User:Tohline/PGE/FirstLawOfThermodynamics - Revision history

Tohline: /* Second Elaboration */

2018-11-01T19:55:13Z

Second Elaboration

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

2018-10-31T22:01:13Z

Second Elaboration

← Older revision		Revision as of 22:01, 31 October 2018
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	====Second Elaboration====		====Second Elaboration====
	In an ''optically thick'' environment, <math>~E_\mathrm{rad} = a_\mathrm{rad}T^4</math>, so,		In an ''optically thick'' environment where local thermodynamic equilibrium is achieved, <math>~E_\mathrm{rad} = a_\mathrm{rad}T^4</math>, so,
	<div align="center">		<div align="center">
	<math>~\Gamma = c\kappa_E a_\mathrm{rad}T^4 = \biggl( \frac{\kappa_E}{\kappa_p} \biggr) \Lambda \, .</math>		<math>~\Gamma = c\kappa_E a_\mathrm{rad}T^4 = \biggl( \frac{\kappa_E}{\kappa_p} \biggr) \Lambda \, .</math>
	</div>		</div>
			In such an environment, we also expect <math>~\kappa_E \leftrightarrow \kappa_p</math>, so the heating and cooling terms will cancel each other and, as a result, will disappear from the ''separate'' expressions for <math>~ds_\mathrm{fluid}/dt</math> and <math>~ds_\mathrm{rad}/dt</math>.


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Tohline: /* Example C */

2018-10-31T21:53:54Z

Example C

← Older revision		Revision as of 21:53, 31 October 2018
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	</div>		</div>

	When, for example, the fluid (gas) is exposed to a radiation ~~field — see descriptions of the numerical algorithms designed to simulate radiation-hydrodynamic flows presented by Hayes et al. (2001) or by Marcello & Tohline (2012) —~~ heating of the fluid by the radiation is handled by setting,		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">		<div align="center">
	<math>~\Gamma = c\kappa_E E_\mathrm{rad} \, ,</math>		<math>~\Gamma = c\kappa_E E_\mathrm{rad} \, ,</math>
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	and the fluid cools — returning energy to the radiation field — according to the reciprocating expression,		and the fluid cools — returning energy to the radiation field — according to the reciprocating expression,
	<div align="center">		<div align="center">
	<math>~\Lambda = 4\pi \kappa_p B_p \, .</math>		<math>~\Lambda = 4\pi \kappa_p B_p = 4 \kappa_p \sigma T^4 \, ,</math>
	</div>		</div>
	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."		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 ~~key~~ to ~~our present~~ point ~~of interest~~, 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 & J. E. Tohline (2012)].)		[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 & J. E. Tohline (2012)].]

	====First Elaboration====		====First Elaboration====
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	In an ''optically thick'' environment, <math>~E_\mathrm{rad} = a_\mathrm{rad}T^4</math>, so,		In an ''optically thick'' environment, <math>~E_\mathrm{rad} = a_\mathrm{rad}T^4</math>, so,
	<div align="center">		<div align="center">
	<math>~\Gamma = c\kappa_E a_\mathrm{rad}T^4 \, .</math>		<math>~\Gamma = c\kappa_E a_\mathrm{rad}T^4 = \biggl( \frac{\kappa_E}{\kappa_p} \biggr) \Lambda \, .</math>
	</div>		</div>

Tohline: /* Example C */

2018-10-31T21:25:56Z

Example C

← Older revision		Revision as of 21:25, 31 October 2018
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	====First Elaboration====		====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> — see, for example, "Example B" above — 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, <math>~E_\mathrm{rad} = a_\mathrm{rad}T^4</math>, so,
			<div align="center">
			<math>~\Gamma = c\kappa_E a_\mathrm{rad}T^4 \, .</math>
			</div>



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Tohline: /* Right-Hand Side (RHS) */

2018-10-31T21:04:45Z

Right-Hand Side (RHS)

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	==Right-Hand Side (RHS)==		==Right-Hand Side (RHS)==

	===Example 1===		===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,		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">		<div align="center">
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	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> — see his equations (3.19) and (3.20) on p. 28 — 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 — see on p. 23 immediately following his equation (2.36) — 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>"		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> — see his equations (3.19) and (3.20) on p. 28 — 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 — see on p. 23 immediately following his equation (2.36) — 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 2===		===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,		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">		<div align="center">
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	</td></tr></table>		</td></tr></table>

	===Example 3===		===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 1" right-hand-side expression, then, we would expect to see,		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">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>
	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 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 key to our present point of interest, 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 & J. E. Tohline (2012)].)

			====First Elaboration====


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Tohline: /* Right-Hand Side (RHS) */

2018-10-31T20:22:31Z

Right-Hand Side (RHS)

← Older revision		Revision as of 20:22, 31 October 2018
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	===Example 3===		===Example 3===
	In astrophysical discussions of the time-rate-of-change of the fluid entropy, it is not unusual to include ~~on the right-hand-side~~ 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 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 1" 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 a radiation field — see descriptions of the numerical algorithms designed to simulate radiation-hydrodynamic flows presented by Hayes et al. (2001) or by Marcello & Tohline (2012) — 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 — returning energy to the radiation field — according to the reciprocating expression,
			<div align="center">
			<math>~\Lambda = 4\pi \kappa_p B_p \, .</math>
			</div>
			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."


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Tohline: /* Right-Hand Side (RHS) */

2018-10-31T19:20:45Z

Right-Hand Side (RHS)

← Older revision		Revision as of 19:20, 31 October 2018
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	==Right-Hand Side (RHS)==		==Right-Hand Side (RHS)==

			===Example 1===
	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,		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">		<div align="center">
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	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> — see his equations (3.19) and (3.20) on p. 28 — 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 — see on p. 23 immediately following his equation (2.36) — 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>"		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> — see his equations (3.19) and (3.20) on p. 28 — 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 — see on p. 23 immediately following his equation (2.36) — 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 2===
	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,		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">		<div align="center">
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	<tr><td align="left"><sup>&dagger;</sup>Text in a green font has been taken directly from Volume I, §2, p. 17 of [<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>].		<tr><td align="left"><sup>&dagger;</sup>Text in a green font has been taken directly from Volume I, §2, p. 17 of [<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>].
	</td></tr></table>		</td></tr></table>

			===Example 3===
			In astrophysical discussions of the time-rate-of-change of the fluid entropy, it is not unusual to include on the right-hand-side 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.


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Tohline: /* Right-Hand Side (RHS) */

2018-10-31T18:37:53Z

Right-Hand Side (RHS)

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	</tr>		</tr>
	</table>		</table>
	[<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>], §3.4, p. 56, Eq. (65)		[<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>], §3.4, p. 56, Eq. (65)<br />
			[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], Vol. II, §4, p. 53, Eq. (4.40)
	</div>		</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>.)		(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>.)
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], §2, p. 17, Eq. (2.17)		[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], Vol. I, §2, p. 17, Eq. (2.17)
	</td>		</td>
	<td align="left" colspan="2">and    [<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>], §3.4, p. 57, Eq. (67)		<td align="left" colspan="2">and    [<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>], §3.4, p. 57, Eq. (67)
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	<table border="1" cellpadding="15" align="center" width="80%"><tr><td align="left">		<table border="1" cellpadding="15" align="center" width="80%"><tr><td align="left">
	Alternatively,<sup>&dagger;</sup> "<font color="#007700">… 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 <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>		Alternatively,<sup>&dagger;</sup> "<font color="#007700">… 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">		<table border="0" cellpadding="5" align="center">

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	<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>"		<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>		</td></tr>
	<tr><td align="left"><sup>&dagger;</sup>Text in a green font has been taken directly from §2, p. 17 of [<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>].		<tr><td align="left"><sup>&dagger;</sup>Text in a green font has been taken directly from Volume I, §2, p. 17 of [<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>].
	</td></tr></table>		</td></tr></table>

Tohline: /* Nonadiabatic Environments */

2018-10-31T16:48:20Z

Nonadiabatic Environments

← Older revision		Revision as of 16:48, 31 October 2018
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	=Nonadiabatic Environments=		=Nonadiabatic Environments=

	~~==Fluid==~~		==Left-Hand Side (LHS)==

	===Left-Hand Side (LHS)===
	There are several potentially useful expressions for the time-rate of change of fluid entropy.		There are several potentially useful expressions for the time-rate of change of fluid entropy.
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	</table>		</table>

	===Right-Hand Side (RHS)===		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)==

	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,		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,
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\rho T \frac{ds}{dt}</math>		<math>~\rho T \frac{ds_\mathrm{fluid}}{dt}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	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> — see his equations (3.19) and (3.20) on p. 28 — 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 — see on p. 23 immediately following his equation (2.36) — 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>"		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> — see his equations (3.19) and (3.20) on p. 28 — 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 — see on p. 23 immediately following his equation (2.36) — 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>"

	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 ~~arises~~ in discussions of stellar structure and evolution. Specifically, on p. 56, his equation (65) states,		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">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\rho T \frac{ds}{dt}</math>		<math>~\rho T \frac{ds_\mathrm{tot}}{dt} = \rho T \frac{d}{dt}\biggl( s_\mathrm{fluid} + s_\mathrm{rad} \biggr)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td></tr></table>		</td></tr></table>

	~~==Radiation==~~

	<br />		<br />
	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Right-Hand Side (RHS) */

2018-10-31T15:49:25Z

Right-Hand Side (RHS)

← Older revision		Revision as of 15:49, 31 October 2018
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	</div>		</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> — see his equations (3.19) and (3.20) on p. 28 — he implicitly zeroes out the 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 — see on p. 23 immediately following his equation (2.36) — 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>"		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> — see his equations (3.19) and (3.20) on p. 28 — 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 — see on p. 23 immediately following his equation (2.36) — 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>"

	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 arises in discussions of stellar structure and evolution. Specifically, on p. 56, his equation (65) states,		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 arises in discussions of stellar structure and evolution. Specifically, on p. 56, his equation (65) states,
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	[<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>], §3.4, p. 57, Eq. (68)		[<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>], §3.4, p. 57, Eq. (68)
	</div>		</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 ~~by [<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>]~~ 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 ("Fourier's law") — whether by heat conduction (electrons) or radiative diffusion (photons).		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]") — whether by heat conduction (electrons) or radiative diffusion (photons).


	<table border="1" cellpadding="15" align="center" width="80%"><tr><td align="left">		<table border="1" cellpadding="15" align="center" width="80%"><tr><td align="left">
	~~On the other hand~~,<sup>&dagger;</sup> "<font color="#007700">… 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 <b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>] <font color="#007700">has the general form for diffusive fluxes (Fick's law):</font>		Alternatively,<sup>&dagger;</sup> "<font color="#007700">… 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 <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">		<table border="0" cellpadding="5" align="center">