User:Tohline/PGE/ConservingMass

Continuity Equation

Among the principal governing equations we have included the

Standard Lagrangian Representation
of the Continuity Equation,

<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 (see the linked Wikipedia discussion, and references therein, to understand how the so-called material derivative serves as a link between Lagrangian and Eulerian descriptions of fluid motion), we directly obtain what is commonly referred to as the

Conservative Form
of the Continuity Equation,

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

© 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