VisTrails Home

User:Tohline/Cylindrical 3D

From VisTrailsWiki

Jump to: navigation, search

Contents

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 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation

Personal tools