User:Tohline/PGE

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\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>

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

<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math>

Poisson Equation

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

In this last expression, <math>~G</math> is the Newtonian gravitational constant.

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | 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