Editing PGE/FirstLawOfThermodynamics (section)

==Right-Hand Side (RHS)==

===Example A===
One physically reasonable pair of sources/sinks of entropy in the fluid arise in the context of what [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>] identify as the ''general equation of heat transfer'', namely,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>\rho T \frac{ds_\mathrm{fluid}}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \nabla\cdot \vec{F}_\mathrm{cond} + \Psi \, .
</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], &sect;49, p. 185, Eq. (49.4)<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, &sect;3, p. 30, Eq. (3.26)<br />
[<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. I, &sect;8.4, p. 369, Eq. (8.35)
</div>

In this expression, 
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  <td align="right">
<math>\vec{F}_\mathrm{cond}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\mathcal{K}_\mathrm{cond} \nabla T \, ,</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, &sect;3, p. 28, Eq. (3.19)
</div>
where, <span title="Coefficient of thermal conductivity"><math>\mathcal{K}_\mathrm{cond}</math></span> is the coefficient of ''thermal'' conductivity;
and the ''rate of viscous dissipation'',
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>\Psi</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\pi_{ik} \frac{\partial v_i}{\partial x_k} \, ,
</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, &sect;3, p. 29, following Eq. (3.25)
</div>

where <span title="Viscous stress tensor"><math>\pi_{ik}</math></span> is the "viscous stress tensor," as defined, for example: by equation (15.3) on p. 48 of [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>]; by equation (44) on p. 52 of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]; and by equation (8.34) on p. 369 of [<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>].  Note that when [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] defines <span title="Viscous stress tensor"><math>\pi_{ik}</math></span> &#8212; see his equations (3.19) and (3.20) on p. 28  &#8212; he implicitly zeroes out the coefficient of bulk viscosity component, keeping only the shear viscosity component because it is the piece that is usually of interest in astrophysical discussions.  [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] goes on to explain &#8212; see on p. 23 immediately following his equation (2.36) &#8212; together, the pair of terms on the right-hand-side express the "<font color="#007700">time rate of adding heat (through heat conduction and the viscous conversion of ordered energy in differential fluid motions to disordered energy in random particle motions).</font>"

===Example B===
In addition to the pair of source/sink terms that arise from the ''general equation of heat transfer'', [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] includes another pair of terms that often arise in discussions of stellar structure and evolution.  Specifically, on p. 56, his equation (65) states,
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  <td align="right">
<math>\rho T \frac{ds_\mathrm{tot}}{dt} = \rho T \frac{d}{dt}\biggl( s_\mathrm{fluid} + s_\mathrm{rad} \biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Psi - \nabla\cdot \vec{F}_\mathrm{cond} + \rho \epsilon_\mathrm{nuc} - \nabla \cdot \vec{F}_\mathrm{rad} \, .
</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 56, Eq. (65)<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, &sect;4, p. 53, Eq. (4.40)
</div>
(Note, that [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] uses the variable notation <math>~\Phi_v</math> in place of <span title="Rate of viscous dissipation"><math>\Psi</math></span>.)  
In this expression, <math>\epsilon_\mathrm{nuc}(\rho,T)</math> expresses the rate at which (specific) energy is released via thermonuclear reactions, and
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>\vec{F}_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{c}{3\rho\kappa_R} \nabla (a_\mathrm{rad}T^4) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>-\chi_\mathrm{rad} \nabla T \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. I, &sect;2, p. 17, Eq. (2.17)
  </td>
  <td align="left" colspan="2">and &nbsp; &nbsp;[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (67)
  </td>
</tr>
</table>
</div>
where [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] refers to 
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  <td align="right">
<math>\chi_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{4c a_\mathrm{rad} T^3}{3\kappa \rho} \, ,
</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (68)
</div>
as the coefficient of ''radiative'' conductivity.  The expression for the radiation flux, <math>\vec{F}_\mathrm{rad}</math>, presented by [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] is identical ''in form'' to the expression presented above for the flux due to heat conduction, <math>\vec{F}_\mathrm{cond}</math>.  This highlights the similarities between the manner in which nature handles transport processes ("[https://en.wikipedia.org/wiki/Thermal_conduction#Fourier's_law Fourier's law]") &#8212; whether by heat conduction (electrons) or radiative diffusion (photons).  


<table border="1" cellpadding="15" align="center" width="80%"><tr><td align="left">
Alternatively,<sup>&dagger;</sup> "<font color="#007700">&hellip; recognizing <math>aT^4</math> as the energy density of blackbody radiation, we see that</font> [the expression for <math>\vec{F}_\mathrm{rad}</math> that appears as equation (2.17) in Volume I of <b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] <font color="#007700">has the general form for diffusive fluxes ([https://en.wikipedia.org/wiki/Fick's_laws_of_diffusion#Fick's_first_law Fick's law]):</font> 
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  <td align="right">
diffusive flux
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \mathcal{D} \nabla</math>(density of quantity being diffused),
  </td>
</tr>
</table>
<font color="#007700">where <math>\mathcal{D}</math> is the diffusivity.  Indeed, this comparison allows us to identify the radiative diffusivity as having the characteristic formula,</font>
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<math>~\mathcal{D}_\mathrm{rad} = \frac{1}{3} c \ell \, ,</math>
</div>
<font color="#007700">where <math>~\ell \equiv 1/\rho\kappa_R</math> is the (Rosseland) mean-free path of the diffusing particles (photons).  A 'random walk' slows down the free-flight speed {{ Template:Math/C_SpeedOfLight }} by a typical factor of <math>\ell/R_\odot</math>, so that the time <math>R_\odot^2/\mathcal{D}_\mathrm{rad}</math> for photons to diffuse to the surface of the Sun is roughly <math>3R_\odot/\ell</math> times longer than the free-flight time <math>R_\odot/c</math> of 2 s.  This process prevents the Sun from releasing its considerable internal reservoir of photons in one powerful blast, but instead regulates it to the stately observed luminosity of <math>L_\odot = 3.86 \times 10^{33}</math> erg s<sup>-1</sup>.</font>"
</td></tr>
<tr><td align="left"><sup>&dagger;</sup>Text in a green font has been taken directly from Volume I, &sect;2, p. 17 of [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>].
</td></tr></table>

===Example C===
In astrophysical discussions of the time-rate-of-change of the fluid entropy, it is not unusual to include a scalar function, <math>\Gamma</math>, that accounts in a generic manner for ''volumetric gains'' of energy due to local sources, and another scalar function, <math>\Lambda</math>, that accounts in a generic manner for ''volumetric loses'' of energy due to local sinks.  In place of the above "Example A" right-hand-side expression, then, we would expect to see,
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<tr>
  <td align="right">
<math>~\rho T \frac{ds_\mathrm{fluid}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \nabla\cdot \vec{F}_\mathrm{cond} + \Psi + \Gamma - \Lambda \, .
</math>
  </td>
</tr>
</table>
</div>

When, for example, the fluid (gas) is exposed to photon radiation, heating of the fluid by the radiation is handled by setting,
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<math>~\Gamma = c\kappa_E E_\mathrm{rad} \, ,</math>
</div>
and the fluid cools &#8212; returning energy to the radiation field &#8212; according to the reciprocating expression,
<div align="center">
<math>~\Lambda = 4\pi \kappa_p B_p = 4 \kappa_p \sigma T^4 \, ,</math>
</div>
where, <math>\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math> is the Stefan-Boltzmann constant.  In such a case, the right-hand-side of the equation describing the corresponding time-rate-of-change of the entropy of the ''radiation'' field, <math>s_\mathrm{rad}</math>, would necessarily contain the same two terms, but in both cases with ''opposite signs.''  That is, the entropy of the radiation field sees <math>\Lambda</math> as a ''source'' while it sees <math>\Gamma</math> as a "sink."

[In addition to <math>~\Gamma</math> and <math>~\Lambda</math>, other terms involving spatial variations in the velocity field and in the radiation energy density also appear on the right-hand-side of the expression for <math>~ds_\mathrm{rad}/dt</math>.  For simplicity, and because these other terms are not relevant to the principal point we are making, we have opted not to detail the entire expression for <math>~ds_\mathrm{rad}/dt</math> here.  The additional terms and details can be found in, for example, {{ ZEUS-MP2006 }} or {{ MT2012 }}.]

====First Elaboration====

When the expressions for <math>ds_\mathrm{fluid}/dt</math> and <math>ds_\mathrm{rad}/dt</math> are added together to obtain a prescription for the time-rate-of-change of <math>s_\mathrm{tot}</math> &#8212; see, for example, "Example B" above &#8212; neither of the functions, <math>\Gamma</math> or <math>\Lambda</math>, will appear explicitly because they have opposite signs in the two separate expressions.  This will be the case whether the environment is ''optically thin'' or ''optically thick.''

====Second Elaboration====
In an ''optically thick'' environment where local thermodynamic equilibrium has been achieved, <math>E_\mathrm{rad} = a_\mathrm{rad}T^4</math>, so,
<div align="center">
<math>~\Gamma = c\kappa_E a_\mathrm{rad}T^4 = \biggl( \frac{\kappa_E}{\kappa_p} \biggr) \Lambda \, .</math>
</div>
In such an environment, we also expect <math>~\kappa_E \leftrightarrow \kappa_p</math>, so the heating and cooling terms will cancel out each other.  As a result, the quantity <math>~(\Lambda - \Gamma) </math> will disappear from the ''separate'' expressions for <math>~ds_\mathrm{fluid}/dt</math> and <math>~ds_\mathrm{rad}/dt</math>.


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