User:Tohline/PGE/ConservingMass

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Whitworth's (1981) Isothermal Free-Energy Surface
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Continuity Equation

Among the principal governing equations we have included the

Standard Lagrangian Representation
of the Continuity Equation,

LSU Key.png

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

Note that this equation also may be written in the form,

<math> \frac{d \ln \rho}{dt} = - \nabla\cdot \vec{v} \, . </math>

By replacing the Lagrangian time derivative <math>d\rho/dt</math> in the first expression by its Eulerian counterpart, we directly obtain what is commonly referred to as the

Conservative Form
of the Continuity Equation,

LSU Key.png

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


Whitworth's (1981) Isothermal Free-Energy Surface

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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