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==Principal Governing Equations== | ==Principal Governing Equations== | ||
According to | 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"> | ||
< | <font color="#770000">'''Euler's Equation'''</font><br> | ||
('''Momentum Conservation''') | |||
<math>\frac{ | <math>\frac{D\vec{v}}{Dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math> | ||
<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
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>
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