User:Tohline/PGE/ConservingMass
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Continuity Equation
Among the principal governing equations we have included the
Standard Lagrangian Representation
of the Continuity Equation,
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<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{\partial\rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0</math>
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© 2014 - 2021 by Joel E. Tohline |