User:Tohline/PGE/ConservingMomentum - Revision history

Tohline: /* Lagrangian Frame of Reference */ Put "hydrostatic balance" into subsection title

2010-01-31T20:29:35Z

Lagrangian Frame of Reference: Put "hydrostatic balance" into subsection title

← Older revision		Revision as of 20:29, 31 January 2010
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	==Time-independent Behavior==		==Time-independent Behavior==
	===Lagrangian Frame of Reference===		===Lagrangian Frame of Reference (hydrostatic balance)===

	If you are riding along with a fluid element — viewing the system from a ''Lagrangian'' frame of reference — the velocity {{User:Tohline/Math/VAR_VelocityVector01}} 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 velocity {{User:Tohline/Math/VAR_VelocityVector01}} of your fluid element will, by definition, remain unchanged over time if,
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	<math>\frac{1}{\rho} \nabla P = - \nabla \Phi</math> .		<math>\frac{1}{\rho} \nabla P = - \nabla \Phi</math> .
	</div>		</div>
	That is to say, every fluid element within a star will experience no net acceleration if the gradient of the pressure balances the gradient in the gravitational field throughout the star.		That is to say, every fluid element within a star will experience no net acceleration if the gradient of the pressure balances the gradient in the gravitational field throughout the star.

	===Eulerian Frame of Reference (steady-state flow field)===		===Eulerian Frame of Reference (steady-state flow field)===

Tohline: /* Eulerian Frame of Reference */ put "steady-state" in subsection title

2010-01-31T20:28:47Z

Eulerian Frame of Reference: put "steady-state" in subsection title

← Older revision		Revision as of 20:28, 31 January 2010
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	That is to say, every fluid element within a star will experience no net acceleration if the gradient of the pressure balances the gradient in the gravitational field throughout the star.		That is to say, every fluid element within a star will experience no net acceleration if the gradient of the pressure balances the gradient in the gravitational field throughout the star.

	===Eulerian Frame of Reference===		===Eulerian Frame of Reference (steady-state flow field)===

	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 velocity 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 velocity of the fluid at your location in space will, by definition, always be the same if,

Tohline: /* Time-independent Behavior */ replace mass density discussion by velocity and momentum density discussion

2010-01-31T20:24:03Z

Time-independent Behavior: replace mass density discussion by velocity and momentum density discussion

← Older revision		Revision as of 20:24, 31 January 2010
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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~~ {{User:Tohline/Math/~~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 velocity {{User:Tohline/Math/VAR_VelocityVector01}} 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\vec{v}}{dt} = 0</math> .
	</div>		</div>

	From the above "~~Standard~~ Lagrangian Representation" of the ~~continuity~~ equation, this ~~condition~~ also ~~implies that~~,		From the above "Lagrangian Representation" of the Euler equation, this also leads to what is often referred to in discussions of stellar structure as the statement of,
	<div align="center">		<div align="center">
	<math>\nabla\~~cdot~~ \~~vec{v} = 0~~</math> .		<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

			<math>\frac{1}{\rho} \nabla P = - \nabla \Phi</math> .
	</div>		</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~~.		That is to say, every fluid element within a star will experience no net acceleration if the gradient of the pressure balances the gradient in the gravitational field throughout the star.

	===Eulerian Frame of Reference===		===Eulerian Frame of Reference===

	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 velocity of the fluid at your location in space will, by definition, always be the same if,
	<div align="center">		<div align="center">
	<math>\frac{\partial\~~rho~~}{\partial t} = 0</math> .		<math>\frac{\partial\vec{v}}{\partial t} = 0</math> .
	</div>		</div>

	From the above "Eulerian Representation" of the ~~continuity~~ equation, this condition also implies that,		From the above "Eulerian Representation" of the Euler equation, this condition also implies that a '''steady-state''' velocity field must obey the relation,
			<div align="center">
			<math>(\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \Phi</math> .
			</div>
			Or, a '''steady-state''' momentum density field must obey the relation,
	<div align="center">		<div align="center">
	<math>\nabla\cdot (\rho \vec{v}) = 0</math> .		<math>\nabla\cdot [(\rho\vec{v})\vec{v}]= - \nabla P - \rho \nabla \Phi</math>
	</div>		</div>



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Tohline: /* Eulerian Representation */

2010-01-31T19:52:09Z

Eulerian Representation

← Older revision		Revision as of 19:52, 31 January 2010
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	</div>		</div>

	The second term on the left-hand-side of this last expression represents the divergence of the "dyadic product" of the vector momentum density ({{User:Tohline/Math/VAR_Density01}}{{User:Tohline/Math/VAR_VelocityVector01}}) and the velocity vector {{User:Tohline/Math/VAR_VelocityVector01}} and is sometimes written as, <math>\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>; it may help the reader to see this term written out in component form or subscript notation, as presented in the following [http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics) linked Wikipedia discussion].		The second term on the left-hand-side of this last expression represents the divergence of the "dyadic product" of the vector momentum density ({{User:Tohline/Math/VAR_Density01}}{{User:Tohline/Math/VAR_VelocityVector01}}) and the velocity vector {{User:Tohline/Math/VAR_VelocityVector01}} and is sometimes written as, <math>\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>; it may help the reader to see this term written out in component form or subscript notation, as presented in the following [http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)#Conservation_and_component_form linked Wikipedia discussion].

	==Time-independent Behavior==		==Time-independent Behavior==

Tohline: /* Eulerian Representation */ Explanation of dyadic product

2010-01-31T19:50:58Z

Eulerian Representation: Explanation of dyadic product

← Older revision		Revision as of 19:50, 31 January 2010
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	{{User:Tohline/Math/EQ_Euler03}}		{{User:Tohline/Math/EQ_Euler03}}
	</div>		</div>

			The second term on the left-hand-side of this last expression represents the divergence of the "dyadic product" of the vector momentum density ({{User:Tohline/Math/VAR_Density01}}{{User:Tohline/Math/VAR_VelocityVector01}}) and the velocity vector {{User:Tohline/Math/VAR_VelocityVector01}} and is sometimes written as, <math>\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>; it may help the reader to see this term written out in component form or subscript notation, as presented in the following [http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics) linked Wikipedia discussion].

	==Time-independent Behavior==		==Time-independent Behavior==

Tohline: /* Eulerian Representation */ Add conservative form

2010-01-31T19:25:16Z

Eulerian Representation: Add conservative form

← Older revision		Revision as of 19:25, 31 January 2010
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	===Eulerian Representation===		===Eulerian Representation===

	By replacing the so-called Lagrangian (or "material") time derivative <math>d\vec{v}/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\vec{v}/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 the

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

			{{User:Tohline/Math/EQ_Euler02}}
			</div>

			As was done above in the context of the Lagrangian representation of the Euler equation, we can multiply this expression through by {{User:Tohline/Math/VAR_Density01}} and combine it with the continuity equation to derive what is commonly referred to as the,

			<div align="center">
	<span id="ConservingMomentum:Conservative"><font color="#770000">'''Conservative Form'''</font></span><br />		<span id="ConservingMomentum:Conservative"><font color="#770000">'''Conservative Form'''</font></span><br />
	of the Euler Equation,		of the Euler Equation,

	{{User:Tohline/Math/~~EQ_Euler02~~}}		{{User:Tohline/Math/EQ_Euler03}}
	</div>		</div>

Tohline: Create "conserving momentum" page, copying style from "conserving mass" page

2010-01-31T18:56:10Z

Create "conserving momentum" page, copying style from "conserving mass" page

New page

{{LSU_HBook_header}}

=Euler Equation=

==Various Forms==

===Lagrangian Representation===

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

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

{{User:Tohline/Math/EQ_Euler01}}
</div>

Multiplying this equation through by the mass density {{User:Tohline/Math/VAR_Density01}} produces the relation,
<div align="center">
<math>\rho\frac{d\vec{v}}{dt} = - \nabla P - \rho\nabla \Phi</math> ,
</div>
which may be rewritten as,
<div align="center">
<math>\frac{d(\rho\vec{v})}{dt}- \vec{v}\frac{d\rho}{dt} = - \nabla P - \rho\nabla \Phi</math> .
</div>
Combining this with the [http://www.vistrails.org/index.php/User:Tohline/PGE/ConservingMass#ConservingMass:Lagrangian Standard Lagrangian Representation of the Continuity Equation], we derive,
<div align="center">
<math>\frac{d(\rho\vec{v})}{dt}+ (\rho\vec{v})\nabla\cdot\vec{v} = - \nabla P - \rho\nabla \Phi</math> .
</div>

===Eulerian Representation===

By replacing the so-called Lagrangian (or "material") time derivative <math>d\vec{v}/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">
<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
or<br />
<span id="ConservingMomentum:Conservative"><font color="#770000">'''Conservative Form'''</font></span><br />
of the Euler Equation,

{{User:Tohline/Math/EQ_Euler02}}
</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 {{User:Tohline/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 "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===

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

{{LSU_HBook_footer}}