User:Tohline/SSC/Stability Eulerian Perspective - Revision history

Tohline: /* Euler Equation */

2017-03-08T21:12:53Z

Euler Equation

← Older revision		Revision as of 21:12, 8 March 2017
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	where, the binomial theorem has been used to obtain the expression on the righthand side of the second line and, in the last line, the sum of the first pair of terms has been set to zero because the initial configuration is assumed to be in equilibrium. Combining these simplification steps, we have the,		where, the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial\|binomial theorem]] has been used to obtain the expression on the righthand side of the second line and, in the last line, the sum of the first pair of terms has been set to zero because the initial configuration is assumed to be in equilibrium. Combining these simplification steps, we have the,
	<div align="center">		<div align="center">
	<font color="#770000">'''Linearized Euler Equation'''</font><br />		<font color="#770000">'''Linearized Euler Equation'''</font><br />

Tohline: /* Part 4 */

2015-07-09T19:26:13Z

Part 4

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	<font size="+50">⏎</font>		<font size="+50">⏎</font>
			<p> </p>
			<p> </p>

			<font size="+2"><math>~\Rightarrow</math></font>
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Tohline: /* Part 4 */

2015-07-09T19:04:09Z

Part 4

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	==Part 4==		==Part 4==



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			<p> </p>
			<p> </p>
			<font size="+10">↵</font>
			<p> </p>
			<p> </p>
			<font size="+50">⏎</font>
			</div>

Tohline: /* Part 3 */ Display Rosseland's derived equation 2.18

2015-07-09T18:45:30Z

Part 3: Display Rosseland's derived equation 2.18

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	[[File:LedouxWalraven1958Eq57.23.png\|600px\|center\|Ledoux & Walraven (1958)]]		[[File:LedouxWalraven1958Eq57.23.png\|600px\|center\|Ledoux & Walraven (1958)]]
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			</div>


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			Wave Equation for Self-Gravitating Fluids as Derived and Presented by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf S. Rosseland (1969)]
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			[[File:RosselandEq2.18.png\|600px\|center\|Rosseland (1969)]]
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	<math>~\Downarrow</math>		<math>~\Downarrow</math>
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	==Part 4==		==Part 4==

Tohline: /* Part 4 */

2015-07-09T18:41:17Z

Part 4

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			<div align="center"><math>~dP/dt</math></div>


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Tohline: /* Part 4 */

2015-07-09T16:38:56Z

Part 4

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			For any scalar variable, <math>~q(\vec{r},t)</math>, the relationship between a ''Lagrangian'' (total) and ''Eulerian'' (partial) time-derivative is,

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			<math>
			~\frac{dq}{dt} = \frac{\partial q}{\partial t} + \vec{v}\cdot \nabla q \, .
			</math>
			</div>



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			<math>~\frac{d}{dt}\biggl[ \nabla_r P \biggr]</math>
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			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\partial }{\partial t}\biggl[ \nabla_r P \biggr] + v_r \frac{\partial }{\partial r} \biggl[ \nabla_r P \biggr]
			= \frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) + \frac{\partial }{\partial r} \biggl[ v_r (\nabla_r P )\biggr] - \nabla_r P \biggl( \frac{\partial v_r}{\partial r} \biggr)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr)
			+ \frac{\partial }{\partial r} \biggl[ v_r (\nabla_r P )\biggr]
			- \nabla_r P \biggl( \frac{\partial v_r}{\partial r} \biggr) </math>
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			</tr>
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			<table border="0" cellpadding="5" align="center">

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			<math>~\frac{d}{dt}\biggl[ \nabla_r P \biggr]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr)
			+ \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt} - \frac{\partial P}{\partial t} \biggr]
			- \rho \biggl[ \frac{\nabla_r P}{\rho} \biggr]\frac{\partial v_r}{\partial r}
			= \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt} \biggr]
			+ \rho \biggl[ \frac{dv_r}{dt} + g \biggr]\frac{\partial v_r}{\partial r}
			</math>
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			</div>


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Tohline: /* Part 4 */

2015-07-08T23:59:46Z

Part 4

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			<math>
			~\frac{\nabla_r P}{\rho} \biggl[ \frac{1}{\rho} \frac{d\rho}{dt}\biggr] = -\frac{\nabla_r P}{\rho} \biggl[\nabla\cdot\vec{v}\biggr]
			= -\frac{\nabla_r P}{\rho} \biggl[\frac{\partial v_r}{\partial r} + \frac{2v_r}{r}\biggr]
			= \biggl[ \frac{dv_r}{dt} + g \biggr] \biggl[\frac{\partial v_r}{\partial r} + \frac{2v_r}{r}\biggr]
			</math>
			</div>



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Tohline: /* Part 4 */

2015-07-08T23:25:15Z

Part 4

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		<math>
		~g \equiv \nabla_r \Phi = \frac{GM_r}{r^2}
		</math>
		</div>




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		<math>
		~g_0 = -\frac{1}{\rho_0} \nabla_r P_0~~~~~~~~\frac{d}{dt}\biggl[\nabla_r \Phi \biggr]
		</math>
		</div>

Tohline: /* Scratch Work */

2015-07-08T17:24:42Z

Scratch Work

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	<math>~\Downarrow</math>		<math>~\Downarrow</math>
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			==Part 4==


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			<math>\frac{d^2 v}{dt^2} = - \frac{1}{\rho} \frac{d}{dt} \biggl[\nabla_r P\biggr] + \frac{\nabla_r P}{\rho^2} \frac{d\rho}{dt}
			- \frac{d}{dt}\biggl[\nabla_r \Phi\biggr]</math>
			</div>






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Tohline: /* Part 1 */

2015-07-06T20:51:04Z

Part 1

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			<math>~
			\biggl(\frac{dP}{d\rho}\biggr)_0 = \biggl( \frac{\gamma_g P}{\rho}\biggr)_0
			</math>
	</div>		</div>