Difference between revisions of "User:Tohline/PGE"

From VistrailsWiki
Jump to navigation Jump to search
(Transfering opening text from old H_Book)
(Principal equations inserted and spaced apart on the page)
Line 5: Line 5:
==Principal Governing Equations==
==Principal Governing Equations==


According to Landau and Lifshitz's (1975) eloquent discussion of the broad subject of ''Fluid Mechanics'', <FONT COLOR="#007700">the state of a moving fluid is determined by five quantities:  the three components of the velocity</FONT> '''<math>\vec{v}</math>''' <FONT COLOR="#007700">and, for example, the pressure</FONT> <math>P</math> <FONT COLOR="#007700">and the density</FONT> <math> \rho </math> <FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential <math> \Phi </math>. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>
According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">the state of a moving fluid is determined by five quantities:  the three components of the velocity</FONT> <math>\vec{v}</math> <FONT COLOR="#007700">and, for example, the pressure</FONT> <math>P</math> <FONT COLOR="#007700">and the density</FONT> <math> \rho </math> <FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential <math> \Phi </math>. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>


<div align="center">
<div align="center">
<math>x \implies y</math>
<font color="#770000">'''Euler's Equation'''</font><br>
('''Momentum Conservation''')


<math>\frac{x^3}{5+6}</math>
<math>\frac{D\vec{v}}{Dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>


<math>\int_{1}^{3}\frac{e^3/x}{x^2}\, dx</math>
 
<font color="#770000">'''Equation of Continuity'''</font><br>
('''Mass Conservation''')
 
<math>\frac{D\vec{v}}{Dt} + \rho \nabla \cdot \vec{v} = 0</math>
 
 
'''Adiabatic Form of the'''<br>
<font color="#770000">'''First Law of Thermodynamics'''</font><br>
('''Specific Entropy Conservation''')
 
<math>\frac{D\vec{v}}{Dt} + \rho \nabla \cdot \vec{v} = 0</math>
 
 
<font color="#770000">'''Poisson Equation'''</font><br>
 
<math>\nabla^2 \Phi = 4\pi G \rho</math>
</div>
</div>



Revision as of 04:35, 18 January 2010

H Book title.gif


Principal Governing Equations

According to the eloquent discussion of the broad subject of Fluid Mechanics presented by Landau and Lifshitz (1975), the state of a moving fluid is determined by five quantities: the three components of the velocity <math>\vec{v}</math> and, for example, the pressure <math>P</math> and the density <math> \rho </math> . For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential <math> \Phi </math>. Accordingly, a complete system of equations of fluid dynamics should be six in number. For an ideal fluid these are:

Euler's Equation
(Momentum Conservation)

<math>\frac{D\vec{v}}{Dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>


Equation of Continuity
(Mass Conservation)

<math>\frac{D\vec{v}}{Dt} + \rho \nabla \cdot \vec{v} = 0</math>


Adiabatic Form of the
First Law of Thermodynamics
(Specific Entropy Conservation)

<math>\frac{D\vec{v}}{Dt} + \rho \nabla \cdot \vec{v} = 0</math>


Poisson Equation

<math>\nabla^2 \Phi = 4\pi G \rho</math>



Home |