User:Tohline/AxisymmetricConfigurations/Equilibria - Revision history

Tohline: /* Axisymmetric Configurations (Steady-State Structures) */

2019-08-04T00:15:28Z

Axisymmetric Configurations (Steady-State Structures)

← Older revision		Revision as of 00:15, 4 August 2019
Line 3:		Line 3:

	=Axisymmetric Configurations (Steady-State Structures)=		=Axisymmetric Configurations (Steady-State Structures)=
	<!-- [[~~Image~~:~~LSU_Structure_still~~.~~gif~~\|~~74px\|left~~]] ~~-->~~		Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29\|identified set of simplified governing equations]].
	~~{{LSU_HBook_header}}~~


	Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29\|identified set of simplified governing equations]].		{{LSU_HBook_header}}

	==Cylindrical Coordinate Base==		==Cylindrical Coordinate Base==

Tohline: /* Axisymmetric Configurations (Structure — Part II) */

2019-08-04T00:14:10Z

Axisymmetric Configurations (Structure — Part II)

← Older revision		Revision as of 00:14, 4 August 2019
Line 2:		Line 2:
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->

	=Axisymmetric Configurations (~~Structure — Part II~~)=		=Axisymmetric Configurations (Steady-State Structures)=
	<!-- [[Image:LSU_Structure_still.gif\|74px\|left]] -->		<!-- [[Image:LSU_Structure_still.gif\|74px\|left]] -->
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
Line 248:		Line 248:
	Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:		Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:

	<table align="center" border="1" cellpadding="10"><tr><td align="center">		<table align="center" border="1" cellpadding="10">
			<tr>
			<th align="center">Spherical Coordinate Base</th>
			</tr>
			<tr><td align="center">
	<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />		<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

Tohline: /* Spherical Coordinate Base */

2019-08-04T00:11:05Z

Spherical Coordinate Base

Show changes

Tohline: Created page with 'FORCETOC =Axisymmetric Configurations (Structure — Part II)= {{LSU_HBook_hea…'

2019-08-04T00:09:42Z

Created page with '__FORCETOC__  =Axisymmetric Configurations (Structure — Part II)=  {{LSU_HBook_hea…'

New page

__FORCETOC__


=Axisymmetric Configurations (Structure — Part II)=

{{LSU_HBook_header}}

Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|identified set of simplified governing equations]].

==Cylindrical Coordinate Base==
We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:

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

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

<span id="PGE:Euler">The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
</span>
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~
\cancelto{0}{\frac{\partial \dot\varpi}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] +
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3}
</math>
</td>
</tr>
<tr>
<td align="right">
<math>~
\cancelto{0}{\frac{\partial \dot{z}}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] +
\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]
</math>
</td>
</tr>
</table>

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>~
\biggl\{\cancel{\frac{\partial \epsilon}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \epsilon}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \epsilon}{\partial z} \biggr]\biggr\} +
P \biggl\{\cancel{\frac{\partial }{\partial t}\biggl(\frac{1}{\rho}\biggr)} +
\biggl[ \dot\varpi \frac{\partial }{\partial\varpi}\biggl(\frac{1}{\rho}\biggr) \biggr] +
\biggl[ \dot{z} \frac{\partial }{\partial z}\biggl(\frac{1}{\rho}\biggr) \biggr] \biggr\} = 0
</math>

<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho .
</math><br />
</div>

The steady-state flow field that will be adopted to satisfy both an axisymmetric geometry and the time-independent constraint is, <math>~\vec{v} = \hat{e}_\varphi (\varpi \dot\varphi)</math>. That is, <math>~\dot\varpi = \dot{z} = 0</math> but, in general, <math>~\dot\varphi</math> is not zero and can be an arbitrary function of <math>~\varpi</math> and <math>~z</math>, that is, <math>~\dot\varphi = \dot\varphi(\varpi,z)</math>. We will seek solutions to the above set of coupled equations for various chosen spatial distributions of the angular velocity <math>~\dot\varphi(\varpi,z)</math>, or of the specific angular momentum, <math>~j(\varpi,z) = \varpi^2 \dot\varphi(\varpi,z)</math>.

<span id="2DgoverningEquations">After setting the radial and vertical velocities to zero,</span> we see that the <math>1^\mathrm{st}</math> (continuity) and <math>4^\mathrm{th}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> & <math>3^\mathrm{rd}</math> (Euler) and <math>5^\mathrm{th}</math> (Poisson) give, respectively,

<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0</math>
</td>
</tr>
<tr>
<td align="right">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0</math>
</td>
</tr>
<tr>
<td align="right">
<math>~
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi G \rho \, .</math>
</td>
</tr>
</table>

</div>

As has been outlined in our discussion of [[User:Tohline/SR#Time-Independent_Problems|supplemental relations for time-independent problems]], in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}.

==Spherical Coordinate Base==

=See Also=
* Part I of ''Axisymmetric Configurations'': [[User:Tohline/AxisymmetricConfigurations/PGE|Simplified Governing Equations]]

{{LSU_HBook_footer}}

User:Tohline/AxisymmetricConfigurations/Equilibria - Revision history

Tohline: /* Axisymmetric Configurations (Steady-State Structures) */

Tohline: /* Axisymmetric Configurations (Structure — Part II) */

Tohline: /* Spherical Coordinate Base */

Tohline: Created page with '__FORCETOC__ =Axisymmetric Configurations (Structure — Part II)= {{LSU_HBook_hea…'

Tohline: Created page with 'FORCETOC =Axisymmetric Configurations (Structure — Part II)= {{LSU_HBook_hea…'