PGE/RotatingFrame - Revision history

Jet53man: /* Overview */

2024-04-28T00:49:29Z

Overview

← Older revision		Revision as of 20:49, 27 April 2024
Line 132:		Line 132:
	<td align="right">		<td align="right">
	<math>		<math>
	\frac{\partial\vec{u}}{dt} + (\vec{u} \cdot \nabla)\vec{u}		\frac{\partial\vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{\partial\vec{u}}{dt} + (\vec{u} \cdot \nabla)\vec{u}		\frac{\partial\vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{\partial}{dt} \biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]		\frac{\partial}{\partial t} \biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]
	+		+
	(\vec{u} \cdot \nabla)\biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]		(\vec{u} \cdot \nabla)\biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]
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	<math>		<math>
	\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]		\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
	+ \frac{\partial}{dt} \biggl[- \vec\Omega \times \vec{x} \biggr]		+ \frac{\partial}{\partial t} \biggl[- \vec\Omega \times \vec{x} \biggr]
	+		+
	(\vec{u} \cdot \nabla)\biggl[- \vec\Omega \times \vec{x} \biggr]		(\vec{u} \cdot \nabla)\biggl[- \vec\Omega \times \vec{x} \biggr]
Line 229:		Line 229:
	\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]		\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
	+		+
	\vec{x}\times \frac{d\vec\Omega}{dt} - \vec\Omega \times \vec{v}		\vec{x}\times \frac{d\vec\Omega}{\partial t} - \vec\Omega \times \vec{v}
	</math>		</math>
	</td>		</td>

Jet53man: /* Overview */

2024-04-28T00:26:32Z

Overview

@@ Line 264: / Line 264: @@
 + \vec{u}\times \vec\Omega
 + \vec\Omega \times (\vec{x}\times \vec\Omega )
 </math>
    </td>

Jet53man: /* Overview */

2024-04-28T00:21:16Z

Overview

@@ Line 182: / Line 182: @@
++
 (\vec{u} \cdot \nabla)\biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]
 </math>
    </td>

Jet53man: /* Overview */

2024-04-28T00:05:38Z

Overview

@@ Line 148: / Line 148: @@
 </table>
-Along the way, and being guided by Chandrasekhar's presentation in Chapter 4, &sect;25 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we appreciate that it can be useful to highlight a ''hybrid'' representation of the Euler Equation that involves a mixture of the velocity variables, <math>\vec{u}</math> along with <math>\vec{v}</math>;  specifically,
+Along the way, and being guided by Chandrasekhar's presentation in Chapter 4, &sect;25 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we appreciate that it can be useful to highlight a ''hybrid'' representation of the Euler Equation that involves a mixture of the velocity variables, <math>\vec{u}</math> along with <math>\vec{v}</math>.  For example, beginning with this last expressions, we can write,
 <table border="0" align="center" cellpadding="5">

Jet53man: /* Overview */

2024-04-27T22:20:59Z

Overview

← Older revision		Revision as of 18:20, 27 April 2024
Line 148:		Line 148:
	</table>		</table>

	Along the way, and being guided by Chandrasekhar's presentation in EFE, we appreciate that it can be useful to highlight a ''hybrid'' representation of the Euler Equation that involves a mixture of the velocity variables, <math>\vec{u}</math> along with <math>\vec{v}</math>; specifically,		Along the way, and being guided by Chandrasekhar's presentation in Chapter 4, §25 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], we appreciate that it can be useful to highlight a ''hybrid'' representation of the Euler Equation that involves a mixture of the velocity variables, <math>\vec{u}</math> along with <math>\vec{v}</math>; specifically,

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

Jet53man: /* Overview */

2024-04-21T18:28:31Z

Overview

@@ Line 258: / Line 258: @@
 <div align="center"><font color="red"><b>CAUTION!</b></font></div>
-If our interpretation of Chandrasekhar's EFE &sect;25 discussion of "moving frames" is correct,
+If our interpretation of Chandrasekhar's discussion of "moving frames" is correct &#8212; see, Chap. 4, &sect;25 of  [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; the RHS of his Eq. (18) should match the LHS of our "hybrid" equation, but it does not:  the pair of vector velocities in his advection term are swapped.  That is, based on our interpretation, the RHS of his Eq. (18) reads,
 </td></tr></table>

Jet53man: /* Overview */

2024-04-21T18:00:58Z

Overview

← Older revision		Revision as of 14:00, 21 April 2024
Line 254:		Line 254:
	</tr>		</tr>
	</table>		</table>

			<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

			<div align="center"><font color="red"><b>CAUTION!</b></font></div>
			If our interpretation of Chandrasekhar's EFE §25 discussion of "moving frames" is correct,
			</td></tr></table>

	==Coordinate Transformation==		==Coordinate Transformation==

Jet53man: /* Overview */

2024-04-21T17:53:30Z

Overview

@@ Line 250: / Line 250: @@
 - ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]
 \, .
 </math>
    </td>

Jet53man: /* Overview */

2024-04-20T17:35:14Z

Overview

@@ Line 280: / Line 280: @@
 + \hat\jmath(\Omega_3 x - \Omega_1 z) + \hat{k}(\Omega_1 y - \Omega_2 x)\biggr]
 \cdot \nabla \biggr\}\vec{v}
 </math>
    </td>

Jet53man: /* Overview */

2024-04-19T16:48:23Z

Overview

← Older revision		Revision as of 12:48, 19 April 2024
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	<math>\frac{\partial\vec{v}}{\partial t}		<math>\frac{\partial\vec{v}}{\partial t}
	+ \biggl[\vec{u} \cdot \nabla \biggr]\vec{v}		+ \biggl[\vec{u} \cdot \nabla \biggr]\vec{v}
	+ \biggl[(\~~vec~~\~~Omega~~ \~~times~~ \~~vec~~{x})\cdot \nabla \biggr]\vec{v}		+ \biggl\{\biggl[\hat\imath(\Omega_2 z - \Omega_3 y)
			+ \hat\jmath(\Omega_3 x - \Omega_1 z) + \hat{k}(\Omega_1 y - \Omega_2 x)\biggr]
			\cdot \nabla \biggr\}\vec{v}
	</math>		</math>
	</td>		</td>