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Joel2: Created page with " =Equations Cast in Cylindrical Coordinates=

Spatial Operators in Cylindrical Coordinates

$\nabla f$ = ${\hat{e}}_\varpi \biggl[ \frac{\p..."$
2023-12-11T22:04:07Z

Created page with "  =Equations Cast in Cylindrical Coordinates= <table align="center" border="0" cellpadding="5"> <tr> <td colspan="3" align="center"> <font color="#770000"><b>Spatial Operators in Cylindrical Coordinates</b></font> </td> </tr> <tr> <td align="right"> <math> \nabla f </math> </td> <td align="center"> = </td> <td align="left"> <math> {\hat{e}}_\varpi \biggl[ \frac{\p..."

New page


=Equations Cast in Cylindrical Coordinates=

<table align="center" border="0" cellpadding="5">
<tr>
<td colspan="3" align="center">
<font color="#770000"><b>Spatial Operators in Cylindrical Coordinates</b></font>
</td>
</tr>

<tr>
<td align="right">
<math>
\nabla f
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<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>
</td>
</tr>

<tr>
<td align="right">
<math>
\nabla^2 f
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<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>
</td>
</tr>

<tr>
<td align="right">
<math>
(\vec{v}\cdot\nabla)f
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<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>
</td>
</tr>

<tr>
<td align="right">
<math>
\nabla \cdot \vec{F}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<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>
</td>
</tr>
</table>

<table align="center" border="0" cellpadding="5">
<tr>
<td colspan="3" align="center">
<font color="#770000"><b>Vector Time-Derivatives in Cylindrical Coordinates</b></font>
</td>
</tr>

<tr>
<td align="right">
<math>
\frac{d}{dt}\vec{F}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<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>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
=
</td>
<td align="left">
<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>
</td>
</tr>

<tr>
<td align="right">
<math>
\vec{v} = \frac{d\vec{x}}{dt} = \frac{d}{dt}\biggl[ \hat{e}_\varpi \varpi + \hat{e}_z z \biggr]
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
{\hat{e}}_\varpi \biggl[ \dot\varpi \biggr] +
{\hat{e}}_\varphi \biggl[ \varpi \dot\varphi \biggr] +
{\hat{e}}_z \biggl[ \dot{z} \biggr] .
</math>
</td>
</tr>
</table>

==Governing Equations==

Introducing the above expressions into the [[PGE|principal governing equations]] gives,

<div align="center">
<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] + \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \varpi \dot\varphi \biggr]
+ \rho \frac{\partial}{\partial z} \biggl[ \dot{z} \biggr] = 0 </math><br />

<span id="PGE:Euler">
<font color="#770000">'''Euler Equation'''</font>
</span><br />

<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><br />

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{Math/EQ_FirstLaw02}}

<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<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><br />
</div>

==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>,

<div align="center">
<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\varphi \frac{\partial f}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial f}{\partial z} \biggr] .
</math>
</div>

Hence,

<div align="center">
<span id="EulerContinuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>
\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
</math><br />

<math>
\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
</math><br />
</div>

Assuming that the initial (subscript <i>i</i>) configuration is axisymmetric and that, following perturbation, each physical parameter, <math>Q</math>, behaves according to the relation,
<div align="center">
<math>
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 \, ,
</math>
</div>
the linearized form of the continuity equation becomes:
<div align="center">
<table border="1" cellpadding="5" width="95%">
<tr>
<td align="center" bgcolor="lightblue" colspan="3">
(This has been obtained by combining the expressions highlighted with a lightblue background color from the accompanying table.)
</td>
</tr>
<tr>
<td align="right">
<math>e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr] </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\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]
</math>
 
<math>
+ e^{im\varphi} \biggl\{ im \biggl[ \rho_i ( \delta\dot\varphi) + \dot\varphi_i (\delta\rho) \biggr] \biggr\}
</math>

<math>
+ 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}
</math>
<math>
+ (\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}
</math>
<math>
+ \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\}
</math>
</td>
</tr>

</table>
</div>


<table border="1" cellpadding="5">
<tr>
<td align="center" colspan="5">
<b>Linearize each term of the <font color="darkblue">Continuity Equation</font> assuming ...</b>
</td>
</tr>
<tr>
<td align="center" colspan="3">
<math>
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
</math>
</td>
<td align="center" colspan="2">
<math>
\mathrm{and} ~ \dot\varpi_i = \dot z_i = 0
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\partial\rho}{\partial t}</math>
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left">
<math>
\cancel{ \frac{\partial (\rho_i) }{\partial t} } + e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]
</math>
</td>
<td align="center" colspan="2">
 
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left" bgcolor="lightblue">
<math>
e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]
</math>
</td>
<td align="center" colspan="1">
<math>~ \rightarrow ~</math>
</td>
<td align="left" bgcolor="lightgreen">
<math>
e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\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}
</math>
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left">
<math>
\frac{ (\rho_i + e^{im\varphi} \delta\rho) ( {\dot\varpi_i} + e^{im\varphi} \delta\dot\varpi)}{\varpi}
</math>

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

<math>
+ ( {\dot\varpi_i} + e^{im\varphi} \delta\dot\varpi) \frac{\partial (\rho_i + e^{im\varphi} \delta\rho)}{\partial\varpi}
</math>
</td>
<td align="center" colspan="2">
 
</td>
</tr>

<tr>
<td align="center">
 
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left">
<math>
\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]
</math>

<math>
+ (\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]
</math>

<math>
+ ( {\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]
</math>
</td>
<td align="center" colspan="2">
 
</td>
</tr>

<tr>
<td align="center">
 
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left" bgcolor="lightblue">
<math>
\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}
</math>

<math>
+ 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}
</math>

<math>
+ (\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]
</math>
</td>

<td align="center" colspan="1">
<math>~~ \rightarrow </math>
</td>
<td align="left">
<math>
\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} }
</math>

<math>
+ 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} }
</math>

<math>
+ (\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]
</math>
</td>
</tr>

<tr>
<td colspan="3">
 
</td>

<td align="center" colspan="1">
<math> \rightarrow </math>
</td>
<td align="left" bgcolor="lightgreen">
<math>
+ e^{im\varphi} \biggl\{ \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \rho_i (\delta \dot\varpi) \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\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}
</math>
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left">
<math>
(\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}
</math>
</td>
<td align="center" colspan="2">
 
</td>
</tr>

<tr>
<td align="center">
 
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left">
<math>
(\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)
</math>

<math>
+ ( {\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)
</math>
</td>
<td align="center" colspan="2">
 
</td>
</tr>

<tr>
<td align="center">
 
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left" bgcolor="lightblue">
<math>
im e^{im\varphi} \biggl[ \rho_i ( \delta\dot\varphi) + \dot\varphi_i (\delta\rho) \biggr]
</math>
</td>
<td align="center" colspan="1">
<math>~ \rightarrow ~</math>
</td>
<td align="left" bgcolor="lightgreen">
<math>
im e^{im\varphi} \biggl[ \rho_i ( \delta\dot\varphi) + \dot\varphi_i (\delta\rho) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr]</math>
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left">
<math>
(\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}
</math>
</td>
<td align="center" colspan="2">
 
</td>
</tr>

<tr>
<td align="center">
 
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left">
<math>
(\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}
</math>

<math>
+ ( {\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}
</math>
</td>
<td align="center" colspan="2">
 
</td>
</tr>

<tr>
<td align="center">
 
</td>
<td align="center">
<math> \rightarrow </math>
</td>
<td align="left" bgcolor="lightblue">
<math>
\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]
</math>
</td>
<td align="center" colspan="1">
<math>~ \rightarrow ~</math>
</td>
<td align="left">
<math>
\rho_i \cancel{ \frac{\partial \dot z_i }{\partial z} } + \cancel{ \dot{z}_i } \frac{\partial \rho_i}{\partial z}
</math>

<math>
+ 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]
</math>
</td>
</tr>

<tr>
<td align="center">
 
</td>
<td align="center">
 
</td>
<td align="left">
 
</td>
<td align="center" colspan="1">
<math>~ \rightarrow ~</math>
</td>
<td align="left" bgcolor="lightgreen">
<math>
e^{im\varphi} \biggl\{ \frac{\partial}{\partial z} \biggl[ \rho_i (\delta \dot z ) \biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
Combining all terms:
</td>
<td align="center">
<math>~ \rightarrow ~</math>
</td>
<td align="left" bgcolor="lightblue">
<math>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]
</math>
   
<math>
+ 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}
</math>

<math>
+ (\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}
</math>

<math>
+ im \biggl[ \rho_i ( \delta\dot\varphi) + \dot\varphi_i (\delta\rho) \biggr]
</math>

<math>
+ \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\}
</math>
</td>
<td align="center" colspan="1">
<math>~ \rightarrow ~</math>
</td>
<td align="left" bgcolor="lightgreen">
<math>
+ e^{im\varphi} \biggl\{ \frac{\partial}{\partial z} \biggl[ \rho_i (\delta \dot z ) \biggr] \biggr\}
</math>
</td>
</tr>

</table>


<div align="center">
<span id="PGE:Euler:R">
<font color="#770000">'''<math>\varpi</math> Component of Euler Equation'''</font>
</span><br />

<math>
\frac{d \dot\varpi}{dt} - \varpi {\dot\varphi}^2 = - \frac{1}{\rho}\frac{\partial P}{\partial\varpi} - \frac{\partial \Phi}{\partial\varpi}
</math><br />

<math>
\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}
</math><br />

<span id="PGE:Euler:Phi">
<font color="#770000">'''<math>\varphi</math> Component of Euler Equation'''</font>
</span><br />

<math>
\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]
</math><br />

<math>
\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]
</math><br />

<span id="PGE:Euler:Z">
<font color="#770000">'''<math>z</math> Component of Euler Equation'''</font>
</span><br />

<math>
\frac{d \dot{z} }{dt} = - \frac{1}{\rho}\frac{\partial P}{\partial z} - \frac{\partial \Phi}{\partial z}
</math><br />

<math>
\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}
</math><br />

</div>

=See Also=

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