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=Continuity Equation=

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
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<font size="-1">[[H_BookTiledMenu#Context|<b>Continuity</b>]]</font>
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==Various Forms==
===Lagrangian Representation===

Among the [[PGE#Principal_Governing_Equations|principal governing equations]] we have included the
 <br />
 <br />
 <br />
 <br />

<div align="center">
<span id="ConservingMass:Lagrangian"><font color="#770000">'''Standard Lagrangian Representation'''</font></span><br />
of the Continuity Equation,

{{Template:Math/EQ_Continuity01}}

[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 13, Eq. (1.53)
</div>

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

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

===Eulerian Representation===

By replacing the so-called Lagrangian (or "material") time derivative, <math>~d\rho/dt</math>, in the first expression by its Eulerian counterpart (see, for example, the wikipedia discussion titled, "[https://en.wikipedia.org/wiki/Material_derivative Material_derivative]," to understand how the Lagrangian and Eulerian descriptions of fluid motion differ from one another conceptually as well as how to mathematically transform from one description to the other), we directly obtain what is commonly referred to as the

<div align="center">
<span id="ConservingMass:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
or<br />
<span id="ConservingMass:Conservative"><font color="#770000">'''Conservative Form'''</font></span><br />
of the Continuity Equation,

{{Template:Math/EQ_Continuity02}}

[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 7, Eq. (1.24)
</div>

==Time-independent Behavior==
===Lagrangian Frame of Reference===

If you are riding along with a fluid element — viewing the system from a ''Lagrangian'' frame of reference — the mass density {{Template:Math/VAR_Density01}} of your fluid element will, by definition, remain unchanged over time if,
<div align="center">
<math>\frac{d\rho}{dt} = 0</math> .
</div>

From the above "[[#ConservingMass:Lagrangian|Standard Lagrangian Representation]]" of the continuity equation, this condition also implies that,
<div align="center">
<math>\nabla\cdot \vec{v} = 0</math> .
</div>
Looking at it a different way, if while riding along with a fluid element you move through a region of space where <math>\nabla\cdot \vec{v} = 0</math>, your mass density will remain unchanged as you move through this region.

===Eulerian Frame of Reference (steady-state mass distribution)===

On the other hand, if you are standing at a fixed location in your coordinate frame watching the fluid flow past you — viewing the system from an ''Eulerian'' frame of reference — the mass density of the fluid at your location in space will, by definition, always be the same if,
<div align="center">
<math>\frac{\partial\rho}{\partial t} = 0</math> .
</div>

From the above "[[#ConservingMass:Eulerian|Eulerian Representation]]" of the continuity equation, this condition also implies that a '''steady-state''' mass distribution will be governed by the relation,
<div align="center">
<math>\nabla\cdot (\rho \vec{v}) = 0</math> .
</div>

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