User:Tohline/PGE/Euler - Revision history

Tohline: /* Double Check Vector Identities */

2019-08-07T21:10:42Z

Double Check Vector Identities

← Older revision		Revision as of 21:10, 7 August 2019
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	</tr>		</tr>
	</table>		</table>
	and we explicitly demonstrate that ~~they~~ they are among the set of velocity profiles that can also be expressed in terms of the gradient of a "centrifugal potential," <math>~\nabla\Psi</math>.		and we explicitly demonstrate that they are among the set of velocity profiles that can also be expressed in terms of the gradient of a "centrifugal potential," <math>~\nabla\Psi</math>.

	=Related Discussions=		=Related Discussions=

Tohline: /* Double Check Vector Identies */

2019-08-07T21:09:22Z

Double Check Vector Identies

← Older revision		Revision as of 21:09, 7 August 2019
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	</div>		</div>

	==Double Check Vector ~~Identies~~==		==Double Check Vector Identities==

	In a subsection of an accompanying chapter titled, [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Double_Check_Vector_Identities\|''Double Check Vector Identities,'']] we explicitly demonstrate for four separate "simple rotation profiles" that these two separate terms involving a nonlinear velocity expression do indeed generate identical mathematical relations, namely.		In a subsection of an accompanying chapter titled, [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Double_Check_Vector_Identities\|''Double Check Vector Identities,'']] we explicitly demonstrate for four separate "simple rotation profiles" that these two separate terms involving a nonlinear velocity expression do indeed generate identical mathematical relations, namely.

Tohline: /* Double Check Vector Identies */

2019-08-07T21:08:49Z

Double Check Vector Identies

← Older revision		Revision as of 21:08, 7 August 2019
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	==Double Check Vector Identies==		==Double Check Vector Identies==

	In a subsection of an accompanying chapter titled, [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Double_Check_Vector_Identities\|''Double Check Vector Identities,'']] we explicitly demonstrate for four separate "simple rotation profiles" that these ~~three~~ separate ~~steady-state balance expressions~~ do indeed generate identical mathematical relations.		In a subsection of an accompanying chapter titled, [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Double_Check_Vector_Identities\|''Double Check Vector Identities,'']] we explicitly demonstrate for four separate "simple rotation profiles" that these two separate terms involving a nonlinear velocity expression do indeed generate identical mathematical relations, namely.
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\vec\zeta \times \vec{v} + \frac{1}{2}\nabla (v^2) \, ;</math>
			</td>
			</tr>
			</table>
			and we explicitly demonstrate that they they are among the set of velocity profiles that can also be expressed in terms of the gradient of a "centrifugal potential," <math>~\nabla\Psi</math>.

	=Related Discussions=		=Related Discussions=

Tohline at 21:02, 7 August 2019

2019-08-07T21:02:56Z

Show changes

Tohline: /* j-Constant Rotation */

2019-08-07T17:31:44Z

j-Constant Rotation

← Older revision		Revision as of 17:31, 7 August 2019
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	-~\hat{e}_\varpi \frac{\omega_c^2}{\varpi} \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]^2		-~\hat{e}_\varpi \frac{\omega_c^2}{\varpi} \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]^2
	=		=
	-~\hat{e}_\varpi \biggl[ \frac{\omega_c^2A^4 \varpi}{(A^2 + \varpi^2)^2} \biggr]		-~\hat{e}_\varpi \biggl[ \frac{\omega_c^2A^4 \varpi}{(A^2 + \varpi^2)^2} \biggr] \, .
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	-~ \hat{e}_\varpi \biggl[ \frac{ \omega_c^2 A^4\varpi }{ (A^2 + \varpi^2)^{2}} \biggr]		-~ \hat{e}_\varpi \biggl[ \frac{ \omega_c^2 A^4\varpi }{ (A^2 + \varpi^2)^{2}} \biggr] \, .
	</math>		</math>
	</td>		</td>

Tohline: /* j-Constant Rotation */

2019-08-07T17:30:37Z

j-Constant Rotation

← Older revision		Revision as of 17:30, 7 August 2019
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	</tr>		</tr>
	</table>		</table>

			[K}  Alternatively,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec\zeta \times \vec{v} + \frac{1}{2}\nabla (v^2)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\hat{e}_z~ \biggl[2\omega_c A^4 (A^2 + \varpi^2)^{-2} \biggr] \times \hat{e}_\varphi ~\omega_c \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr] + \frac{1}{2} \hat{e}_\varpi \frac{\partial}{\partial \varpi}\biggl[ \omega_c^2 A^4\varpi^2 (A^2 + \varpi^2)^{-2}\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \hat{e}_\varpi ~ \biggl[ \frac{2\omega_c^2 A^6 \varpi }{(A^2 + \varpi^2)^{3}} \biggr] + \hat{e}_\varpi \biggl[ \omega_c^2 A^4\varpi (A^2 + \varpi^2)^{-2} - 2 \omega_c^2 A^4\varpi^3 (A^2 + \varpi^2)^{-3}\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\hat{e}_\varpi \biggl[ \frac{ \omega_c^2 A^4\varpi }{ (A^2 + \varpi^2)^{2}} - \frac{2 \omega_c^2 A^4\varpi^3}{ (A^2 + \varpi^2)^{3} }
			- \frac{2\omega_c^2 A^6 \varpi }{(A^2 + \varpi^2)^{3}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\hat{e}_\varpi \biggl[ (A^2 + \varpi^2) - 2 \varpi^2 - 2A^2 \biggr] \frac{ \omega_c^2 A^4\varpi }{ (A^2 + \varpi^2)^{3}}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~ \hat{e}_\varpi \biggl[ \frac{ \omega_c^2 A^4\varpi }{ (A^2 + \varpi^2)^{2}} \biggr]
			</math>
			</td>
			</tr>
			</table>

			[L}  Or,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla \Psi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\hat{e}_\varpi ~ \frac{1}{2} \frac{\partial}{\partial \varpi}\biggl[ \omega_c^2 A^4 (A^2 + \varpi^2)^{-1} \biggr]
			= - \hat{e}_\varpi \biggl[ \frac{ \omega_c^2 A^4 \varpi }{ (A^2 + \varpi^2)^{2}} \biggr] \, .</math>
			</td>
			</tr>
			</table>
			This demonstrates that, in the case of a j-constant rotation profile, all three expressions are identical.

	=Related Discussions=		=Related Discussions=

Tohline: /* j-Constant Rotation */

2019-08-07T16:59:27Z

j-Constant Rotation

← Older revision		Revision as of 16:59, 7 August 2019
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	~~~\Rightarrow~~~ \frac{j^2}{\varpi^3} = \frac{(\varpi v_\varphi)^2}{\varpi^3} =		~~~\Rightarrow~~~ \frac{j^2}{\varpi^3} = \frac{(\varpi v_\varphi)^2}{\varpi^3} =
	\frac{\omega_c^2}{\varpi} \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]^2 =		\frac{\omega_c^2}{\varpi} \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]^2 =
	~~\omega_c^2~~ \biggl[ \frac{A^4\varpi}{(A^2 + \varpi^2)^2}\biggr]		\biggl[ \frac{\omega_c^2 A^4\varpi}{(A^2 + \varpi^2)^2}\biggr]
	\, ,		\, ,
	</math>		</math>
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	<math>~		<math>~
	\hat{e}_z~ \biggl[2\omega_c A^4 (A^2 + \varpi^2)^{-2} \biggr] \, .		\hat{e}_z~ \biggl[2\omega_c A^4 (A^2 + \varpi^2)^{-2} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

			[J]   Hence,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\hat{e}_\varpi \biggl[ - \frac{v_\varphi \cdot v_\varphi}{\varpi} \biggr] </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~\hat{e}_\varpi \frac{\omega_c^2}{\varpi} \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]^2
			=
			-~\hat{e}_\varpi \biggl[ \frac{\omega_c^2A^4 \varpi}{(A^2 + \varpi^2)^2} \biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* j-Constant Rotation */

2019-08-05T23:49:07Z

j-Constant Rotation

← Older revision		Revision as of 23:49, 5 August 2019
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	</math>		</math>
	</div>		</div>

	where, <math>~\omega_c</math>, and the characteristic length, <math>~A</math>, are both independent of radial position. This also means that,		where, <math>~\omega_c</math>, and the characteristic length, <math>~A</math>, are both independent of radial position. This also means that,

	<div align="center">		<div align="center">
	<math>		<math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\hat{e}_z \biggl[ \frac{1}{\varpi} \frac{\~~partial~~ (\varpi^2 \~~omega_0~~ )}{\~~partial~~ \varpi} \biggr]		\hat{e}_z \biggl\{ \frac{\omega_c}{\varpi} \frac{\partial }{\partial \varpi} \biggl[ \frac{A^2\varpi^2}{A^2 + \varpi^2}\biggr]\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\hat{e}_z~ \frac{\omega_c}{\varpi} \biggl\{ \biggl[ 2A^2\varpi(A^2 + \varpi^2)^{-1} \biggr] - \biggl[ 2A^2\varpi^3(A^2 + \varpi^2)^{-2} \biggr]\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\hat{e}_z~ \biggl[2\omega_c A^4 (A^2 + \varpi^2)^{-2} \biggr] \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Double Check Vector Identities */

2019-08-05T23:37:56Z

Double Check Vector Identities

← Older revision		Revision as of 23:37, 5 August 2019
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	</table>		</table>
	This demonstrates that, in the case of a constant <math>~v_\varphi</math> profile, all three expressions are identical.		This demonstrates that, in the case of a constant <math>~v_\varphi</math> profile, all three expressions are identical.


			==j-Constant Rotation==
			In the case of so-called j-constant rotation, we have,
			<div align="center">
			<math>~\vec{v} = \hat{e}_\varphi (v_\varphi) = \hat{e}_\varphi ~\omega_c \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]
			~~~\Rightarrow~~~ \frac{j^2}{\varpi^3} = \frac{(\varpi v_\varphi)^2}{\varpi^3} =
			\frac{\omega_c^2}{\varpi} \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]^2 =
			\omega_c^2 \biggl[ \frac{A^4\varpi}{(A^2 + \varpi^2)^2}\biggr]
			\, ,
			</math>
			</div>
			where, <math>~\omega_c</math>, and the characteristic length, <math>~A</math>, are both independent of radial position. This also means that,
			<div align="center">
			<math>
			\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi = +\frac{1}{2}\biggl[ \frac{\omega_c^2 A^4}{(A^2 + \varpi^2)}\biggr]~;
			</math>
			</div>
			and,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec\zeta = \nabla \times \vec{v}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\hat{e}_\varpi \biggl[ -\cancel{ \frac{\partial v_\varphi}{\partial z} }\biggr] + \hat{e}_z \biggl[ \frac{1}{\varpi} \frac{\partial (\varpi v_\varphi)}{\partial \varpi} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\hat{e}_z \biggl[ \frac{1}{\varpi} \frac{\partial (\varpi^2 \omega_0 )}{\partial \varpi} \biggr]
			</math>
			</td>
			</tr>
			</table>

	=Related Discussions=		=Related Discussions=

Tohline: /* Uniform vφ */

2019-08-05T23:12:47Z

Uniform vφ

← Older revision		Revision as of 23:12, 5 August 2019
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	</tr>		</tr>
	</table>		</table>

			[I}  Or,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla \Psi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\hat{e}_\varpi\frac{\partial}{\partial\varpi} \biggl\{- v_0^2 \ln \biggl( \frac{\varpi}{\varpi_0} \biggr)\biggr\}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-~ \hat{e}_\varpi v_0^2 \biggl(\frac{\varpi}{\varpi_0} \biggr)^{-1} \frac{1}{\varpi_0}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-~ \hat{e}_\varpi \biggl( \frac{v_0^2}{\varpi} \biggr) \, .</math>
			</td>
			</tr>
			</table>
			This demonstrates that, in the case of a constant <math>~v_\varphi</math> profile, all three expressions are identical.

	=Related Discussions=		=Related Discussions=