PGE/ConservingMass - Revision history

Jet53man at 22:53, 6 October 2021

2021-10-06T22:53:49Z

← Older revision		Revision as of 18:53, 6 October 2021
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	=Continuity Equation=		=Continuity Equation=

	~~==Various Forms==~~

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

Jet53man: /* Continuity Equation */

2021-07-28T03:09:59Z

Continuity Equation

← Older revision		Revision as of 23:09, 27 July 2021
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	Among the [[PGE#Principal_Governing_Equations\|principal governing equations]] we have included the		Among the [[PGE#Principal_Governing_Equations\|principal governing equations]] we have included the
			<br />
			<br />
			<br />
			<br />

	<div align="center">		<div align="center">

Jet53man: /* Lagrangian Representation */

2021-07-28T03:08:02Z

Lagrangian Representation

← Older revision		Revision as of 23:08, 27 July 2021
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	==Various Forms==		==Various Forms==

			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#Context\|<b>Continuity</b>]]</font>
			\|}
	===Lagrangian Representation===		===Lagrangian Representation===

Jet53man: /* Lagrangian Frame of Reference */

2021-06-30T22:54:34Z

Lagrangian Frame of Reference

← Older revision		Revision as of 18:54, 30 June 2021
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	===Lagrangian Frame of Reference===		===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 {{~~TemplateMath~~/VAR_Density01}} of your fluid element will, by definition, remain unchanged over time if,		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">		<div align="center">
	<math>\frac{d\rho}{dt} = 0</math> .		<math>\frac{d\rho}{dt} = 0</math> .

Jet53man: Created page with "FORCETOC =Continuity Equation= ==Various Forms== ===Lagrangian Representation=== Among the PGE#Principal_Governing_Equations|p..."

2021-06-30T22:52:16Z

Created page with "__FORCETOC__  =Continuity Equation= ==Various Forms== ===Lagrangian Representation=== Among the PGE#Principal_Governing_Equations|p..."

New page

__FORCETOC__


=Continuity Equation=

==Various Forms==

===Lagrangian Representation===

Among the [[PGE#Principal_Governing_Equations|principal governing equations]] we have included the

<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 {{TemplateMath/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>

{{ SGFfooter }}

PGE/ConservingMass - Revision history

Jet53man at 22:53, 6 October 2021

Jet53man: /* Continuity Equation */

Jet53man: /* Lagrangian Representation */

Jet53man: /* Lagrangian Frame of Reference */

Jet53man: Created page with "__FORCETOC__ =Continuity Equation= ==Various Forms== ===Lagrangian Representation=== Among the PGE#Principal_Governing_Equations|p..."

Jet53man: Created page with "FORCETOC =Continuity Equation= ==Various Forms== ===Lagrangian Representation=== Among the PGE#Principal_Governing_Equations|p..."