\frac{1}{r}\,\frac{\partial v_z}{\partial \theta}\right)^2\right].$	(1.156)