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

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==Principal Governing Equations==
==Principal Governing Equations==
===Ignoring the Effects of Magnetic Fields===
===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)] — see also, [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,
<div align="center">
<div align="center">
<span id="Poisson"><font color="#770000">'''Poisson Equation'''</font></span>
<span id="Poisson"><font color="#770000">'''Poisson Equation'''</font></span>

Revision as of 16:02, 21 October 2018


Radiation-Hydrodynamics

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

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

<math>~=</math>

<math>~ - \frac{P}{\rho}\nabla \cdot \vec{v} + \frac{1}{\rho} \biggl[c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p\biggr] \, , </math>

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

<math>~=</math>

<math>~ - \frac{1}{\rho} \biggl[ \nabla \cdot \vec{F} + \vec{\bold{P}}:\nabla{\vec{v}} + c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \biggr] \, . </math>

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>



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Whitworth's (1981) Isothermal Free-Energy Surface

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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