Difference between revisions of "User:Tohline/PGE/FirstLawOfThermodynamics"

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==Overview==
Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible &#8212; specifically, polytropic and isothermal &#8212; fluids as presented in a series of papers by Jeremiah P. Ostriker:
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. Ostriker (1964, ApJ, 140, 1056)] &#8212; ''The Equilibrium of Polytropic and Isothermal Cylinders''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. Ostriker (1964, ApJ, 140, 1067)] &#8212; ''The Equilibrium of Self-Gravitating Rings''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1529O J. Ostriker (1964, ApJ, 140, 1529)] &#8212; ''On the Oscillations and the Stability of a Homogeneous Compressible Cylinder''
* [http://adsabs.harvard.edu/abs/1965ApJS...11..167O J. Ostriker (1965, ApJ Supplements, 11, 167)] &#8212; ''Cylindrical Emden and Associated Functions''
I believe that much, if not all, of this material was drawn from Ostriker's doctoral dissertation research at the University of Chicago (and Yerkes Observatory) under the guidance of [https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar S. Chandrasekhar].


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, "<font color="#007700">for an infinitesimal quasi-statical change of state,</font>" the change <math>~dQ</math> in the total heat content <math>~Q</math> of a fluid element is given by, what we will label as the,


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


<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)
</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.  Now, if it is understood that the specified changes are occurring over a certain interval of time <math>~dt</math>, then from this expression we can directly derive what will henceforth be referred to as,


&nbsp;<br />
&nbsp;<br />
{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Revision as of 20:54, 23 October 2018

First Law of Thermodynamics

Whitworth's (1981) Isothermal Free-Energy Surface
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Following the detailed discussion of the laws of thermodynamics that can be found, for example in Chapter I of [C67] 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, what we will label as the,

Standard Form
of the First Law of Thermodyamics

<math>~dQ</math>

<math>~=</math>

<math>~ d\epsilon + PdV \, , </math>

[C67], Chapter II, Eq. (2)

where, <math>~\epsilon</math> is the specific internal energy, <math>~P</math> is the pressure, and <math>~V</math><math>~= 1/</math><math>~\rho</math> is the specific volume of the fluid element. Now, if it is understood that the specified changes are occurring over a certain interval of time <math>~dt</math>, then from this expression we can directly derive what will henceforth be referred to as,

 

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation

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