User:Tohline/PGE/ConservingMass - Revision history

Tohline: /* Lagrangian Representation */

2018-05-05T03:41:46Z

Lagrangian Representation

← Older revision		Revision as of 03:41, 5 May 2018
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	{{User:Tohline/Math/EQ_Continuity01}}		{{User:Tohline/Math/EQ_Continuity01}}

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

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

	===Eulerian Representation===		===Eulerian Representation===

Tohline: /* Eulerian Representation */

2018-05-05T03:28:13Z

Eulerian Representation

← Older revision		Revision as of 03:28, 5 May 2018
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	{{User:Tohline/Math/EQ_Continuity02}}		{{User:Tohline/Math/EQ_Continuity02}}

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

Tohline: /* Eulerian Frame of Reference (steady-state mass distribution) */

2016-11-18T19:17:24Z

Eulerian Frame of Reference (steady-state mass distribution)

← Older revision		Revision as of 19:17, 18 November 2016
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	</div>		</div>

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

Tohline: /* Lagrangian Frame of Reference */

2016-11-18T19:16:19Z

Lagrangian Frame of Reference

← Older revision		Revision as of 19:16, 18 November 2016
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	</div>		</div>

	From the above "Standard Lagrangian Representation" of the continuity equation, this condition also implies that,		From the above "[[#ConservingMass:Lagrangian\|Standard Lagrangian Representation]]" of the continuity equation, this condition also implies that,
	<div align="center">		<div align="center">
	<math>\nabla\cdot \vec{v} = 0</math> .		<math>\nabla\cdot \vec{v} = 0</math> .

Tohline: /* Eulerian Representation */

2016-11-18T19:12:19Z

Eulerian Representation

← Older revision		Revision as of 19:12, 18 November 2016
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	===Eulerian Representation===		===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, "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		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">		<div align="center">

Tohline: /* Eulerian Representation */

2015-07-10T22:38:39Z

Eulerian Representation

← Older revision		Revision as of 22:38, 10 July 2015
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	===Eulerian Representation===		===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 the ~~linked [http://en.~~wikipedia~~.org/wiki/Material_derivative Wikipedia~~ discussion], ~~and references therein~~, 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		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, "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">		<div align="center">

Tohline: /* Various Forms */

2015-07-10T22:27:41Z

Various Forms

← Older revision		Revision as of 22:27, 10 July 2015
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	===Lagrangian Representation===		===Lagrangian Representation===

	Among the [~~http://www.vistrails.org/index.php/~~User:Tohline/PGE principal governing equations] we have included the		Among the [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] we have included the

	<div align="center">		<div align="center">
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	===Eulerian Representation===		===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 the linked [http://en.wikipedia.org/wiki/Material_derivative Wikipedia discussion], and references therein, 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		By replacing the so-called Lagrangian (or "material") time derivative, <math>~d\rho/dt</math>, in the first expression by its Eulerian counterpart (see the linked [http://en.wikipedia.org/wiki/Material_derivative Wikipedia discussion], and references therein, 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">		<div align="center">

Tohline: Move title image

2012-11-06T17:41:43Z

Move title image

← Older revision		Revision as of 17:41, 6 November 2012
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			=Continuity Equation=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	~~=Continuity Equation=~~

	==Various Forms==		==Various Forms==

Tohline: /* Eulerian Frame of Reference */ add "steady-state" concept

2010-01-31T20:32:28Z

Eulerian Frame of Reference: add "steady-state" concept

← Older revision		Revision as of 20:32, 31 January 2010
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	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.		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===		===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,		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,
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	</div>		</div>

	From the above "Eulerian Representation" of the continuity equation, this condition also implies that,		From the above "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">		<div align="center">
	<math>\nabla\cdot (\rho \vec{v}) = 0</math> .		<math>\nabla\cdot (\rho \vec{v}) = 0</math> .

Tohline: /* Various Forms */ Insert subsubsection headings

2010-01-31T18:57:20Z

Various Forms: Insert subsubsection headings

← Older revision		Revision as of 18:57, 31 January 2010
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	==Various Forms==		==Various Forms==

			===Lagrangian Representation===

	Among the [http://www.vistrails.org/index.php/User:Tohline/PGE principal governing equations] we have included the		Among the [http://www.vistrails.org/index.php/User:Tohline/PGE principal governing equations] we have included the
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	</math>		</math>
	</div>		</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 the linked [http://en.wikipedia.org/wiki/Material_derivative Wikipedia discussion], and references therein, 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		By replacing the so-called Lagrangian (or "material") time derivative <math>d\rho/dt</math> in the first expression by its Eulerian counterpart (see the linked [http://en.wikipedia.org/wiki/Material_derivative Wikipedia discussion], and references therein, 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