Radiation-Hydrodynamics

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

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

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

the,

Continuity Equation

<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{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 Realizations

First Law

By combining the continuity equation with the

First Law of Thermodynamics

<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 1^st 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>~c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, .</math>

Energy-Density of Radiation Field

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).

Thermodynamic Equilibrium

In an optically thick environment that is in thermodynamic equilibrium at temperature, <math>~T</math>, the energy-density of the radiation field is,

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

<math>~a_\mathrm{rad}T^4 \, ,</math>

and each fluid element will radiate — and, hence lose some of its internal energy to the surrounding radiation field — at a rate that is governed by the integrated Planck function,

<math>~B_p = \frac{\sigma}{\pi}T^4 </math>

<math>~\frac{ca_\mathrm{rad}}{4\pi} T^4 \, ,</math>

where, <math>~\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math>, is the Stefan-Boltzmann constant, and the radiation constant — which is included in an associated appendix among our list of key physical constants — is,

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

<math>~\equiv</math>

Also under these conditions, it can be shown that — see, for example, discussion associated with equations (12) and (18) in Marcello & J. E. Tohline (2012) —

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

<math>~\rightarrow</math>

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

and,

<math>~\rightarrow</math>

<math>~- \frac{1}{3}\biggl(\frac{c}{\chi}\biggr) \nabla E_\mathrm{rad} \, ,</math>

which implies,

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

<math>~\rightarrow</math>

<math>~-\nabla P_\mathrm{rad} \, ,</math>

where we have recognized that the radiation pressure,

<math>~P_\mathrm{rad} = \frac{1}{3}E_\mathrm{rad}</math>

<math>~\frac{1}{3}a_\mathrm{rad}T^4 \, .</math>

Hence, the modified Euler equation becomes,

<math>~ - \nabla (P+P_\mathrm{rad}) - \rho \nabla \Phi \, , </math>

and the equation governing the time-dependent behavior of <math>~E_\mathrm{rad}</math> becomes,

<math>~\frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) + \frac{1}{3}E_\mathrm{rad} \nabla \cdot \vec{v} </math>

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

Related Discussions

Euler equation viewed from a rotating frame of reference.
An earlier draft of this "Euler equation" presentation.

User:Tohline/Appendix/Ramblings/RadiationHydro

Contents

Radiation-Hydrodynamics

Governing Equations

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

Various Realizations

First Law

Energy-Density of Radiation Field

Thermodynamic Equilibrium

Optically Thick Regime

Related Discussions

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