User:Tohline/Cylindrical 3D - Revision history

Tohline: /* Eulerian Formulation */ Write summary form of linearized continuity equation

2013-03-17T22:41:11Z

Eulerian Formulation: Write summary form of linearized continuity equation

← Older revision		Revision as of 22:41, 17 March 2013
Line 199:		Line 199:
	+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0		+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0
	</math><br />		</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>

	<!--		<!--
Line 218:		Line 268:
	<td align="center" colspan="3">		<td align="center" colspan="3">
	<math>		<math>
	Q(\varpi, \~~phi~~, z, t) = [q_i(\varpi, z) + \delta q(\varpi, z, t) e^{i m \varphi}] ~~~ \mathrm{and} ~~~ \delta q/q_i \ll 1		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>		</math>
	</td>		</td>
Line 588:		Line 638:


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

Tohline: /* Eulerian Formulation */ Add 2nd line of tabulated equations to make more consistent with rest of table

2013-03-17T18:20:59Z

Eulerian Formulation: Add 2nd line of tabulated equations to make more consistent with rest of table

← Older revision		Revision as of 18:20, 17 March 2013
Line 231:		Line 231:
	<td align="right">		<td align="right">
	<math>\frac{\partial\rho}{\partial t}</math>		<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>
	<td align="center">		<td align="center">
Line 237:		Line 254:
	<td align="left" bgcolor="lightblue">		<td align="left" bgcolor="lightblue">
	<math>		<math>
	~~\cancel{ \frac{\partial (\rho_i) }{\partial t} } +~~ e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]		e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]
	</math>		</math>
	</td>		</td>
Line 529:		Line 546:
	+ \frac{\partial}{\partial z} \biggl[ \rho_i \dot z_i \biggr]		+ \frac{\partial}{\partial z} \biggl[ \rho_i \dot z_i \biggr]
	</math>		</math>

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

Tohline: /* Eulerian Formulation */ Fix z-dot term in continuity equation and start putting summary equations together

2013-03-17T03:31:35Z

Eulerian Formulation: Fix z-dot term in continuity equation and start putting summary equations together

← Older revision		Revision as of 03:31, 17 March 2013
Line 200:		Line 200:
	</math><br />		</math><br />

	<!--		<!--
	TABLE TO LINEARIZE CONTINUITY EQUATION		XXX
			XXX
			XXX
			XXX TABLE TO LINEARIZE CONTINUITY EQUATION
			XXX
			XXX
			XXX
	-->		-->
	<table border="1" cellpadding="5">		<table border="1" cellpadding="5">
Line 430:		Line 436:
	<td align="left">		<td align="left">
	<math>		<math>
	(\rho_i + e^{im\varphi} \delta\rho) \frac{\partial ( {\dot{z}_i} + e^{im\varphi} \delta\dot{z})}{\partial~~\varphi~~}		(\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~~\varphi~~}		+ ( {\dot{z}_i} +e^{im\varphi} \delta\dot{z}) \frac{\partial (\rho_i + e^{im\varphi} \delta\rho)}{\partial z}
	</math>		</math>
	</td>		</td>
Line 448:		Line 454:
	<td align="left">		<td align="left">
	<math>		<math>
	(\rho_i + e^{im\varphi} \delta\rho) ~~\cancel~~{ \frac{\partial ( {\dot{z}_i} )}{\partial~~\varphi~~} }		(\rho_i + e^{im\varphi} \delta\rho) { \frac{\partial ( {\dot{z}_i} )}{\partial z} }
	+ im e^{im\varphi} (\rho_i + e^{im\varphi} \cancel{ \delta\rho })( \delta\dot{z})		+ e^{im\varphi} (\rho_i + e^{im\varphi} \cancel{{ \delta\rho } } ) \frac{\partial ( \delta\dot{z})}{\partial z}
	</math>		</math>

	<math>		<math>
	+ ( {\dot{z}_i} +e^{im\varphi} \delta\dot{z}) ~~\cancel{~~ \frac{\partial (\rho_i )}{\partial~~\varphi}~~ }		+ ( {\dot{z}_i} +e^{im\varphi} \delta\dot{z}) \frac{\partial (\rho_i )}{\partial z}
	+ im e^{im\varphi} ( {\dot{z}_i} +e^{im\varphi} \cancel{ \delta\dot{z} }) (\delta\rho)		+ e^{im\varphi} ( {\dot{z}_i} +e^{im\varphi} \cancel{ \delta\dot{z} } ) \frac{\partial (\delta\rho)}{\partial z}
	</math>		</math>
	</td>		</td>
Line 471:		Line 477:
	<td align="left" bgcolor="lightblue">		<td align="left" bgcolor="lightblue">
	<math>		<math>
	im e^{im\varphi} \biggl[ \rho_i ( \delta\dot{z}) + \dot{z}_i (\delta\rho) \biggr]		\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>		</math>
	</td>		</td>
Line 479:		Line 488:
	<td align="left">		<td align="left">
	<math>		<math>
	im e^{im\varphi} \biggl[ \rho_i ( \delta\dot{z}) + \cancel{\dot{z}_i }(\delta\rho) \biggr]		\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>		</math>
	</td>		</td>
Line 499:		Line 513:
	<td align="left" bgcolor="lightgreen">		<td align="left" bgcolor="lightgreen">
	<math>		<math>
	im e^{im\varphi} \biggl[ \rho_i ( \delta\dot{z}) \biggr]		e^{im\varphi} \biggl\{ \frac{\partial}{\partial z} \biggl[ \rho_i (\delta \dot z ) \biggr] \biggr\}
	</math>		</math>
	</td>		</td>
	</tr>		</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>		</table>
	<!-- END CONTINUITY EQUATION TABLE -->		<!--
			XXX
			XXX
			XXX
			XXX END CONTINUITY EQUATION TABLE
			XXX
			XXX
			XXX
			-->

Tohline: /* Eulerian Formulation / Work on rhoz derivative

2013-03-17T00:08:46Z

Eulerian Formulation: Work on rho*z derivative

← Older revision		Revision as of 00:08, 17 March 2013
Line 229:		Line 229:
	<math>~~ \rightarrow ~~</math>		<math>~~ \rightarrow ~~</math>
	</td>		</td>
	<td align="left">		<td align="left" bgcolor="lightblue">
	<math>		<math>
	\cancel{ \frac{\partial (\rho_i) }{\partial t} } + e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]		\cancel{ \frac{\partial (\rho_i) }{\partial t} } + e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]
Line 237:		Line 237:
	<math>~~~ \rightarrow ~~~</math>		<math>~~~ \rightarrow ~~~</math>
	</td>		</td>
	<td align="left">		<td align="left" bgcolor="lightgreen">
	<math>		<math>
	e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]		e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]
Line 304:		Line 304:
	<math>~~ \rightarrow ~~</math>		<math>~~ \rightarrow ~~</math>
	</td>		</td>
	<td align="left">		<td align="left" bgcolor="lightblue">
	<math>		<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}		\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}
Line 348:		Line 348:
	<math>~~~~ \rightarrow ~~~~</math>		<math>~~~~ \rightarrow ~~~~</math>
	</td>		</td>
	<td align="left">		<td align="left" bgcolor="lightgreen">
	<math>		<math>
	+ e^{im\varphi} \biggl\{ \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \rho_i (\delta \dot\varpi) \biggr] \biggr\}		+ e^{im\varphi} \biggl\{ \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \rho_i (\delta \dot\varpi) \biggr] \biggr\}
Line 401:		Line 401:
	<td align="center">		<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>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>		<math>
	im e^{im\varphi} \~~biggl[~~ \rho_i ( \delta\dot\varphi) + \dot\~~varphi_i~~ (\delta\rho) \biggr]		(\rho_i + e^{im\varphi} \delta\rho) \frac{\partial ( {\dot{z}_i} + e^{im\varphi} \delta\dot{z})}{\partial\varphi}
			+ ( {\dot{z}_i} +e^{im\varphi} \delta\dot{z}) \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{z}_i} )}{\partial\varphi} }
			+ im e^{im\varphi} (\rho_i + e^{im\varphi} \cancel{ \delta\rho })( \delta\dot{z})
			</math>

			<math>
			+ ( {\dot{z}_i} +e^{im\varphi} \delta\dot{z}) \cancel{ \frac{\partial (\rho_i )}{\partial\varphi} }
			+ im e^{im\varphi} ( {\dot{z}_i} +e^{im\varphi} \cancel{ \delta\dot{z} }) (\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{z}) + \dot{z}_i (\delta\rho) \biggr]
	</math>		</math>
	</td>		</td>
Line 415:		Line 479:
	<td align="left">		<td align="left">
	<math>		<math>
	im e^{im\varphi} \biggl[ \rho_i ( \delta\dot~~\varphi~~) + \dot\~~varphi_i~~ (\delta\~~rho~~) \biggr]		im e^{im\varphi} \biggl[ \rho_i ( \delta\dot{z}) + \cancel{\dot{z}_i }(\delta\rho) \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>
			im e^{im\varphi} \biggl[ \rho_i ( \delta\dot{z}) \biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* Eulerian Formulation */ Work on next term in continuity equation

2013-03-11T02:46:16Z

Eulerian Formulation: Work on next term in continuity equation

← Older revision		Revision as of 02:46, 11 March 2013
Line 234:		Line 234:
	</math>		</math>
	</td>		</td>
	<td align="center" colspan="2">		<td align="center" colspan="1">
	~~ ~~		<math>~~~ \rightarrow ~~~</math>
			</td>
			<td align="left">
			<math>
			e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]
			</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="left">		<td align="left">
	<math>		<math>
	e^{im\varphi} \~~biggl[~~ \frac{ (\~~rho_i~~ + \delta\~~rho~~) ( {\dot\~~varpi_i~~} + \delta\dot\~~varpi~~)}{\~~varpi~~}		(\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>		</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>		<math>
	+ (\rho_i + \delta\rho) \frac{\partial ( {\dot\~~varpi_i~~} + \~~delta~~\~~dot~~\~~varpi)~~}{\~~partial~~\~~varpi~~}		(\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>

	<math>		<math>
	+ ( {\dot\~~varpi_i~~} + \delta\dot\~~varpi~~) \frac{\partial (\rho_i + \~~delta~~\~~rho)~~}{\~~partial~~\~~varpi~~} \~~biggr]~~		+ ( {\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>		</math>
	</td>		</td>
Line 375:		Line 396:

	</td>		</td>
			</tr>

			<tr>
			<td align="center">

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

Tohline: /* Eulerian Formulation */ Finished one term in continuity equation

2013-03-11T02:14:43Z

Eulerian Formulation: Finished one term in continuity equation

← Older revision		Revision as of 02:14, 11 March 2013
Line 306:		Line 306:
	<math>		<math>
	+ e^{im\varphi} \biggl[ \frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \frac{ \dot\varpi_i }{\varpi} ( \delta\rho )		+ 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\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]		+ ( \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + ( {\dot\varpi_i} ) \frac{\partial (\delta\rho)}{\partial\varpi} \biggr]
	</math>		</math>
Line 316:		Line 320:
	<td align="left">		<td align="left">
	<math>		<math>
	~~e^{im\varphi} \biggl[~~ \cancel{ \frac{ \rho_i \dot\varpi_i}{\varpi} } + \~~frac~~{ \rho_i }{\~~varpi~~} ( \~~delta\dot~~\varpi ) + \cancel{ \frac{ \~~dot~~\~~varpi_i~~ }{\varpi} ~~( \delta\rho )~~ }		\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>

	<math>		<math>
	+ \~~cancel~~{ (\rho_i + \delta\~~rho~~) \~~frac~~{\~~partial~~ {\dot\varpi_i} }{~~\partial~~\varpi} } + (\~~rho_i~~ ) \frac{\partial ~~( \delta~~\dot\~~varpi)~~}{\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} }
	</math>		</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">
	<math>		<math>
	+ ~~( \cancel~~{\~~dot~~\~~varpi_i} +~~ \~~delta~~\~~dot~~\varpi) \frac{\partial ~~\rho_i~~ }{\partial\varpi} + \~~cancel{ ( {~~\~~dot~~\~~varpi_i} ) \frac{\partial~~ (\delta\~~rho)}{\partial~~\varpi~~} }~~ \biggr]		+ e^{im\varphi} \biggl\{ \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \rho_i (\delta \dot\varpi) \biggr] \biggr\}
	</math>		</math>
	</td>		</td>

Tohline: /* Eulerian Formulation */ More table entries

2013-03-11T01:44:16Z

Eulerian Formulation: More table entries

← Older revision		Revision as of 01:44, 11 March 2013
Line 275:		Line 275:
	<td align="left">		<td align="left">
	<math>		<math>
	~~e^{im\varphi} \biggl[~~ \frac{ \rho_i \dot\varpi_i}{\varpi} + \frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \frac{ \dot\varpi_i }{\varpi} ( \delta\rho ) + \cancel{ \frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} }		\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>

	<math>		<math>
	+ (\rho_i + \delta\rho) \frac{\partial {\dot\varpi_i} }{\partial\varpi} + (\rho_i + \cancel{\delta\rho}) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi}		+ (\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>

	<math>		<math>
	+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + ( {\dot\varpi_i} + \cancel{\delta\dot\varpi}) \frac{\partial (\delta\rho)}{\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]
			</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} + \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} + (\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>		</math>
	</td>		</td>

Tohline: /* Eulerian Formulation */ Finish implementation of zero radial and vertical velocities

2013-03-11T01:06:36Z

Eulerian Formulation: Finish implementation of zero radial and vertical velocities

← Older revision		Revision as of 01:06, 11 March 2013
Line 230:		Line 230:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{\partial (\delta\rho) }{\partial t}</math>		<math>
			\cancel{ \frac{\partial (\rho_i) }{\partial t} } + e^{im\varphi} \biggr[ \frac{\partial (\delta\rho) }{\partial t} \biggr]
			</math>
	</td>		</td>
	<td align="center" colspan="2">		<td align="center" colspan="2">
Line 248:		Line 250:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{ (\rho_i + \delta\rho) ( {\dot\varpi_i} + \delta\dot\varpi)}{\varpi}		\frac{ (\rho_i + e^{im\varphi} \delta\rho) ( {\dot\varpi_i} + e^{im\varphi} \delta\dot\varpi)}{\varpi}
	+ (\rho_i + \delta\rho) \frac{\partial ( {\dot\varpi_i} + \delta\dot\varpi)}{\partial\varpi}		</math>
	+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial (\rho_i + \delta\rho)}{\partial\varpi}
			<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>		</math>
	</td>		</td>
Line 267:		Line 275:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{ \rho_i \dot\varpi_i}{\varpi} + \frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \frac{ \dot\varpi_i }{\varpi} ( \delta\rho ) + \cancel{ \frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} }		e^{im\varphi} \biggl[ \frac{ \rho_i \dot\varpi_i}{\varpi} + \frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \frac{ \dot\varpi_i }{\varpi} ( \delta\rho ) + \cancel{ \frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} }
			</math>

			<math>
	+ (\rho_i + \delta\rho) \frac{\partial {\dot\varpi_i} }{\partial\varpi} + (\rho_i + \cancel{\delta\rho}) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi}		+ (\rho_i + \delta\rho) \frac{\partial {\dot\varpi_i} }{\partial\varpi} + (\rho_i + \cancel{\delta\rho}) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi}
	</math>		</math>

	<math>		<math>
	+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + ( {\dot\varpi_i} + \cancel{\delta\dot\varpi}) \frac{\partial (\delta\rho)}{\partial\varpi}		+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + ( {\dot\varpi_i} + \cancel{\delta\dot\varpi}) \frac{\partial (\delta\rho)}{\partial\varpi} \biggr]
	</math>		</math>
	</td>		</td>

	<td align="center" colspan="1">		<td align="center" colspan="1">
	<math>~~~~ \rightarrow ~~~~</math>		<math>~~~~ \rightarrow ~~~~</math>
Line 280:		Line 292:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{ \rho_i \dot\varpi_i}{\varpi} + \frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \frac{ \dot\varpi_i }{\varpi} ( \delta\rho ) + \cancel{ ~~\frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} }~~		e^{im\varphi} \biggl[ \cancel{ \frac{ \rho_i \dot\varpi_i}{\varpi} } + \frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \cancel{ \frac{ \dot\varpi_i }{\varpi} ( \delta\rho ) }
	+ (\rho_i + \delta\rho) \frac{\partial {\dot\varpi_i} }{\partial\varpi} + (\rho_i ~~+ \cancel{\delta\rho}~~) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi}		</math>

			<math>
			+ \cancel{ (\rho_i + \delta\rho) \frac{\partial {\dot\varpi_i} }{\partial\varpi} } + (\rho_i ) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi}
	</math>		</math>

	<math>		<math>
	+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + ( {\dot\varpi_i~~} + \cancel{\delta\dot\varpi~~}) \frac{\partial (\delta\rho)}{\partial\varpi}		+ ( \cancel{\dot\varpi_i} + \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + \cancel{ ( {\dot\varpi_i} ) \frac{\partial (\delta\rho)}{\partial\varpi} } \biggr]
	</math>		</math>
	</td>		</td>
	</tr>		</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>
			e^{im\varphi} \biggl[ \frac{ (\rho_i + \delta\rho) ( {\dot\varpi_i} + \delta\dot\varpi)}{\varpi}
			</math>

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

			<math>
			+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial (\rho_i + \delta\rho)}{\partial\varpi} \biggr]
			</math>
			</td>
			<td align="center" colspan="2">

			</td>
			</tr>


	</table>		</table>

Tohline: /* Eulerian Formulation */ Improve and finish linearization of continuity equation

2013-03-10T18:50:58Z

Eulerian Formulation: Improve and finish linearization of continuity equation

← Older revision		Revision as of 18:50, 10 March 2013
Line 200:		Line 200:
	</math><br />		</math><br />

			<!--
			TABLE TO LINEARIZE CONTINUITY EQUATION
			-->
	<table border="1" cellpadding="5">		<table border="1" cellpadding="5">
	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="5">
	<b>Linearize the Continuity Equation assuming</b>		<b>Linearize each term of the <font color="darkblue">Continuity Equation</font> assuming ...</b>
	</td>		</td>
	</tr>		</tr>
Line 209:		Line 212:
	<td align="center" colspan="3">		<td align="center" colspan="3">
	<math>		<math>
	Q(\varpi, \phi, z, t) = [q_i(\varpi, z) + \delta q(\varpi, z, t) e^{i m \varphi}]; \delta q/q_i \ll 1; \dot\varpi_i = \dot z_i = 0</math>		Q(\varpi, \phi, 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>		</td>
	</tr>		</tr>
Line 222:		Line 231:
	<td align="left">		<td align="left">
	<math>\frac{\partial (\delta\rho) }{\partial t}</math>		<math>\frac{\partial (\delta\rho) }{\partial t}</math>
			</td>
			<td align="center" colspan="2">

	</td>		</td>
	</tr>		</tr>
Line 236:		Line 248:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{ (\rho_i + \delta\rho) ( ~~\cancel~~{\dot\varpi_i} + \delta\dot\varpi)}{\varpi}		\frac{ (\rho_i + \delta\rho) ( {\dot\varpi_i} + \delta\dot\varpi)}{\varpi}
	+ (\rho_i + \delta\rho) \frac{\partial ( ~~\cancel~~{\dot\varpi_i} + \delta\dot\varpi)}{\partial\varpi}		+ (\rho_i + \delta\rho) \frac{\partial ( {\dot\varpi_i} + \delta\dot\varpi)}{\partial\varpi}
	+ ( ~~\cancel~~{\dot\varpi_i} + \delta\dot\varpi) \frac{\partial (\rho_i + \delta\rho)}{\partial\varpi}		+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial (\rho_i + \delta\rho)}{\partial\varpi}
	</math>		</math>
	</td>		</td>
			<td align="center" colspan="2">

			</td>
	</tr>		</tr>

Line 252:		Line 267:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \cancel{ \frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} }		\frac{ \rho_i \dot\varpi_i}{\varpi} + \frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \frac{ \dot\varpi_i }{\varpi} ( \delta\rho ) + \cancel{ \frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} }
	+ (\rho_i) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi} + \cancel {(\delta\rho) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi} }		+ (\rho_i + \delta\rho) \frac{\partial {\dot\varpi_i} }{\partial\varpi} + (\rho_i + \cancel{\delta\rho}) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi}
	+ ( \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + \cancel {( \delta\dot\varpi) \frac{\partial (\delta\rho)}{\partial\varpi} }		</math>

			<math>
			+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + ( {\dot\varpi_i} + \cancel{\delta\dot\varpi}) \frac{\partial (\delta\rho)}{\partial\varpi}
			</math>
			</td>
			<td align="center" colspan="1">
			<math>~~~~ \rightarrow ~~~~</math>
			</td>
			<td align="left">
			<math>
			\frac{ \rho_i \dot\varpi_i}{\varpi} + \frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \frac{ \dot\varpi_i }{\varpi} ( \delta\rho ) + \cancel{ \frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} }
			+ (\rho_i + \delta\rho) \frac{\partial {\dot\varpi_i} }{\partial\varpi} + (\rho_i + \cancel{\delta\rho}) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi}
			</math>

			<math>
			+ ( {\dot\varpi_i} + \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + ( {\dot\varpi_i} + \cancel{\delta\dot\varpi}) \frac{\partial (\delta\rho)}{\partial\varpi}
	</math>		</math>
	</td>		</td>
Line 260:		Line 291:

	</table>		</table>
			<!-- END CONTINUITY EQUATION TABLE -->

Tohline: /* Eulerian Formulation */ Establish linearization table for continuity equation

2013-03-10T04:41:11Z

Eulerian Formulation: Establish linearization table for continuity equation

← Older revision		Revision as of 04:41, 10 March 2013
Line 199:		Line 199:
	+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0		+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0
	</math><br />		</math><br />

			<table border="1" cellpadding="5">
			<tr>
			<td align="center" colspan="3">
			<b>Linearize the Continuity Equation assuming</b>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			<math>
			Q(\varpi, \phi, z, t) = [q_i(\varpi, z) + \delta q(\varpi, z, t) e^{i m \varphi}]; \delta q/q_i \ll 1; \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>\frac{\partial (\delta\rho) }{\partial t}</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 + \delta\rho) ( \cancel{\dot\varpi_i} + \delta\dot\varpi)}{\varpi}
			+ (\rho_i + \delta\rho) \frac{\partial ( \cancel{\dot\varpi_i} + \delta\dot\varpi)}{\partial\varpi}
			+ ( \cancel{\dot\varpi_i} + \delta\dot\varpi) \frac{\partial (\rho_i + \delta\rho)}{\partial\varpi}
			</math>
			</td>
			</tr>

			<tr>
			<td align="center">

			</td>
			<td align="center">
			<math>~~ \rightarrow ~~</math>
			</td>
			<td align="left">
			<math>
			\frac{ \rho_i }{\varpi} ( \delta\dot\varpi ) + \cancel{ \frac{ (\delta\rho) ( \delta\dot\varpi)}{\varpi} }
			+ (\rho_i) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi} + \cancel {(\delta\rho) \frac{\partial ( \delta\dot\varpi)}{\partial\varpi} }
			+ ( \delta\dot\varpi) \frac{\partial \rho_i }{\partial\varpi} + \cancel {( \delta\dot\varpi) \frac{\partial (\delta\rho)}{\partial\varpi} }
			</math>
			</td>
			</tr>

			</table>