User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4 - Revision history

Tohline: /* Try Shifting to (Quasi-) Inertial Frame */

2021-05-28T22:59:17Z

Try Shifting to (Quasi-) Inertial Frame

← Older revision		Revision as of 22:59, 28 May 2021
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( y'\cos\theta )		\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( \cos\theta )
	- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( y'\sin\theta ) \biggr]\biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]		- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( \sin\theta ) - \Omega' \biggr]\biggl[\frac{2x' y'}{x^2_\mathrm{max}} \biggr]
	+		+
	\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( z_0\tan\theta )		\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( z_0\tan\theta )
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	+		+
	\biggl[\beta \Omega_2 \sin\theta		\biggl[\beta \Omega_2 \sin\theta
	- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2y' }{y^2_\mathrm{max}} \biggr]		- \gamma \Omega_3 \cos\theta + \Omega'\biggr]x' \biggl[\frac{2y' }{y^2_\mathrm{max}} \biggr]
	-		-
	\biggl[\beta \Omega_2 \sin\theta		\biggl[\beta \Omega_2 \sin\theta
	- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2y_c }{y^2_\mathrm{max}} \biggr]		- \gamma \Omega_3 \cos\theta + \Omega' \biggr]x' \biggl[\frac{2y_c }{y^2_\mathrm{max}} \biggr]
	</math>		</math>
	</td>		</td>
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	<math>~ 2x'y' \biggl\{		<math>~ 2x'y' \biggl\{
	\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( \cos\theta )		\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( \cos\theta )
	- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( \sin\theta ) \biggr]\biggl[\frac{1}{x^2_\mathrm{max}} \biggr]		- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( \sin\theta ) - \Omega'\biggr]\biggl[\frac{1}{x^2_\mathrm{max}} \biggr]
	+		+
	\biggl[\beta \Omega_2 \sin\theta		\biggl[\beta \Omega_2 \sin\theta
	- \gamma \Omega_3 \cos\theta \biggr] \biggl[\frac{1 }{y^2_\mathrm{max}} \biggr]		- \gamma \Omega_3 \cos\theta + \Omega' \biggr] \biggl[\frac{1 }{y^2_\mathrm{max}} \biggr]
	\biggr\}		\biggr\}
	</math>		</math>
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	-		-
	\biggl[\beta \Omega_2 \sin\theta		\biggl[\beta \Omega_2 \sin\theta
	- \gamma \Omega_3 \cos\theta \biggr]\biggl[\frac{y_c }{y^2_\mathrm{max}} \biggr]		- \gamma \Omega_3 \cos\theta + \Omega' \biggr]\biggl[\frac{y_c }{y^2_\mathrm{max}} \biggr]
	\biggr\} \, .		\biggr\} \, .
	</math>		</math>

Tohline: /* Test Orthogonality */

2021-05-28T22:51:19Z

Test Orthogonality

← Older revision		Revision as of 22:51, 28 May 2021
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	</tr>		</tr>
	</table>		</table>

			===Stick With Rotating Frame===
	The dot product of these two vectors is,		The dot product of these two vectors is,

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	</tr>		</tr>
	</table>		</table>

			===Try Shifting to (Quasi-) Inertial Frame===

			Let's adjust both remaining components of <math>~\boldsymbol{u'}_\mathrm{EFE}</math> to see what their equivalent rotating-frame expressions are. The relevant shift is,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\boldsymbol{v'}_\mathrm{shift} \equiv \boldsymbol{\Omega'} \times \boldsymbol{x'}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\boldsymbol{\hat{k}} \Omega' \times \biggl[ \boldsymbol{\hat\imath'}x' + \boldsymbol{\hat\jmath'}y' \biggr]
			=
			- \boldsymbol{\hat\imath'} \Omega' y' + \boldsymbol{\hat\jmath'} \Omega' x' \, ,
			</math>
			</td>
			</tr>
			</table>
			where,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Omega'</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \Omega_2^2 + \Omega_3^2 \biggr]^{1 / 2} \, .</math>
			</td>
			</tr>
			</table>
			We have, then,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\boldsymbol{u'}_\mathrm{inertial} \equiv \boldsymbol{u'}_\mathrm{EFE} + \boldsymbol{v'}_\mathrm{shift}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\boldsymbol{\hat\imath'} \biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( y'\cos\theta + z_0\tan\theta )
			- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( y'\sin\theta ) - \Omega' y'\biggr]
			+
			\boldsymbol{\hat\jmath'} \biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta + \Omega'\biggr]x'
			</math>
			</td>
			</tr>
			</table>
			The dot product of this expression with <math>~\nabla f</math> is,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla f \cdot \boldsymbol{u'}_\mathrm{inertial}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( y'\cos\theta + z_0\tan\theta )
			- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( y'\sin\theta ) - \Omega' y' \biggr]\biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]
			+
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta + \Omega'\biggr]x' \biggl[\frac{2(y' - y_c)}{y^2_\mathrm{max}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( y'\cos\theta )
			- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( y'\sin\theta ) \biggr]\biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]
			+
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( z_0\tan\theta )
			\biggr]\biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2y' }{y^2_\mathrm{max}} \biggr]
			-
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2y_c }{y^2_\mathrm{max}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ 2x'y' \biggl\{
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( \cos\theta )
			- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( \sin\theta ) \biggr]\biggl[\frac{1}{x^2_\mathrm{max}} \biggr]
			+
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta \biggr] \biggl[\frac{1 }{y^2_\mathrm{max}} \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ 2x' \biggl\{
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 \tan\theta
			\biggr]\biggl[\frac{z_0}{x^2_\mathrm{max}} \biggr]
			-
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta \biggr]\biggl[\frac{y_c }{y^2_\mathrm{max}} \biggr]
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>
			Now, drawing from the various above "boxed" relations and examining the two terms separately, we first find,

	=See Also=		=See Also=

Tohline: /* Test Orthogonality */

2021-05-26T19:06:38Z

Test Orthogonality

← Older revision		Revision as of 19:06, 26 May 2021
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	- \frac{c^2 \tan^2\theta}{\sin\theta( c^2 + b^2 \tan^2\theta )}		- \frac{c^2 \tan^2\theta}{\sin\theta( c^2 + b^2 \tan^2\theta )}
	=		=
	- \frac{c^2 \sin\theta}{( c^2\cos^2\theta + b^2 \sin^2\theta )} \, .		- \frac{c^2 \sin\theta}{( c^2\cos^2\theta + b^2 \sin^2\theta )}
			=
			- \frac{\sin\theta}{b^2 \kappa^2} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Test Orthogonality */

2021-05-26T17:36:53Z

Test Orthogonality

Show changes

Tohline: /* Test Orthogonality */

2021-05-26T16:43:47Z

Test Orthogonality

← Older revision		Revision as of 16:43, 26 May 2021
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	\boldsymbol{\hat\imath'} \biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]		\boldsymbol{\hat\imath'} \biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]
	+		+
	\boldsymbol{\hat\jmath'} \biggl[\frac{2(y' - y_c)}{y^2_\mathrm{max}} \biggr] \, ,		\boldsymbol{\hat\jmath'} \biggl[\frac{2(y' - y_c)}{y^2_\mathrm{max}} \biggr]
			-
			\boldsymbol{\hat{k}'} \biggl[\frac{2(y' - y_c)}{y^2_\mathrm{max}} \biggr] \frac{\partial y_c}{\partial z'} \, ,
	</math>		</math>
	</td>		</td>

Tohline: /* Test Orthogonality */

2021-05-25T23:17:20Z

Test Orthogonality

← Older revision		Revision as of 23:17, 25 May 2021
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	- \gamma \Omega_3 \biggr]\biggl[\frac{ x^2_\mathrm{max}}{y^2_\mathrm{max}} \biggr]		- \gamma \Omega_3 \biggr]\biggl[\frac{ x^2_\mathrm{max}}{y^2_\mathrm{max}} \biggr]
	\biggr\} \cos\theta		\biggr\} \cos\theta
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \frac{2x'y'}{x^2_\mathrm{max}} \biggl\{
			\biggl[ c^2 \gamma \Omega_3
			- b^2 \beta \Omega_2 ( \tan\theta ) \biggr]
			+
			\biggl[\beta \Omega_2 \tan\theta
			- \gamma \Omega_3 \biggr] b^2 c^2 \kappa^2
			\biggr\} \frac{a^2 \cos\theta}{b^2 c^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \frac{2x'}{x^2_\mathrm{max}} \biggl\{
			\biggl[ \gamma \Omega_3 \cdot \frac{\tan\theta }{\cos\theta}
			\biggr] \biggl[ - \frac{c^2\kappa^2 y_c}{\sin\theta} \biggr]
			-y_c
			\biggl[\beta \Omega_2 \tan\theta
			- \gamma \Omega_3 \biggr]b^2 \kappa^2
			\biggr\} \frac{a^2 \cos\theta}{b^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \frac{2x'y'}{x^2_\mathrm{max}} \biggl\{
			c^2 \gamma \Omega_3 [1-b^2\kappa^2]
			+
			b^2 \beta \Omega_2 \tan\theta [c^2 \kappa^2 -1]
			\biggr\} \frac{a^2 \cos\theta}{b^2 c^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \frac{2x'}{x^2_\mathrm{max}} \biggl\{
			- \biggl[ \gamma \Omega_3 \cdot \frac{1}{\cos^2\theta}
			\biggr] c^2
			-
			\biggl[\beta \Omega_2 \tan\theta
			- \gamma \Omega_3 \biggr]b^2
			\biggr\} \frac{a^2 \kappa^2 \cdot y_c \cos\theta}{b^2}
	</math>		</math>
	</td>		</td>

Tohline: /* Test Orthogonality */

2021-05-25T22:57:12Z

Test Orthogonality

← Older revision		Revision as of 22:57, 25 May 2021
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	\biggl[\beta \Omega_2 \sin\theta		\biggl[\beta \Omega_2 \sin\theta
	- \gamma \Omega_3 \cos\theta \biggr]\biggl[\frac{y_c }{y^2_\mathrm{max}} \biggr]		- \gamma \Omega_3 \cos\theta \biggr]\biggl[\frac{y_c }{y^2_\mathrm{max}} \biggr]
	\biggr\}		\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>
			Now, drawing from the various above "boxed" relations,


			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla f \cdot \boldsymbol{u'}_\mathrm{EFE}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \frac{2x'y'}{x^2_\mathrm{max}} \biggl\{
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3
			- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( \tan\theta ) \biggr]
			+
			\biggl[\beta \Omega_2 \tan\theta
			- \gamma \Omega_3 \biggr] \biggl[\frac{x^2_\mathrm{max} }{y^2_\mathrm{max}} \biggr]
			\biggr\} \cos\theta
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \frac{2x'}{x^2_\mathrm{max}} \biggl\{
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 \cdot \frac{\tan\theta }{\cos\theta}
			\biggr]z_0
			-y_c
			\biggl[\beta \Omega_2 \tan\theta
			- \gamma \Omega_3 \biggr]\biggl[\frac{ x^2_\mathrm{max}}{y^2_\mathrm{max}} \biggr]
			\biggr\} \cos\theta
	</math>		</math>
	</td>		</td>

Tohline: /* Test Orthogonality */

2021-05-25T22:37:04Z

Test Orthogonality

← Older revision		Revision as of 22:37, 25 May 2021
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	\biggl[\beta \Omega_2 \sin\theta		\biggl[\beta \Omega_2 \sin\theta
	- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2(y' - y_c)}{y^2_\mathrm{max}} \biggr]		- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2(y' - y_c)}{y^2_\mathrm{max}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( y'\cos\theta )
			- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( y'\sin\theta ) \biggr]\biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]
			+
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( z_0\tan\theta )
			\biggr]\biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2y' }{y^2_\mathrm{max}} \biggr]
			-
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2y_c }{y^2_\mathrm{max}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ 2x'y' \biggl\{
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( \cos\theta )
			- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( \sin\theta ) \biggr]\biggl[\frac{1}{x^2_\mathrm{max}} \biggr]
			+
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta \biggr] \biggl[\frac{1 }{y^2_\mathrm{max}} \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ 2x' \biggl\{
			\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 \tan\theta
			\biggr]\biggl[\frac{z_0}{x^2_\mathrm{max}} \biggr]
			-
			\biggl[\beta \Omega_2 \sin\theta
			- \gamma \Omega_3 \cos\theta \biggr]\biggl[\frac{y_c }{y^2_\mathrm{max}} \biggr]
			\biggr\}
	</math>		</math>
	</td>		</td>

Tohline: /* Test Orthogonality */

2021-05-25T22:25:17Z

Test Orthogonality

← Older revision		Revision as of 22:25, 25 May 2021
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	</td></tr></table>		</td></tr></table>

	Next, ~~we pull~~ from [[#Riemann_Flow\|Step #3, above]],
			Next, after [[#ThetaDef\|setting]] <math>~\tan\theta = - (\beta\Omega_2/\gamma\Omega_3)</math>, and pulling from [[#Riemann_Flow\|Step #3, above]], we have,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	~~\boldsymbol{\hat\imath'}~~ \biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( y'\cos\theta + z_0\tan\theta )		\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 ( y'\cos\theta + z_0\tan\theta )
	- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( y'\sin\theta ) \biggr]		- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( y'\sin\theta ) \biggr]\biggl[\frac{2x'}{x^2_\mathrm{max}} \biggr]
	+		+
	~~\boldsymbol{\hat\jmath'}~~ \biggl[\beta \Omega_2 \sin\theta		\biggl[\beta \Omega_2 \sin\theta
	- \gamma \Omega_3 \cos\theta \biggr]x' \~~, .~~		- \gamma \Omega_3 \cos\theta \biggr]x' \biggl[\frac{2(y' - y_c)}{y^2_\mathrm{max}} \biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* Riemann Flow */

2021-05-25T22:18:49Z

Riemann Flow

← Older revision		Revision as of 22:18, 25 May 2021
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	</tr>		</tr>
	</table>		</table>
	In order for the <math>~\boldsymbol{k}'</math> component to be zero in the tipped plane, we must choose the tipping angle such that,		<span id="ThetaDef">In order</span> for the <math>~\boldsymbol{k}'</math> component to be zero in the tipped plane, we must choose the tipping angle such that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">