Difference between revisions of "User:Tohline/Cylindrical 3D"

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(Create this section by copying AxisymmetricConfigurations/PGE)
 
(→‎Governing Equations: Insert terms that were initially canceled because of assumed axisymmetry)
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<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />


<math>\frac{d\rho}{dt} + \frac{\rho}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \dot\varpi \biggr]  
<math>\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[ \rho \dot{z} \biggr] = 0 </math><br />
+ \rho \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math><br />


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<math>
<math>
{\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{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}}_z \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \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]  
</math><br />
</math><br />


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<math>
<math>
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho .  
\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 .  
</math><br />
</math><br />
</div>
</div>


==Eulerian Formulation==
==Eulerian Formulation==

Revision as of 23:09, 9 March 2013

Equations Cast in Cylindrical Coordinates

Spatial Operators in Cylindrical Coordinates

<math> \nabla f </math>

=

<math> {\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] ; </math>

<math> \nabla^2 f </math>

=

<math> \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} ; </math>

<math> (\vec{v}\cdot\nabla)f </math>

=

<math> \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] ; </math>

<math> \nabla \cdot \vec{F} </math>

=

<math> \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} ; </math>

Vector Time-Derivatives in Cylindrical Coordinates

<math> \frac{d}{dt}\vec{F} </math>

=

<math> {\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} </math>

 

=

<math> {\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} ; </math>

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

=

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

Governing Equations

Introducing the above expressions into the principal governing equations gives,

Equation of Continuity

<math>\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[ \rho \dot{z} \biggr] = 0 </math>


Euler Equation

<math> {\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] </math>


Adiabatic Form of the
First Law of Thermodynamics

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


Poisson Equation

<math> \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 . </math>

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, <math>f</math>,


<math> \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{z} \frac{\partial f}{\partial z} \biggr] . </math>

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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