Cylindrical3D

Equations Cast in Cylindrical Coordinates

Spatial Operators in Cylindrical Coordinates
$\mathrm{\nabla }f$	=	${\hat{e}}_{\varpi }\left[\frac{\partial f}{\partial \varpi }\right]+{\hat{e}}_{\phi }\left[\frac{1}{\varpi }\frac{\partial f}{\partial \phi }\right]+{\hat{e}}_z\left[\frac{\partial f}{\partial z}\right];$
${\mathrm{\nabla }}^{2}f$	=	$\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[\varpi \frac{\partial f}{\partial \varpi }\right]+\frac{1}{{\varpi }^2}\frac{{\partial }^{2}f}{\partial {\phi }^2}+\frac{{\partial }^{2}f}{\partial z^2};$
$(\overrightarrow{v}\cdot \mathrm{\nabla })f$	=	$\left[v_{\varpi }\frac{\partial f}{\partial \varpi }\right]+\left[\frac{v_{\phi }}{\varpi }\frac{\partial f}{\partial \phi }\right]+\left[v_z\frac{\partial f}{\partial z}\right];$
$\mathrm{\nabla }\cdot \overrightarrow{F}$	=	$\frac{1}{\varpi }\frac{\partial (\varpi F_{\varpi })}{\partial \varpi }+\frac{1}{\varpi }\frac{\partial F_{\phi }}{\partial \phi }+\frac{\partial F_z}{\partial z};$

Vector Time-Derivatives in Cylindrical Coordinates
$\frac{d}{dt}\overrightarrow{F}$	=	${\hat{e}}_{\varpi }\frac{d{F}_{\varpi }}{dt}+F_{\varpi }\frac{d{\hat{e}}_{\varpi }}{dt}+{\hat{e}}_{\phi }\frac{d{F}_{\phi }}{dt}+F_{\phi }\frac{d{\hat{e}}_{\phi }}{dt}+{\hat{e}}_z\frac{d{F}_z}{dt}+F_z\frac{d{\hat{e}}_z}{dt}$
	=	${\hat{e}}_{\varpi }[\frac{d{F}_{\varpi }}{dt}-F_{\phi }\dot{\phi }]+{\hat{e}}_{\phi }[\frac{d{F}_{\phi }}{dt}+F_{\varpi }\dot{\phi }]+{\hat{e}}_z\frac{d{F}_z}{dt};$
$\overrightarrow{v}=\frac{d\overrightarrow{x}}{dt}=\frac{d}{dt}[{\hat{e}}_{\varpi }\varpi +{\hat{e}}_{z}z]$	=	${\hat{e}}_{\varpi }\left[\dot{\varpi }\right]+{\hat{e}}_{\phi }\left[\varpi \dot{\phi }\right]+{\hat{e}}_z\left[\dot{z}\right].$

Governing Equations

Introducing the above expressions into the principal governing equations gives,

Equation of Continuity

$\frac{d\rho }{dt}+\frac{\rho }{\varpi }\frac{\partial }{\partial \varpi }\left[\varpi \dot{\varpi }\right]+\frac{1}{\varpi }\frac{\partial }{\partial \phi }\left[\varpi \dot{\phi }\right]+\rho \frac{\partial }{\partial z}\left[\dot{z}\right]=0$

Euler Equation

${\hat{e}}_{\varpi }[\frac{d\dot{\varpi }}{dt}-\varpi {\dot{\phi }}^2]+{\hat{e}}_{\phi }[\frac{d(\varpi \dot{\phi })}{dt}+\dot{\varpi }\dot{\phi }]+{\hat{e}}_z\left[\frac{d\dot{z}}{dt}\right]=-{\hat{e}}_{\varpi }[\frac{1}{\rho }\frac{\partial P}{\partial \varpi }+\frac{\partial \mathrm{\Phi }}{\partial \varpi }]-{\hat{e}}_{\phi }\frac{1}{\varpi }[\frac{1}{\rho }\frac{\partial P}{\partial \phi }+\frac{\partial \mathrm{\Phi }}{\partial \phi }]-{\hat{e}}_z[\frac{1}{\rho }\frac{\partial P}{\partial z}+\frac{\partial \mathrm{\Phi }}{\partial z}]$

Adiabatic Form of the
First Law of Thermodynamics

$\frac{dϵ}{dt}+P\frac{d}{dt}\left(\frac{1}{\rho }\right)=0$

Poisson Equation

$\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[\varpi \frac{\partial \mathrm{\Phi }}{\partial \varpi }\right]+\frac{1}{{\varpi }^2}\frac{{\partial }^2\mathrm{\Phi }}{\partial {\phi }^2}+\frac{{\partial }^2\mathrm{\Phi }}{\partial z^2}=4\pi G\rho .$

Eulerian Formulation

Each of the above simplified governing equations has been written in terms of Lagrangian time derivatives. An Eulerian formulation of each equation can be obtained by replacing each Lagrangian time derivative by its Eulerian counterpart. Specifically, for any scalar function, $f$,

$\frac{df}{dt}\to \frac{\partial f}{\partial t}+(\overrightarrow{v}\cdot \mathrm{\nabla })f=\frac{\partial f}{\partial t}+\left[\dot{\varpi }\frac{\partial f}{\partial \varpi }\right]+\left[\dot{\phi }\frac{\partial f}{\partial \phi }\right]+\left[\dot{z}\frac{\partial f}{\partial z}\right].$

Hence,

Equation of Continuity

$\frac{\partial \rho }{\partial t}+\left[\dot{\varpi }\frac{\partial \rho }{\partial \varpi }\right]+\frac{\rho }{\varpi }\frac{\partial }{\partial \varpi }\left[\varpi \dot{\varpi }\right]+\left[\dot{\phi }\frac{\partial \rho }{\partial \phi }\right]+\frac{1}{\varpi }\frac{\partial }{\partial \phi }\left[\varpi \dot{\phi }\right]+\left[\dot{z}\frac{\partial \rho }{\partial z}\right]+\rho \frac{\partial }{\partial z}\left[\dot{z}\right]=0$

$\Rightarrow \frac{\partial \rho }{\partial t}+\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[\rho \varpi \dot{\varpi }\right]+\frac{1}{\varpi }\frac{\partial }{\partial \phi }\left[\rho \varpi \dot{\phi }\right]+\frac{\partial }{\partial z}\left[\rho \dot{z}\right]=0$

Assuming that the initial (subscript i) configuration is axisymmetric and that, following perturbation, each physical parameter, $Q$, behaves according to the relation,

$Q(\varpi ,\phi ,z,t)=[q_i(\varpi ,z)+\delta q(\varpi ,z,t)e^{im\phi }]\mathrm{a}\mathrm{n}\mathrm{d}\delta q/q_i\ll 1,$

the linearized form of the continuity equation becomes:

(This has been obtained by combining the expressions highlighted with a lightblue background color from the accompanying table.)
$e^{im\phi }\left[\frac{\partial (\delta \rho )}{\partial t}\right]$	$=$	$\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[{\rho }_i\varpi {\dot{\varpi }}_i\right]+\frac{\partial }{\partial z}\left[{\rho }_i{\dot{z}}_i\right]$   $+e^{im\phi }\left\{im\right[{\rho }_i(\delta \dot{\phi })+{\dot{\phi }}_i(\delta \rho )\left]\right\}$ $+e^{im\phi }\{\frac{{\rho }_i}{\varpi }(\delta \dot{\varpi })+\frac{{\dot{\varpi }}_i}{\varpi }(\delta \rho )+(\delta \rho )\frac{\partial {\dot{\varpi }}_i}{\partial \varpi }$ $+({\rho }_i)\frac{\partial (\delta \dot{\varpi })}{\partial \varpi }+(\delta \dot{\varpi })\frac{\partial {\rho }_i}{\partial \varpi }+({\dot{\varpi }}_i)\frac{\partial (\delta \rho )}{\partial \varpi }$ $+{\rho }_i\frac{\partial (\delta \dot{z})}{\partial z}+\delta \rho \frac{\partial {\dot{z}}_i}{\partial z}+{\dot{z}}_i\frac{\partial (\delta \rho )}{\partial z}+(\delta \dot{z})\frac{\partial {\rho }_i}{\partial z}\}$

Linearize each term of the Continuity Equation assuming ...
$Q(\varpi ,\phi ,z,t)=[q_i(\varpi ,z)+\delta q(\varpi ,z,t)e^{im\phi }]\mathrm{a}\mathrm{n}\mathrm{d}\delta q/q_i\ll 1$			$\mathrm{a}\mathrm{n}\mathrm{d}{\dot{\varpi }}_i={\dot{z}}_i=0$
$\frac{\partial \rho }{\partial t}$	$\to$	$\cancel{\frac{\partial ({\rho }_i)}{\partial t}}+e^{im\phi }\left[\frac{\partial (\delta \rho )}{\partial t}\right]$
	$\to$	$e^{im\phi }\left[\frac{\partial (\delta \rho )}{\partial t}\right]$	$\to$	$e^{im\phi }\left[\frac{\partial (\delta \rho )}{\partial t}\right]$
$\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[\rho \varpi \dot{\varpi }\right]=\frac{\rho \dot{\varpi }}{\varpi }+\rho \frac{\partial \dot{\varpi }}{\partial \varpi }+\dot{\varpi }\frac{\partial \rho }{\partial \varpi }$	$\to$	$\frac{({\rho }_i+e^{im\phi }\delta \rho )({\dot{\varpi }}_i+e^{im\phi }\delta \dot{\varpi })}{\varpi }$ $+({\rho }_i+e^{im\phi }\delta \rho )\frac{\partial ({\dot{\varpi }}_i+e^{im\phi }\delta \dot{\varpi })}{\partial \varpi }$ $+({\dot{\varpi }}_i+e^{im\phi }\delta \dot{\varpi })\frac{\partial ({\rho }_i+e^{im\phi }\delta \rho )}{\partial \varpi }$
	$\to$	$\frac{{\rho }_i{\dot{\varpi }}_i}{\varpi }+e^{im\phi }[\frac{{\rho }_i}{\varpi }(\delta \dot{\varpi })+\frac{{\dot{\varpi }}_i}{\varpi }(\delta \rho )]+e^{2im\phi }\left[\cancel{\frac{(\delta \rho )(\delta \dot{\varpi })}{\varpi }}\right]$ $+({\rho }_i+e^{im\phi }\delta \rho )\frac{\partial {\dot{\varpi }}_i}{\partial \varpi }+e^{im\phi }[({\rho }_i+e^{im\phi }\cancel{\delta \rho })\frac{\partial (\delta \dot{\varpi })}{\partial \varpi }]$ $+({\dot{\varpi }}_i+e^{im\phi }\delta \dot{\varpi })\frac{\partial {\rho }_i}{\partial \varpi }+e^{im\phi }[({\dot{\varpi }}_i+e^{im\phi }\cancel{\delta \dot{\varpi }})\frac{\partial (\delta \rho )}{\partial \varpi }]$
	$\to$	$\frac{{\rho }_i{\dot{\varpi }}_i}{\varpi }+{\rho }_i\frac{\partial {\dot{\varpi }}_i}{\partial \varpi }+{\dot{\varpi }}_i\frac{\partial {\rho }_i}{\partial \varpi }$ $+e^{im\phi }[\frac{{\rho }_i}{\varpi }(\delta \dot{\varpi })+\frac{{\dot{\varpi }}_i}{\varpi }(\delta \rho )+(\delta \rho )\frac{\partial {\dot{\varpi }}_i}{\partial \varpi }$ $+({\rho }_i)\frac{\partial (\delta \dot{\varpi })}{\partial \varpi }+(\delta \dot{\varpi })\frac{\partial {\rho }_i}{\partial \varpi }+({\dot{\varpi }}_i)\frac{\partial (\delta \rho )}{\partial \varpi }]$	$\to$	$\cancel{\frac{{\rho }_i{\dot{\varpi }}_i}{\varpi }}+\cancel{{\rho }_i\frac{\partial {\dot{\varpi }}_i}{\partial \varpi }}+\cancel{{\dot{\varpi }}_i\frac{\partial {\rho }_i}{\partial \varpi }}$ $+e^{im\phi }[\frac{{\rho }_i}{\varpi }(\delta \dot{\varpi })+\cancel{\frac{{\dot{\varpi }}_i}{\varpi }(\delta \rho )}+\cancel{(\delta \rho )\frac{\partial {\dot{\varpi }}_i}{\partial \varpi }}$ $+({\rho }_i)\frac{\partial (\delta \dot{\varpi })}{\partial \varpi }+(\delta \dot{\varpi })\frac{\partial {\rho }_i}{\partial \varpi }+\cancel{({\dot{\varpi }}_i)\frac{\partial (\delta \rho )}{\partial \varpi }}]$
			$\to$	$+e^{im\phi }\left\{\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\right[\varpi {\rho }_i(\delta \dot{\varpi })\left]\right\}$
$\frac{1}{\varpi }\frac{\partial }{\partial \phi }\left[\rho \varpi \dot{\phi }\right]=\frac{\rho }{\varpi }\frac{\partial (\varpi \dot{\phi })}{\partial \phi }+\dot{\phi }\frac{\partial \rho }{\partial \phi }$	$\to$	$({\rho }_i+e^{im\phi }\delta \rho )\frac{\partial ({\dot{\phi }}_i+e^{im\phi }\delta \dot{\phi })}{\partial \phi }+({\dot{\phi }}_i+e^{im\phi }\delta \dot{\phi })\frac{\partial ({\rho }_i+e^{im\phi }\delta \rho )}{\partial \phi }$
	$\to$	$({\rho }_i+e^{im\phi }\delta \rho )\cancel{\frac{\partial ({\dot{\phi }}_i)}{\partial \phi }}+im{e}^{im\phi }({\rho }_i+e^{im\phi }\cancel{\delta \rho })(\delta \dot{\phi })$ $+({\dot{\phi }}_i+e^{im\phi }\delta \dot{\phi })\cancel{\frac{\partial ({\rho }_i)}{\partial \phi }}+im{e}^{im\phi }({\dot{\phi }}_i+e^{im\phi }\cancel{\delta \dot{\phi }})(\delta \rho )$
	$\to$	$im{e}^{im\phi }[{\rho }_i(\delta \dot{\phi })+{\dot{\phi }}_i(\delta \rho )]$	$\to$	$im{e}^{im\phi }[{\rho }_i(\delta \dot{\phi })+{\dot{\phi }}_i(\delta \rho )]$
$\frac{\partial }{\partial z}\left[\rho \dot{z}\right]$	$\to$	$({\rho }_i+e^{im\phi }\delta \rho )\frac{\partial ({\dot{z}}_i+e^{im\phi }\delta \dot{z})}{\partial z}+({\dot{z}}_i+e^{im\phi }\delta \dot{z})\frac{\partial ({\rho }_i+e^{im\phi }\delta \rho )}{\partial z}$
	$\to$	$({\rho }_i+e^{im\phi }\delta \rho )\frac{\partial ({\dot{z}}_i)}{\partial z}+e^{im\phi }({\rho }_i+e^{im\phi }\cancel{\delta \rho })\frac{\partial (\delta \dot{z})}{\partial z}$ $+({\dot{z}}_i+e^{im\phi }\delta \dot{z})\frac{\partial ({\rho }_i)}{\partial z}+e^{im\phi }({\dot{z}}_i+e^{im\phi }\cancel{\delta \dot{z}})\frac{\partial (\delta \rho )}{\partial z}$
	$\to$	${\rho }_i\frac{\partial {\dot{z}}_i}{\partial z}+{\dot{z}}_i\frac{\partial {\rho }_i}{\partial z}+e^{im\phi }[{\rho }_i\frac{\partial (\delta \dot{z})}{\partial z}+\delta \rho \frac{\partial {\dot{z}}_i}{\partial z}+{\dot{z}}_i\frac{\partial (\delta \rho )}{\partial z}+(\delta \dot{z})\frac{\partial {\rho }_i}{\partial z}]$	$\to$	${\rho }_i\cancel{\frac{\partial {\dot{z}}_i}{\partial z}}+\cancel{{\dot{z}}_i}\frac{\partial {\rho }_i}{\partial z}$ $+e^{im\phi }[{\rho }_i\frac{\partial (\delta \dot{z})}{\partial z}+\delta \rho \cancel{\frac{\partial {\dot{z}}_i}{\partial z}}+\cancel{{\dot{z}}_i}\frac{\partial (\delta \rho )}{\partial z}+(\delta \dot{z})\frac{\partial {\rho }_i}{\partial z}]$
			$\to$	$e^{im\phi }\left\{\frac{\partial }{\partial z}\right[{\rho }_i(\delta \dot{z})\left]\right\}$
Combining all terms:	$\to$	$e^{im\phi }\left[\frac{\partial (\delta \rho )}{\partial t}\right]=\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[{\rho }_i\varpi {\dot{\varpi }}_i\right]+\frac{\partial }{\partial z}\left[{\rho }_i{\dot{z}}_i\right]$     $+e^{im\phi }\{\frac{{\rho }_i}{\varpi }(\delta \dot{\varpi })+\frac{{\dot{\varpi }}_i}{\varpi }(\delta \rho )+(\delta \rho )\frac{\partial {\dot{\varpi }}_i}{\partial \varpi }$ $+({\rho }_i)\frac{\partial (\delta \dot{\varpi })}{\partial \varpi }+(\delta \dot{\varpi })\frac{\partial {\rho }_i}{\partial \varpi }+({\dot{\varpi }}_i)\frac{\partial (\delta \rho )}{\partial \varpi }$ $+im[{\rho }_i(\delta \dot{\phi })+{\dot{\phi }}_i(\delta \rho )]$ $+{\rho }_i\frac{\partial (\delta \dot{z})}{\partial z}+\delta \rho \frac{\partial {\dot{z}}_i}{\partial z}+{\dot{z}}_i\frac{\partial (\delta \rho )}{\partial z}+(\delta \dot{z})\frac{\partial {\rho }_i}{\partial z}\}$	$\to$	$+e^{im\phi }\left\{\frac{\partial }{\partial z}\right[{\rho }_i(\delta \dot{z})\left]\right\}$

$\varpi$ Component of Euler Equation

$\frac{d\dot{\varpi }}{dt}-\varpi {\dot{\phi }}^2=-\frac{1}{\rho }\frac{\partial P}{\partial \varpi }-\frac{\partial \mathrm{\Phi }}{\partial \varpi }$

$\to \frac{\partial \dot{\varpi }}{\partial t}+\left[\dot{\varpi }\frac{\partial \dot{\varpi }}{\partial \varpi }\right]+\left[\dot{\phi }\frac{\partial \dot{\varpi }}{\partial \phi }\right]+\left[\dot{z}\frac{\partial \dot{\varpi }}{\partial z}\right]-\varpi {\dot{\phi }}^2=-\frac{1}{\rho }\frac{\partial P}{\partial \varpi }-\frac{\partial \mathrm{\Phi }}{\partial \varpi }$

$\phi$ Component of Euler Equation

$\frac{d(\varpi \dot{\phi })}{dt}+\dot{\varpi }\dot{\phi }=-\frac{1}{\varpi }[\frac{1}{\rho }\frac{\partial P}{\partial \phi }+\frac{\partial \mathrm{\Phi }}{\partial \phi }]$

$\to \frac{\partial (\varpi \dot{\phi })}{\partial t}+\left[\dot{\varpi }\frac{\partial (\varpi \dot{\phi })}{\partial \varpi }\right]+\left[\dot{\phi }\frac{\partial (\varpi \dot{\phi })}{\partial \phi }\right]+\left[\dot{z}\frac{\partial (\varpi \dot{\phi })}{\partial z}\right]+\dot{\varpi }\dot{\phi }=-\frac{1}{\varpi }[\frac{1}{\rho }\frac{\partial P}{\partial \phi }+\frac{\partial \mathrm{\Phi }}{\partial \phi }]$

$z$ Component of Euler Equation

$\frac{d\dot{z}}{dt}=-\frac{1}{\rho }\frac{\partial P}{\partial z}-\frac{\partial \mathrm{\Phi }}{\partial z}$

$\to \frac{\partial \dot{z}}{\partial t}+\left[\dot{\varpi }\frac{\partial \dot{z}}{\partial \varpi }\right]+\left[\dot{\phi }\frac{\partial \dot{z}}{\partial \phi }\right]+\left[\dot{z}\frac{\partial \dot{z}}{\partial z}\right]=-\frac{1}{\rho }\frac{\partial P}{\partial z}-\frac{\partial \mathrm{\Phi }}{\partial z}$

See Also