User:Tohline/Appendix/Ramblings/RadiationHydro
Radiation-Hydrodynamics
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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,
the,
Continuity Equation
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<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,
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<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:
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<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> |
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<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.
By combining it with the continuity equation and switching to a material derivative notation, the left-hand side of this last expression can be rewritten in a form that matches equation (4) of Marcello & J. E. Tohline (2012), namely,
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<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math> |
<math>~=</math> |
<math>~ \frac{dE_\mathrm{rad}}{dt} - E_\mathrm{rad}~\frac{d\rho}{dt} </math> |
Optically Thick Regime
In the optically thick regime, the following conditions hold:
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<math>~c\kappa_E E_\mathrm{rad}</math> |
<math>~\rightarrow</math> |
<math>~4\pi \kappa_p B_p \, ,</math> |
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<math>~E_\mathrm{rad}</math> |
<math>~\rightarrow</math> |
<math>~aT^4 \, ,</math> |
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<math>~\biggl(\frac{\chi}{c}\biggr) \vec{F}</math> |
<math>~\rightarrow</math> |
<math>~- \nabla \biggl(\frac{aT^4}{3} \biggr) \, ,</math> |
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<math>~ \vec{\bold{P}}:\nabla{\vec{v}}</math> |
<math>~\rightarrow</math> |
<math>~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, .</math> |
Related Discussions
- Euler equation viewed from a rotating frame of reference.
- An earlier draft of this "Euler equation" presentation.
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© 2014 - 2021 by Joel E. Tohline |