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Equations Cast in Cylindrical Coordinates

Spatial Operators in Cylindrical Coordinates


\nabla f

=

 
{\hat{e}}_\varpi \biggl[ \frac{\partial f}{\partial\varpi} \biggr] + {\hat{e}}_\varphi {\biggl[ \frac{1}{\varpi} \frac{\partial f}{\partial\varphi} \biggr]} +  {\hat{e}}_z \biggl[ \frac{\partial f}{\partial z} \biggr] ;


\nabla^2 f

=


\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial f}{\partial\varpi} \biggr] + {\frac{1}{\varpi^2} \frac{\partial^2 f}{\partial\varphi^2}} + \frac{\partial^2 f}{\partial z^2} ;


(\vec{v}\cdot\nabla)f

=


\biggl[ v_\varpi \frac{\partial f}{\partial\varpi} \biggr] + {\biggl[ \frac{v_\varphi}{\varpi} \frac{\partial f}{\partial\varphi} \biggr]} + 
\biggl[ v_z \frac{\partial f}{\partial z} \biggr] ;


\nabla \cdot \vec{F}

=


\frac{1}{\varpi} \frac{\partial (\varpi F_\varpi)}{\partial\varpi} + {\frac{1}{\varpi} \frac{\partial F_\varphi}{\partial\varphi}} + \frac{\partial F_z}{\partial z} ;

Vector Time-Derivatives in Cylindrical Coordinates


\frac{d}{dt}\vec{F}

=


{\hat{e}}_\varpi \frac{dF_\varpi}{dt} + F_\varpi \frac{d{\hat{e}}_\varpi}{dt} + {\hat{e}}_\varphi \frac{dF_\varphi}{dt} + F_\varphi \frac{d{\hat{e}}_\varphi}{dt} + {\hat{e}}_z \frac{dF_z}{dt} + F_z \frac{d{\hat{e}}_z}{dt}

 

=


{\hat{e}}_\varpi \biggl[ \frac{dF_\varpi}{dt} -  F_\varphi \dot\varphi  \biggr] + {\hat{e}}_\varphi \biggl[ \frac{dF_\varphi}{dt} + F_\varpi \dot\varphi \biggr]  + {\hat{e}}_z \frac{dF_z}{dt} ;


\vec{v} = \frac{d\vec{x}}{dt} = \frac{d}{dt}\biggl[ \hat{e}_\varpi \varpi + \hat{e}_z z \biggr]

=


{\hat{e}}_\varpi \biggl[ \dot\varpi \biggr] + 
{\hat{e}}_\varphi \biggl[ \varpi \dot\varphi \biggr]  + 
{\hat{e}}_z \biggl[ \dot{z} \biggr] .

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} \biggl[ \varpi \dot\varpi \biggr] + \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \varpi \dot\varphi \biggr]
+ \rho \frac{\partial}{\partial z} \biggl[ \dot{z} \biggr] = 0


Euler Equation


{\hat{e}}_\varpi \biggl[ \frac{d \dot\varpi}{dt} -  \varpi {\dot\varphi}^2  \biggr] + {\hat{e}}_\varphi \biggl[ \frac{d(\varpi\dot\varphi)}{dt} + \dot\varpi \dot\varphi \biggr]  + {\hat{e}}_z \biggl[ \frac{d \dot{z}}{dt} \biggr] = -
{\hat{e}}_\varpi \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] 
-  {\hat{e}}_\varphi \frac{1}{\varpi} \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial \varphi} + \frac{\partial \Phi}{\partial \varphi} \biggr] 
-  {\hat{e}}_z \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]


Adiabatic Form of the
First Law of Thermodynamics

~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0


Poisson Equation


\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] 
+ \frac{1}{\varpi^2} \frac{\partial^2 \Phi}{\partial \varphi^2} + \frac{\partial^2 \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} \rightarrow \frac{\partial f}{\partial t} + (\vec{v}\cdot \nabla)f =
\frac{\partial f}{\partial t} + \biggl[ \dot\varpi \frac{\partial f}{\partial\varpi} \biggr] + \biggl[ \dot\varphi \frac{\partial f}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial f}{\partial z} \biggr]   .

Hence,

Equation of Continuity


\frac{\partial\rho}{\partial t} + \biggl[ \dot\varpi \frac{\partial \rho}{\partial\varpi} \biggr] + \frac{\rho}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \dot\varpi \biggr] 
+ \biggl[ \dot\varphi \frac{\partial \rho}{\partial\varphi} \biggr] + \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \varpi \dot\varphi \biggr]
+ \biggl[ \dot{z} \frac{\partial \rho}{\partial z} \biggr] + \rho \frac{\partial}{\partial z} \biggl[ \dot{z} \biggr] = 0


\Rightarrow ~~~ \frac{\partial\rho}{\partial t} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] 
+ \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho \varpi \dot\varphi \biggr]
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 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, \varphi, z, t) = [q_i(\varpi, z) + \delta q(\varpi, z, t) e^{i m \varphi}] ~~~ \mathrm{and} ~~~ \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\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]

=


\frac{1}{\varpi} \frac{ \partial}{\partial\varpi} \biggl[ \rho_i \varpi \dot\varpi_i \biggr] 
+ \frac{\partial}{\partial z} \biggl[ \rho_i \dot z_i \biggr] 
  
+ e^{im\varphi} \biggl\{ im \biggl[ \rho_i ( \delta\dot\varphi) + \dot\varphi_i (\delta\rho) \biggr] \biggr\}


+ e^{im\varphi} \biggl\{ \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} \biggr\}

Linearize each term of the Continuity Equation assuming ...


Q(\varpi, \varphi, z, t) = [q_i(\varpi, z) + \delta q(\varpi, z, t) e^{i m \varphi}] ~~~ \mathrm{and} ~~~ \delta q/q_i \ll 1


\mathrm{and} ~~~  \dot\varpi_i = \dot z_i = 0

\frac{\partial\rho}{\partial t}

~~ \rightarrow ~~


\cancel{ \frac{\partial (\rho_i) }{\partial t} } + e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]

 

 

~~ \rightarrow ~~


e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]

~~~ \rightarrow ~~~


e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]

\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] =
\frac{\rho \dot\varpi}{\varpi} + \rho\frac{\partial \dot\varpi}{\partial\varpi} + \dot\varpi \frac{\partial \rho}{\partial\varpi}

~~ \rightarrow ~~


\frac{ (\rho_i + e^{im\varphi} \delta\rho) ( {\dot\varpi_i} + e^{im\varphi} \delta\dot\varpi)}{\varpi}


+ (\rho_i + e^{im\varphi} \delta\rho) \frac{\partial ( {\dot\varpi_i} + e^{im\varphi} \delta\dot\varpi)}{\partial\varpi}


+ ( {\dot\varpi_i} + e^{im\varphi} \delta\dot\varpi) \frac{\partial (\rho_i + e^{im\varphi} \delta\rho)}{\partial\varpi}

 

 

~~ \rightarrow ~~


\frac{ \rho_i \dot\varpi_i}{\varpi}  + e^{im\varphi} \biggl[ \frac{ \rho_i }{\varpi}  ( \delta\dot\varpi ) + \frac{ \dot\varpi_i }{\varpi}  ( \delta\rho ) \biggr] + 
e^{2im\varphi} \biggl[ \cancel{ \frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} } \biggr]


+ (\rho_i + e^{im\varphi} \delta\rho) \frac{\partial {\dot\varpi_i} }{\partial\varpi} + e^{im\varphi} \biggl[ (\rho_i + e^{im\varphi} \cancel{\delta\rho}) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi} \biggr]


+ ( {\dot\varpi_i} + e^{im\varphi} \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + e^{im\varphi}\biggl[ ( {\dot\varpi_i} + e^{im\varphi} \cancel{\delta\dot\varpi}) \frac{\partial (\delta\rho)}{\partial\varpi}  \biggr]

 

 

~~ \rightarrow ~~


\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\varphi} \biggl[ \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}  \biggr]

~~~~ \rightarrow ~~~~


\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\varphi} \biggl[ \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} }  \biggr]

 

~~~~ \rightarrow ~~~~


+ e^{im\varphi}  \biggl\{ \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \rho_i (\delta \dot\varpi) \biggr] \biggr\}

\frac{1}{\varpi} \frac{\partial}{\partial\varphi} \biggl[ \rho \varpi \dot\varphi \biggr] =
\frac{\rho}{\varpi} \frac{\partial (\varpi \dot\varphi) }{\partial\varphi} + \dot\varphi \frac{\partial \rho}{\partial\varphi}

~~ \rightarrow ~~


(\rho_i + e^{im\varphi} \delta\rho) \frac{\partial ( {\dot\varphi_i} + e^{im\varphi} \delta\dot\varphi)}{\partial\varphi} 
+ ( {\dot\varphi_i} +e^{im\varphi}  \delta\dot\varphi) \frac{\partial (\rho_i + e^{im\varphi} \delta\rho)}{\partial\varphi}

 

 

~~ \rightarrow ~~


(\rho_i + e^{im\varphi} \delta\rho) \cancel{ \frac{\partial ( {\dot\varphi_i} )}{\partial\varphi} } 
+ im  e^{im\varphi} (\rho_i + e^{im\varphi} \cancel{ \delta\rho })( \delta\dot\varphi)


+ ( {\dot\varphi_i} +e^{im\varphi}  \delta\dot\varphi) \cancel{ \frac{\partial (\rho_i )}{\partial\varphi} } 
+  im  e^{im\varphi} ( {\dot\varphi_i} +e^{im\varphi}  \cancel{ \delta\dot\varphi }) (\delta\rho)

 

 

~~ \rightarrow ~~


im  e^{im\varphi} \biggl[ \rho_i ( \delta\dot\varphi) + \dot\varphi_i (\delta\rho) \biggr]

~~~ \rightarrow ~~~


im  e^{im\varphi} \biggl[ \rho_i ( \delta\dot\varphi) + \dot\varphi_i (\delta\rho) \biggr]

\frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr]

~~ \rightarrow ~~


(\rho_i + e^{im\varphi} \delta\rho) \frac{\partial ( {\dot{z}_i} + e^{im\varphi} \delta\dot{z})}{\partial z} 
+ ( {\dot{z}_i} +e^{im\varphi}  \delta\dot{z}) \frac{\partial (\rho_i + e^{im\varphi} \delta\rho)}{\partial z}

 

 

~~ \rightarrow ~~


(\rho_i + e^{im\varphi} \delta\rho) { \frac{\partial ( {\dot{z}_i} )}{\partial z} } 
+ e^{im\varphi} (\rho_i + e^{im\varphi} \cancel{{ \delta\rho } } ) \frac{\partial ( \delta\dot{z})}{\partial z}


+ ( {\dot{z}_i} +e^{im\varphi}  \delta\dot{z}) \frac{\partial (\rho_i )}{\partial z}  
+  e^{im\varphi} ( {\dot{z}_i} +e^{im\varphi} \cancel{ \delta\dot{z} } ) \frac{\partial (\delta\rho)}{\partial z}

 

 

~~ \rightarrow ~~


\rho_i \frac{\partial \dot z_i }{\partial z} + \dot{z}_i \frac{\partial \rho_i}{\partial z}
+
e^{im\varphi} \biggl[ \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} \biggr]

~~~ \rightarrow ~~~


\rho_i \cancel{ \frac{\partial \dot z_i }{\partial z} } + \cancel{ \dot{z}_i } \frac{\partial \rho_i}{\partial z}


+ e^{im\varphi} \biggl[ \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} \biggr]

 

 

 

~~~ \rightarrow ~~~


e^{im\varphi} \biggl\{ \frac{\partial}{\partial z} \biggl[ \rho_i (\delta \dot z )  \biggr] \biggr\}

Combining all terms:

~~~ \rightarrow ~~~

e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr] = \frac{1}{\varpi} \frac{ \partial}{\partial\varpi} \biggl[ \rho_i \varpi \dot\varpi_i \biggr] 
+ \frac{\partial}{\partial z} \biggl[ \rho_i \dot z_i \biggr] 
    
+ e^{im\varphi} \biggl\{ \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 \biggl[ \rho_i ( \delta\dot\varphi) + \dot\varphi_i (\delta\rho) \biggr]


+ \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} \biggr\}

~~~ \rightarrow ~~~


+ e^{im\varphi} \biggl\{ \frac{\partial}{\partial z} \biggl[ \rho_i (\delta \dot z )  \biggr] \biggr\}


\varpi Component of Euler Equation


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


\rightarrow ~~~ \frac{\partial \dot\varpi}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + 
\biggl[ \dot\varphi \frac{\partial \dot\varpi}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] -  \varpi {\dot\varphi}^2   = 
- \frac{1}{\rho}\frac{\partial P}{\partial\varpi} - \frac{\partial \Phi}{\partial\varpi}


\varphi Component of Euler Equation


\frac{d (\varpi\dot\varphi) }{dt} + \dot\varpi \dot\varphi = 
-  \frac{1}{\varpi} \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial \varphi} + \frac{\partial \Phi}{\partial \varphi} \biggr]


\rightarrow ~~~ \frac{\partial (\varpi\dot\varphi)}{\partial t} + \biggl[ \dot\varpi \frac{\partial (\varpi\dot\varphi)}{\partial\varpi} \biggr] + 
\biggl[ \dot\varphi \frac{\partial (\varpi\dot\varphi)}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial (\varpi\dot\varphi)}{\partial z} \biggr] + \dot\varpi \dot\varphi = 
-  \frac{1}{\varpi} \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial \varphi} + \frac{\partial \Phi}{\partial \varphi} \biggr]


z Component of Euler Equation


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


\rightarrow ~~~ \frac{\partial \dot{z}}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] 
+ \biggl[ \dot\varphi \frac{\partial \dot{z}}{\partial\varphi} \biggr] +\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr] = 
- \frac{1}{\rho}\frac{\partial P}{\partial z} - \frac{\partial \Phi}{\partial z}


See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2019 by Joel E. Tohline
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