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=First Law of Thermodynamics=

{{LSU_HBook_header}}

==Standard Presentation==
Following the detailed discussion of the laws of thermodynamics that can be found, for example, in Chapter I of [<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>] we know that, for an infinitesimal quasi-statical change of state, the change <math>~dQ</math> in the total heat content <math>~Q</math> of a fluid element is given by the,

<div align="center">
<span id="FundamentalLaw"><font color="#770000">'''Fundamental Law of Thermodynamics'''</font></span>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dQ</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
d\epsilon + PdV \, ,
</math>
  </td>
</tr>
</table>
[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (2)<br />
[<b>[[User:Tohline/Appendix/References#CH87|<font color="red">H87</font>]]</b>], &sect;1.2, Eq. (1.2)<br />
[<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;4.1, Eq. (4.1)<br />
[<b>[[User:Tohline/Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.2, Eq. (1.10)<br />
[<b>[[User:Tohline/Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;1.6.5, Eq. (1.124)
</div>
where, {{ User:Tohline/Math/VAR_SpecificInternalEnergy01 }} is the specific internal energy, {{ User:Tohline/Math/VAR_Pressure01 }} is the pressure, and {{ User:Tohline/Math/VAR_SpecificVolume01 }}<math>~= 1/</math>{{ User:Tohline/Math/VAR_Density01 }} is the specific volume of the fluid element.  Generally, the change in the total heat content can be rewritten in terms of the gas temperature, {{ User:Tohline/Math/VAR_Temperature01 }}, and the specific entropy of the fluid, {{ User:Tohline/Math/VAR_SpecificEntropy01 }}, via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dQ</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~T ds \, .</math>
  </td>
</tr>
</table>
[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter I, Eq. (76) &amp; Chapter II, Eq. (44)<br />
[<b>[[User:Tohline/Appendix/References#CH87|<font color="red">H87</font>]]</b>], &sect;1.4, p. 16<br />
[<b>[[User:Tohline/Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.2, Eq. (1.10)
</div>


If, in addition, it is understood that the specified changes are occurring over an interval of time <math>~dt</math>, then from this pair of expressions we derive what will henceforth be referred to as the,

<div align="center">
<span id="Standard Form"><font color="#770000">'''Standard Form'''</font></span><br />
of the First Law of Thermodyamics

{{ User:Tohline/Math/EQ_FirstLaw01 }}

[<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, Eq. (64)<br />
[<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Chapter 4, Eq. (4.27)<br />
[<b>[[User:Tohline/Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;7.3.3, Eq. (7.162)
</div>

If the state changes occur in such a way that no heat seeps into or leaks out of the fluid element, then <math>~ds/dt = 0</math> and the changes are said to have been made ''adiabatically.''  For an adiabatically evolving system, therefore, the ''First Law'' assumes what henceforth will be referred to as the,

<div align="center">
<span id="Standard Form"><font color="#770000">'''Adiabatic Form'''</font></span><br />
of the First Law of Thermodyamics

{{ User:Tohline/Math/EQ_FirstLaw02 }}

[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (13)<br />
[<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, Eq. (70)
</div>
Clearly this form of the ''First Law'' also may be viewed as a statement of ''specific entropy conservation.''

==Entropy Tracer==

===Initial Recognition===
Multiplying the ''Adiabatic Form of the First Law of Thermodynamics'' through by {{ User:Tohline/Math/VAR_Density01 }} and rearranging terms, we find that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d(\rho\epsilon)}{dt} - \epsilon \frac{d\rho}{dt} -  \frac{P}{\rho} \frac{d\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon) \frac{1}{\rho}\frac{d\rho}{dt} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon)\frac{d\ln\rho}{dt} \, ,
</math>
  </td>
</tr>
</table>
is an equally valid statement of the conservation of specific entropy in an adiabatic flow.  In combination, first, with
 
<div align="center">
<span id="IdealGasB"><font color="#770000">'''Form B'''</font></span><br />
of the Ideal Gas Equation

{{ User:Tohline/Math/EQ_EOSideal02 }}
</div>

and, second, with the
<div align="center">
<span id="Continuity"><font color="#770000">'''Lagrangian Form'''</font></span><br />
of the Continuity Equation

{{ User:Tohline/Math/EQ_Continuity01 }}
</div>

we may furthermore rewrite this expression as,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d(\rho\epsilon)}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\gamma_g (\rho\epsilon)\frac{d\ln\rho}{dt} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{\gamma_g} \frac{d\ln(\rho\epsilon)}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d\ln\rho}{dt} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \nabla\cdot\vec{v} \, . 
</math>
  </td>
</tr>
</table>

This relation has the classic form of a conservation law.  It certifies that, within the context of adiabatic flows, the ''entropy tracer,''

<div align="center">
<math>~\tau \equiv (\rho\epsilon)^{1/\gamma_g} = \biggl[ \frac{P}{(\gamma_g - 1)} \biggr]^{1/\gamma_g} \, ,</math>
</div>

is the volume density of a conserved quantity.  In this case, that conserved quantity is the entropy of each fluid element.  

===Substantiation===
To further substantiate this claim, we note that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\tau}{\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\epsilon^{1/\gamma_g} \cdot \rho^{1/\gamma_g - 1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \ln\biggl(\frac{\tau}{\rho}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\gamma_g} \biggl[ \ln\epsilon - ( \gamma_g-1)\ln\rho \biggr] \, .
</math>
  </td>
</tr>
</table>

Now, from the first law, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ds</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{T} \biggl[
d\epsilon - \frac{P}{\rho} {d\ln\rho}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
c_V~ d\ln\epsilon - \frac{\Re}{\mu} ~{d\ln\rho}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~ \frac{ds}{c_P}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{c_V}{c_P}~ d\ln\epsilon - \frac{\Re/\mu}{c_P} ~{d\ln\rho}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\gamma_g} \biggl[  d\ln\epsilon - (\gamma_g-1){d\ln\rho} \biggr] \, ,
</math>
  </td>
</tr>
</table>
which, upon integration, gives,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{s}{c_P}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\gamma_g} \biggl[  \ln\epsilon - (\gamma_g-1)\ln\rho \biggr] + \mathrm{constant} \, .
</math>
  </td>
</tr>
</table>

To within an additive constant, this is precisely the expression for the logarithm of the entropy tracer, as provided immediately above.  Hence, we see that,
<div align="center">
<math>~s = c_P \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant}  \, ,</math>
</div>
that is, we see that the variable, <math>~\tau</math>, traces the fluid entropy just as <math>~\rho</math> traces the fluid mass.

We have found one other instance in the literature &#8212; although there are undoubtedly others &#8212; where the role of this ''entropy tracer'' previously has been identified.  In chapter IX of [<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>] we find that, "apart from an unimportant additive constant," the specific entropy is,
<div align="center" id="LL75">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_P \ln \biggl(\frac{P^{1/\gamma_g}}{\rho} \biggr) \, .</math>
  </td>
</tr>
</table>
[<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], &sect;80, Eq. (80.12)
</div>

Given that <math>~\tau \propto P^{1/\gamma_g}</math>, this is clearly the same expression as we have derived for the specific entropy of the fluid.

===Incorporation Into the First Law===
Multiplying the ''Standard Form of the First Law of Thermodynamics'' through by {{ User:Tohline/Math/VAR_Density01 }}, we can now write,
<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>~
\frac{d(\rho\epsilon)}{dt} - \gamma_g (\rho\epsilon) ~\frac{d\ln\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{\rho T}{\gamma_g(\rho\epsilon)} ~\frac{ds}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt} - \frac{d\ln\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 
\frac{1}{c_P} ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d\ln(\tau/\rho)}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 
\frac{1}{c_P}\biggl( \frac{\tau}{\rho} \biggr) ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d(\tau/\rho)}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho} \biggl[ \frac{d\tau}{dt} - \frac{\tau}{\rho}\frac{d\rho}{dt}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho} \biggl[ \frac{d\tau}{dt} + \tau \nabla\cdot\vec{v} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 
\biggl( \frac{\tau}{c_P} \biggr) ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) 
</math>
  </td>
</tr>
</table>

Now,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_V \gamma_g</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{c_V}{\rho\epsilon}\biggr) \gamma_g \tau^{\gamma_g}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{1}{\rho T}\biggr) \gamma_g \tau^{\gamma_g} \, .</math>
  </td>
</tr>
</table>
</div>

Hence,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{1}{\gamma_g \tau^{\gamma_g-1}} \biggr) ~\rho T~\frac{ds}{dt} \, .
</math>
  </td>
</tr>
</table>

[http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello &amp; Tohline (2012)], &sect;2.2, Eq. (31)
</div>


Notice, as well, that we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 
\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_P~
\frac{d\ln(\tau/\rho)}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_V \biggl[
\frac{d\ln(\tau/\rho)^\gamma}{dt}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_V \frac{d}{dt}\biggl[
\ln\biggl( \frac{\rho\epsilon}{\rho^{\gamma_g}}\biggr)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~
\rho T ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho\epsilon ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{\rho\epsilon}{\rho^{\gamma_g}}\biggr)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~
\frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr)
\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho T ~\frac{ds}{dt} \, .
</math>
  </td>
</tr>
</table>

Specifically for the case, <math>~\gamma_g = \tfrac{5}{3}</math>, this gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 
\frac{3}{2} ~P ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{P}{\rho^{5/3}}\biggr)
\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho T ~\frac{ds}{dt} \, .
</math>
  </td>
</tr>
</table>

[<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Chapter 9, Eq. (9.26)
</div>
It is fair to say, therefore, that in this specific case [<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] also recognized the relevance of and the conservative nature of, what we have referred to as, the ''entropy tracer.''

=Nonadiabatic Environments=

==Left-Hand Side (LHS)==
There are several potentially useful expressions for the time-rate of change of fluid entropy.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho T \frac{ds_\mathrm{fluid}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\frac{d\epsilon}{dt} + P\nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\gamma_g \tau^{\gamma_g-1}\biggl[ \frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\epsilon \biggl[
\frac{d\ln(\tau/\rho)^{\gamma_g}}{dt}
\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr)
\biggr] \, .
</math>
  </td>
</tr>
</table>

Clearly to within an additive constant, an expression for the fluid entropy, itself, is

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ c_P \ln\biggl( \frac{\tau}{\rho} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_V \ln \biggl(\frac{P}{\rho^{\gamma_g}} \biggr) \, .</math>
  </td>
</tr>
</table>

In [[User:Tohline/Appendix/Ramblings/RadiationHydro#Optically_Thick_Regime|optically thick environments]] where the radiation field is intermixed and in equilibrium with the fluid (gas), the time-rate-of-change in the entropy of the radiation field is characterized by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho T \frac{ds_\mathrm{rad}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\frac{d}{dt}\biggl(\frac{E_\mathrm{rad}}{\rho}\biggr) + \rho P_\mathrm{rad} \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4a_\mathrm{rad} \rho T}{3} \frac{d}{dt} \biggl( \frac{T^3}{\rho}\biggr)
</math>
  </td>
</tr>
</table>

Hence, to within an additive constant, an expression for the entropy of the radiation field is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~s_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{4 a_\mathrm{rad} T^3}{3\rho} \, .</math>
  </td>
</table>

==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>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>] identify as the ''general equation of heat transfer'', namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<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 \, .
</math>
  </td>
</tr>
</table>
[<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], &sect;49, p. 185, Eq. (49.4)<br />
[<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, &sect;3, p. 30, Eq. (3.26)<br />
[<b>[[User:Tohline/Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. I, &sect;8.4, p. 369, Eq. (8.35)
</div>

In this expression, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, &sect;3, p. 28, Eq. (3.19)
</div>
where, <math>~\mathcal{K}_\mathrm{cond}</math> is the coefficient of ''thermal'' conductivity;
and the ''rate of viscous dissipation'',
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, &sect;3, p. 29, following Eq. (3.25)
</div>

where <math>~\pi_{ik}</math> is the "viscous stress tensor," as defined, for example: by equation (15.3) on p. 48 of [<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>]; by equation (44) on p. 52 of [<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>]; and by equation (8.34) on p. 369 of [<b>[[User:Tohline/Appendix/References#P00|<font color="red">P00</font>]]</b>].  Note that when [<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] defines <math>~\pi_{ik}</math> &#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>[[User:Tohline/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>[[User:Tohline/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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 56, Eq. (65)<br />
[<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, &sect;4, p. 53, Eq. (4.40)
</div>
(Note, that [<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>] uses the variable notation <math>~\Phi_v</math> in place of <math>~\Psi</math>.)  
In this expression, <math>~\epsilon_\mathrm{nuc}(\rho,T)</math> expresses the rate at which (specific) energy is released via thermonuclear reactions, and
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>[[User:Tohline/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>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (67)
  </td>
</tr>
</table>
</div>
where [<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>] refers to 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>[[User:Tohline/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>[[User:Tohline/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>[[User:Tohline/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> 
<table border="0" cellpadding="5" align="center">

<tr>
  <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>
<div align="center">
<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 <math>~c</math> 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 surfacce 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>[[User:Tohline/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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<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,
<div align="center">
<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,  [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006)] or [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello &amp; J. E. Tohline (2012)].]

====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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