Difference between revisions of "User:Tohline/SR"

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(Type initial paragraphs into "Supplemental Relations" page)
 
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**adopt an equation of state, and
**adopt an equation of state, and
** specify initial conditions.
** specify initial conditions.
===Time-Dependent Problems===
====Equation of State====
The equation of state that generally will be adopted for ''time-dependent'' problems is one that describes an ideal gas.  As the accompanying discussion illustrates, the ideal gas equation of state can assume a variety of different forms.  Throughout this H_Book, we frequently will use "Form B" of the ideal gas equation of state to supplement the [http://www.vistrails.org/index.php/User:Tohline/PGE principal governing equations], namely,




<div align="center">
<div align="center">
<span id="PGE:Euler"><font color="#770000">'''Euler's Equation'''</font></span><br />
('''Momentum Conservation''')


{{User:Tohline/Math/EQ_Euler01}}
{{User:Tohline/Math/EQ_EOSideal02}}


</div>




{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Revision as of 04:17, 23 January 2010

Whitworth's (1981) Isothermal Free-Energy Surface
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Supplemental Relations

Apart from the independent variables <math>~t</math> and <math>~\vec{x}</math>, our principal governing equations involve the vector velocity <math>~\vec{v}</math>, and the four scalar variables, <math>~\Phi</math>, <math>~P</math>, <math>~\rho</math>, and <math>~\epsilon</math>. Because the variables outnumber the equations by one, one (additional) supplemental relationship between the physical vaiables must be specified in order to close the set of equations.

Also, in order to complete the unique specification of a particular physical problem, either a steady-state flow field or initial conditions must be specified, depending on whether one is studying a time-independent (structure) or time-dependent (stability or dynamics) problem, respectively. Throughout this H_Book, the following strategy will be adopted in order to complete the physical specification of each examined system:

  • For time-independent problems, we will ...
    • adopt a structural relationship between <math>~P</math> and <math>~\rho</math>, and
    • specify a steady-state flow-field.
  • For time-dependent problems, we will ...
    • adopt an equation of state, and
    • specify initial conditions.


Time-Dependent Problems

Equation of State

The equation of state that generally will be adopted for time-dependent problems is one that describes an ideal gas. As the accompanying discussion illustrates, the ideal gas equation of state can assume a variety of different forms. Throughout this H_Book, we frequently will use "Form B" of the ideal gas equation of state to supplement the principal governing equations, namely,


<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>


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