Difference between revisions of "User:Tohline/Appendix/Ramblings/RadiationHydro"

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{{LSU_HBook_header}}
{{LSU_HBook_header}}


==Principal Governing Equations==
==Governing Equations==
===Hayes et al. (2006) — But Ignoring the Effects of Magnetic Fields===
===Hayes et al. (2006) — But Ignoring the Effects of Magnetic Fields===
First, referencing §2 of [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006, ApJS, 165, 188 - 228)] — alternatively see §2.1 of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp)] — we see that the set of principal governing equations that is typically used in the astrophysics community to include the effects of radiation on self-gravitating fluid flows includes the,
First, referencing §2 of [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006, ApJS, 165, 188 - 228)] — alternatively see §2.1 of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp)] — we see that the set of principal governing equations that is typically used in the astrophysics community to include the effects of radiation on self-gravitating fluid flows includes the,
Line 79: Line 79:


===Various Manipulations===  
===Various Manipulations===  
By combining the left-hand side of this last expression with the continuity equation then replacing the Lagrangian (that is, the [https://en.wikipedia.org/wiki/Material_derivative ''material'']) time derivative by its Eulerian counterpart, the left-hand side can be rewritten as,


By combining the continuity equation with the
<div id="PGE:FirstLaw" align="center">
<font color="#770000">'''First Law of Thermodynamics'''</font>
{{User:Tohline/Math/EQ_FirstLaw01}}
</div>
we can write,
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>
<math>~\rho T\frac{ds}{dt}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 92: Line 100:
   <td align="left">
   <td align="left">
<math>~
<math>~
\frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho}~\frac{d\rho}{dt}
\rho \frac{d\epsilon}{dt} - \frac{P}{\rho} \frac{d\rho}{dt}
</math>
</math>
   </td>
   </td>
Line 106: Line 114:
   <td align="left">
   <td align="left">
<math>~
<math>~
\frac{dE_\mathrm{rad}}{dt} + E_\mathrm{rad}\nabla\cdot \vec{v}
\rho \frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
Given that the specific internal energy <math>~(\epsilon)</math> and the internal energy density <math>~(e)</math> are related via the expression, <math>~\epsilon = e/\rho</math>, we appreciate that the first of the above-identified ''energy-conservation-based'' dynamical equations is simply a restatement of the 1<sup>st</sup> Law of Thermodynamics in the context of a physical system whose fluid elements gain or lose entropy as a result of the (radiation-transport-related) source and sink terms,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\rho T \frac{ds}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, .</math>
  </td>
</tr>
</table>
By combining the left-hand side of the second of the above-identified ''energy-conservation-based'' dynamical equations with the continuity equation, then replacing the Lagrangian (that is, the [https://en.wikipedia.org/wiki/Material_derivative ''material'']) time derivative by its Eulerian counterpart, the left-hand side can be rewritten as,
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 120: Line 149:
   <td align="left">
   <td align="left">
<math>~
<math>~
\frac{\partial E_\mathrm{rad}}{\partial t} + \vec{v}\cdot \nabla E_\mathrm{rad}+ E_\mathrm{rad}\nabla\cdot \vec{v}
\frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho}~\frac{d\rho}{dt}
</math>
</math>
   </td>
   </td>
Line 134: Line 163:
   <td align="left">
   <td align="left">
<math>~
<math>~
\frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) \, ,
\frac{dE_\mathrm{rad}}{dt} + E_\mathrm{rad}\nabla\cdot \vec{v}
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
which provides an alternate form of the expression, as found for example in equation (4) of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello &amp; J. E. Tohline (2012)].
By combining the continuity equation with the
<div id="PGE:FirstLaw" align="center">
<font color="#770000">'''First Law of Thermodynamics'''</font>
{{User:Tohline/Math/EQ_FirstLaw01}}
</div>
we can write,
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\rho T\frac{ds}{dt}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 161: Line 177:
   <td align="left">
   <td align="left">
<math>~
<math>~
\rho \frac{d\epsilon}{dt} - \frac{P}{\rho} \frac{d\rho}{dt}
\frac{\partial E_\mathrm{rad}}{\partial t} + \vec{v}\cdot \nabla E_\mathrm{rad}+ E_\mathrm{rad}\nabla\cdot \vec{v}
</math>
</math>
   </td>
   </td>
Line 175: Line 191:
   <td align="left">
   <td align="left">
<math>~
<math>~
\rho \frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} \, .
\frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) \, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
which provides an alternate form of the expression, as found for example in equation (4) of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello &amp; J. E. Tohline (2012)].


===Optically Thick Regime===
===Optically Thick Regime===

Revision as of 21:47, 21 October 2018


Radiation-Hydrodynamics

Whitworth's (1981) Isothermal Free-Energy Surface
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Governing Equations

Hayes et al. (2006) — But Ignoring the Effects of Magnetic Fields

First, referencing §2 of J. C. Hayes et al. (2006, ApJS, 165, 188 - 228) — alternatively see §2.1 of D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp) — we see that the set of principal governing equations that is typically used in the astrophysics community to include the effects of radiation on self-gravitating fluid flows includes the,

Poisson Equation

LSU Key.png

<math>\nabla^2 \Phi = 4\pi G \rho</math>

Hayes et al. (2006), p. 190, Eq. (15)

the,

Continuity Equation

LSU Key.png

<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>

and — ignoring magnetic fields — a modified version of the,

Lagrangian Representation
of the Euler Equation,

<math>~\frac{d\vec{v}}{dt}</math>

<math>~=</math>

<math>~ - \frac{1}{\rho}\nabla P - \nabla \Phi + \frac{1}{\rho}\biggl(\frac{\chi}{c}\biggr) \vec{F} \, , </math>

plus the following pair of additional energy-conservation-based dynamical equations:

<math>~\rho \frac{d}{dt} \biggl( \frac{e}{\rho}\biggr) + P\nabla \cdot \vec{v} </math>

<math>~=</math>

<math>~ c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, , </math>

<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>

<math>~=</math>

<math>~ - \biggl[ \nabla \cdot \vec{F} + \bold{P}_\mathrm{st}:\nabla{\vec{v}} + c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \biggr] \, , </math>

where, in this last expression, <math>~\bold{P}_\mathrm{st}</math> is the radiation stress tensor.

Various Manipulations

By combining the continuity equation with the

First Law of Thermodynamics

LSU Key.png

<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math>

we can write,

<math>~\rho T\frac{ds}{dt}</math>

<math>~=</math>

<math>~ \rho \frac{d\epsilon}{dt} - \frac{P}{\rho} \frac{d\rho}{dt} </math>

 

<math>~=</math>

<math>~ \rho \frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} \, . </math>

Given that the specific internal energy <math>~(\epsilon)</math> and the internal energy density <math>~(e)</math> are related via the expression, <math>~\epsilon = e/\rho</math>, we appreciate that the first of the above-identified energy-conservation-based dynamical equations is simply a restatement of the 1st Law of Thermodynamics in the context of a physical system whose fluid elements gain or lose entropy as a result of the (radiation-transport-related) source and sink terms,

<math>~\rho T \frac{ds}{dt}</math>

<math>~=</math>

<math>~c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, .</math>


By combining the left-hand side of the second of the above-identified energy-conservation-based dynamical equations with the continuity equation, then replacing the Lagrangian (that is, the material) time derivative by its Eulerian counterpart, the left-hand side can be rewritten as,

<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>

<math>~=</math>

<math>~ \frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho}~\frac{d\rho}{dt} </math>

 

<math>~=</math>

<math>~ \frac{dE_\mathrm{rad}}{dt} + E_\mathrm{rad}\nabla\cdot \vec{v} </math>

 

<math>~=</math>

<math>~ \frac{\partial E_\mathrm{rad}}{\partial t} + \vec{v}\cdot \nabla E_\mathrm{rad}+ E_\mathrm{rad}\nabla\cdot \vec{v} </math>

 

<math>~=</math>

<math>~ \frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) \, , </math>

which provides an alternate form of the expression, as found for example in equation (4) of Marcello & J. E. Tohline (2012).

Optically Thick Regime

In the optically thick regime, the following conditions hold:

<math>~c\kappa_E E_\mathrm{rad}</math>

<math>~\rightarrow</math>

<math>~4\pi \kappa_p B_p \, ,</math>

<math>~E_\mathrm{rad}</math>

<math>~\rightarrow</math>

<math>~aT^4 \, ,</math>

<math>~\biggl(\frac{\chi}{c}\biggr) \vec{F}</math>

<math>~\rightarrow</math>

<math>~- \nabla \biggl(\frac{aT^4}{3} \biggr) \, ,</math>

<math>~ \vec{\bold{P}}:\nabla{\vec{v}}</math>

<math>~\rightarrow</math>

<math>~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, .</math>

Related Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation