User:Tohline/Cylindrical 3D/Linearization - Revision history

Tohline: /* Linearized Equations in Cylindrical Coordinates */ Finished assembling summary table

2016-03-12T05:26:09Z

Linearized Equations in Cylindrical Coordinates: Finished assembling summary table

← Older revision		Revision as of 05:26, 12 March 2016
Line 64:		Line 64:
	These match, for example, equations (3.1) - (3.4) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750), hereafter, PPI.		These match, for example, equations (3.1) - (3.4) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750), hereafter, PPI.

			==Linearization==

	If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become:		If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become:
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	</div>		</div>
	where the logic followed in deriving the last expression from the next-to-last one is directly analogous to [[#Linearizing_Radial_Component_of_Euler_Equation\|the logic used, above]], in obtaining the final expression for the radial component of the linearized Euler equation.		where the logic followed in deriving the last expression from the next-to-last one is directly analogous to [[#Linearizing_Radial_Component_of_Euler_Equation\|the logic used, above]], in obtaining the final expression for the radial component of the linearized Euler equation.

			===Linearizing Continuity Equation===

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

			<tr>
			<td align="right">
			<math>~\frac{\partial\rho^'}{\partial t}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr]
			- \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho_0 \varpi {\dot\varphi}^' + \rho^' \varpi {\dot\varphi}_0 \biggr]
			- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~~\frac{\partial\rho^'}{\partial t} + ( {\dot\varphi}_0 )\frac{\partial \rho^'}{\partial \varphi}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr]
			- \frac{1}{\varpi} \frac{\partial }{\partial \varphi} \biggl[ \rho_0 \varpi {\dot\varphi}^' \biggr]
			- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			===Summary===
			<div align="center">
			<table border="1" cellpadding="5" align="center">
			<tr>
			<th align="center">
			Set of Linearized Principal Governing Equations in Cylindrical Coordinates
			</th>
			</tr>
			<tr><td align="center">

			<table border="0" cellpadding="8" align="center">

			<tr><td align="center" colspan="3"><font color="#770000">'''Continuity Equation'''</font></td></tr>
			<tr>
			<td align="right">
			<math>~\frac{\partial\rho^'}{\partial t} + ( {\dot\varphi}_0 )\frac{\partial \rho^'}{\partial \varphi}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr]
			- \frac{1}{\varpi} \frac{\partial }{\partial \varphi} \biggl[ \rho_0 \varpi {\dot\varphi}^' \biggr]
			- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] \, .
			</math>
			</td>
			</tr>

			<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varpi</math> Component of Euler Equation'''</font></td></tr>
			<tr>
			<td align="right">
			<math>~
			\frac{\partial {\dot\varpi}^'}{\partial t} +
			( {\dot\varphi}_0 ) \frac{\partial {\dot\varpi}^'}{\partial\varphi}
			- 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) - \frac{\partial \Phi^'}{\partial \varpi}
			</math>
			</td>
			</tr>

			<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varphi</math> Component of Euler Equation'''</font></td></tr>
			<tr>
			<td align="right">
			<math>~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} +
			( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
			\frac{{\dot\varpi}^'}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{1}{\varpi} \biggl[ \frac{\partial }{\partial \varphi} \biggl(\frac{P^'}{\rho_0}\biggr)+ \frac{\partial \Phi^'}{\partial \varphi} \biggr]
			</math>
			</td>
			</tr>

			<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component of Euler Equation'''</font></td></tr>
			<tr>
			<td align="right">
			<math>~
			\frac{\partial {\dot{z}}^'}{\partial t}
			+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr)
			- \frac{\partial \Phi^'}{\partial z}
			</math>
			</td>
			</tr>

			<tr><td align="center" colspan="3"><font color="#770000">'''Adiabatic Form of the 1<sup>st</sup> Law of Thermodynamics'''</font></td></tr>
			<tr>
			<td align="right">
			<math>~\frac{P^'}{P_0}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \frac{\gamma \rho^'}{\rho_0} </math>
			</td>
			</tr>

			<tr><td align="center" colspan="3"><font color="#770000">'''Poisson Equation'''</font></td></tr>
			<tr>
			<td align="right">
			<math>~\nabla^2 \Phi^'
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			4\pi G\rho^'
			</math>
			</td>
			</tr>
			</table>

			</td></tr>
			</table>
			</div>

	=See Also=		=See Also=

Tohline: /* Eulerian Formulation of Nonlinear Governing Equations */ Finished deriving vertical component of linearized Euler equation

2016-03-12T00:28:14Z

Eulerian Formulation of Nonlinear Governing Equations: Finished deriving vertical component of linearized Euler equation

← Older revision		Revision as of 00:28, 12 March 2016
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}		<math>~
			- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}
	- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0} \biggr)		- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0} \biggr)
	- \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}</math>		- \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}
			</math>
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>
	</div>		</div>


			===Linearizing Vertical Component of Euler Equation===

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

			<tr>
			<td align="right">
			<math>~
			\frac{\partial {\dot{z}}^'}{\partial t}
			+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial z} - \frac{\partial (\Phi_0+\Phi^')}{\partial z}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{1}{\rho_0}\frac{\partial P^'}{\partial z}
			- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial z}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0} \biggr)
			- \frac{\partial (\Phi_0+\Phi^')}{\partial z}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~~
			\frac{\partial {\dot{z}}^'}{\partial t}
			+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}
			+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial z}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial z}\biggr]
			+ \frac{\partial \Phi^'}{\partial z}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl\{
			- \frac{1}{\rho_0}\frac{\partial P_0 }{\partial z}
			- \frac{\partial \Phi_0}{\partial z} \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~~
			\frac{\partial {\dot{z}}^'}{\partial t}
			+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}
			+ \biggl[ \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr]
			+ \frac{\partial \Phi^'}{\partial z}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where the logic followed in deriving the last expression from the next-to-last one is directly analogous to [[#Linearizing_Radial_Component_of_Euler_Equation\|the logic used, above]], in obtaining the final expression for the radial component of the linearized Euler equation.

	=See Also=		=See Also=

Tohline: Finished linearizing azimuthal component of Euler equation

2016-03-11T23:40:45Z

Finished linearizing azimuthal component of Euler equation

← Older revision		Revision as of 23:40, 11 March 2016
Line 65:		Line 65:


	If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become,		If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become:

			===Linearizing Radial Component of Euler Equation===
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 79:		Line 81:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial\varpi} - \frac{\partial \Phi_0}{\partial\varpi}</math>		<math>~- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial\varpi} - \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}</math>
	</td>		</td>
	</tr>		</tr>
Line 96:		Line 98:
	<math>~- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}		<math>~- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}
	- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0} \biggr)		- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0} \biggr)
	- \frac{\partial \Phi_0}{\partial\varpi}</math>		- \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}</math>
	</td>		</td>
	</tr>		</tr>
Line 104:		Line 106:
	<math>~\Rightarrow~~~~		<math>~\Rightarrow~~~~
	\frac{\partial {\dot\varpi}^'}{\partial t} +		\frac{\partial {\dot\varpi}^'}{\partial t} +
	~~\biggl[~~ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} ~~\biggr]~~		{\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi}
	- 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')		- 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
	+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial\varpi}\biggr]		+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial\varpi}\biggr]
			+ \frac{\partial \Phi^'}{\partial \varpi}
	</math>		</math>
	</td>		</td>
Line 114:		Line 117:
	<td align="left">		<td align="left">
	<math>~\biggl\{ \varpi ( {\dot\varphi}_0)^2		<math>~\biggl\{ \varpi ( {\dot\varphi}_0)^2
	- ~~\biggl[~~\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}~~\biggr]~~		- \frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}
	- \frac{\partial \Phi_0}{\partial\varpi} \biggr\}		- \frac{\partial \Phi_0}{\partial\varpi} \biggr\}
	</math>		</math>
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	<math>~\Rightarrow~~~~		<math>~\Rightarrow~~~~
	\frac{\partial {\dot\varpi}^'}{\partial t} +		\frac{\partial {\dot\varpi}^'}{\partial t} +
	~~\biggl[~~ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} ~~\biggr]~~		{\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi}
	- 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')		- 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
	+ \biggl[ \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr]		+ \biggl[ \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr] + \frac{\partial \Phi^'}{\partial \varpi}
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	</tr>		</tr>
			</table>
			</div>

			This last expression has been obtained by recognizing that, in the next-to-last expression: (1) The terms inside the curly braces on the right-hand side collectively provide a statement of equilibrium (in the radial-coordinate direction) in the initial, unperturbed configuration and therefore the terms sum to zero; and (2) the terms inside square brackets on the left-hand side can be rewritten in a more compact form because we have adopted a polytropic equation of state to build the unperturbed initial equilibrium configuration and are examining only adiabatic perturbations with <math>~\gamma = (n+1)/n</math>, in which case,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\nabla P_0}{P_0} = \frac{(n+1)}{n} \cdot \frac{\nabla \rho_0}{\rho_0} \, ,</math>
			</td>
			<td align="center">
			and
			</td>
			<td align="left">
			<math>~\frac{P^'}{P_0} = \frac{\gamma \rho^'}{\rho_0} \, .</math>
			</td>
			</tr>
			</table>
			</div>


			===Linearizing Azimuthal Component of Euler Equation===
			Keeping in mind that the initial equilibrium configuration is axisymmetric — that is, equilibrium parameters exhibit no variation in the azimuthal direction — and, in addition, <math>~\dot\varphi_0</math> exhibits no variation in the vertical direction, we have,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} + ( {\dot\varpi}^') \frac{\partial (\varpi\dot\varphi_0)}{\partial\varpi} +
			( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
			( {\dot\varpi}^') {\dot\varphi_0} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{1}{\varpi} \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial \varphi} + \frac{\partial \Phi^'}{\partial \varphi} \biggr]</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} +
			( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
			\frac{{\dot\varpi}^'}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{1}{\varpi} \biggl[ \frac{\partial }{\partial \varphi} \biggl(\frac{P^'}{\rho_0}\biggr)+ \frac{\partial \Phi^'}{\partial \varphi} \biggr]
			\, .</math>
			</td>
			</tr>

	</table>		</table>
	</div>		</div>

	=See Also=		=See Also=



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Eulerian Formulation of Nonlinear Governing Equations */ Finished linearizing varpi-component of Euler

2016-03-11T01:37:59Z

Eulerian Formulation of Nonlinear Governing Equations: Finished linearizing varpi-component of Euler

← Older revision		Revision as of 01:37, 11 March 2016
Line 8:		Line 8:

	From our more detailed, [[User:Tohline/Cylindrical_3D#Eulerian_Formulation\|accompanying discussion]] we pull the Eulerian representation of the set of principal governing equations written in cylindrical coordinates.		From our more detailed, [[User:Tohline/Cylindrical_3D#Eulerian_Formulation\|accompanying discussion]] we pull the Eulerian representation of the set of principal governing equations written in cylindrical coordinates.


	~~<div align="center">~~
	~~<span id="EulerContinuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />~~

	~~<math>~~
	~~\frac{\partial\rho}{\partial t} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr]~~
	~~+ \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho \varpi \dot\varphi \biggr]~~
	~~+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0~~
	~~</math><br />~~
	~~</div>~~


Line 63:		Line 52:
	</div>		</div>

			<div align="center">
			<span id="EulerContinuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

			<math>
			\frac{\partial\rho}{\partial t} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr]
			+ \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho \varpi \dot\varphi \biggr]
			+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0
			</math><br />
			</div>

			These match, for example, equations (3.1) - (3.4) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750), hereafter, PPI.


			If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\partial {\dot\varpi}^'}{\partial t} +
			\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr]
			- \varpi ( { {\dot\varphi}_0 + {\dot\varphi}^'})^2 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial\varpi} - \frac{\partial \Phi_0}{\partial\varpi}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~~
			\frac{\partial {\dot\varpi}^'}{\partial t} +
			\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr]
			- \varpi ( {\dot\varphi}_0)^2 - 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}
			- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0} \biggr)
			- \frac{\partial \Phi_0}{\partial\varpi}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~~
			\frac{\partial {\dot\varpi}^'}{\partial t} +
			\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr]
			- 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
			+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial\varpi}\biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl\{ \varpi ( {\dot\varphi}_0)^2
			- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]
			- \frac{\partial \Phi_0}{\partial\varpi} \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~~
			\frac{\partial {\dot\varpi}^'}{\partial t} +
			\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr]
			- 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
			+ \biggl[ \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

	=See Also=		=See Also=

Tohline: Begin new chapter in which the linearized equations are derived in cylindrical coordiinates

2016-03-11T00:19:35Z

Begin new chapter in which the linearized equations are derived in cylindrical coordiinates

New page

__FORCETOC__


=Linearized Equations in Cylindrical Coordinates=
{{LSU_HBook_header}}

==Eulerian Formulation of Nonlinear Governing Equations==

From our more detailed, [[User:Tohline/Cylindrical_3D#Eulerian_Formulation|accompanying discussion]] we pull the Eulerian representation of the set of principal governing equations written in cylindrical coordinates.

<div align="center">
'''Equation of Continuity''' 

<math>
\frac{\partial\rho}{\partial t} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr]
+ \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho \varpi \dot\varphi \biggr]
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0
</math> 
</div>

<div align="center">

'''<math>\varpi</math> Component of Euler Equation'''
 

<math>
\frac{\partial \dot\varpi}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] +
\biggl[ \dot\varphi \frac{\partial \dot\varpi}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] - \varpi {\dot\varphi}^2 =
- \frac{1}{\rho}\frac{\partial P}{\partial\varpi} - \frac{\partial \Phi}{\partial\varpi}
</math> 


'''<math>\varphi</math> Component of Euler Equation'''
 

<math>
\frac{\partial (\varpi\dot\varphi)}{\partial t} + \biggl[ \dot\varpi \frac{\partial (\varpi\dot\varphi)}{\partial\varpi} \biggr] +
\biggl[ \dot\varphi \frac{\partial (\varpi\dot\varphi)}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial (\varpi\dot\varphi)}{\partial z} \biggr] + \dot\varpi \dot\varphi =
- \frac{1}{\varpi} \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial \varphi} + \frac{\partial \Phi}{\partial \varphi} \biggr]
</math> 


'''<math>z</math> Component of Euler Equation'''
 

<math>
\frac{\partial \dot{z}}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr]
+ \biggl[ \dot\varphi \frac{\partial \dot{z}}{\partial\varphi} \biggr] +\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr] =
- \frac{1}{\rho}\frac{\partial P}{\partial z} - \frac{\partial \Phi}{\partial z}
</math> 

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=See Also=

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