PGE - Revision history

Jet53man: /* Principal Governing Equations */

2022-05-01T17:19:04Z

Principal Governing Equations

← Older revision		Revision as of 13:19, 1 May 2022
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	<font size="-1">[[H_BookTiledMenu#Context\|<b>Principle<br />Governing<br />Equations</b>]]<br />(PGEs)</font>		<font size="-1">[[H_BookTiledMenu#Context\|<b>Principle<br />Governing<br />Equations</b>]]<br />(PGEs)</font>
	\|}		\|}
	According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">"the state of a moving fluid is determined by five quantities: the three components of the velocity</FONT> {{ Math/VAR_VelocityVector01 }} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{ Math/VAR_Pressure01 }} <FONT COLOR="#007700">and the density</FONT> {{ Math/VAR_Density01 }}<FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{ Math/VAR_NewtonianPotential01 }}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>		According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975) — hereafter, [<b>[[Appendix/References#LL75\|<font color="red">LL75</font>]]</b>] — <FONT COLOR="#007700">"the state of a moving fluid is determined by five quantities: the three components of the velocity</FONT> {{ Math/VAR_VelocityVector01 }} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{ Math/VAR_Pressure01 }} <FONT COLOR="#007700">and the density</FONT> {{ Math/VAR_Density01 }}<FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{ Math/VAR_NewtonianPotential01 }}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>
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	<br />		<br />

Jet53man: /* See Also */

2021-09-21T15:33:26Z

Jet53man at 03:04, 28 July 2021

2021-07-28T03:04:55Z

← Older revision		Revision as of 23:04, 27 July 2021
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	__FORCETOC__		__FORCETOC__
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->

	=Principal Governing Equations=		=Principal Governing Equations=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	{\| class="PGEclass" style="float:~~right~~; margin-~~left~~: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: ~~150px~~; width: ~~150px~~; background-color:lightgreen;" \|[[H_BookTiledMenu#Context\|<b>Principle<br />Governing<br />Equations</b>]]<br />(PGEs)		! style="height: 125px; width: 125px; background-color:lightgreen;" \|
			<font size="-1">[[H_BookTiledMenu#Context\|<b>Principle<br />Governing<br />Equations</b>]]<br />(PGEs)</font>
	\|}		\|}
	According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">"the state of a moving fluid is determined by five quantities: the three components of the velocity</FONT> {{ Math/VAR_VelocityVector01 }} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{ Math/VAR_Pressure01 }} <FONT COLOR="#007700">and the density</FONT> {{ Math/VAR_Density01 }}<FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{ Math/VAR_NewtonianPotential01 }}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>		According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">"the state of a moving fluid is determined by five quantities: the three components of the velocity</FONT> {{ Math/VAR_VelocityVector01 }} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{ Math/VAR_Pressure01 }} <FONT COLOR="#007700">and the density</FONT> {{ Math/VAR_Density01 }}<FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{ Math/VAR_NewtonianPotential01 }}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>
			<br />
			<br />
			<br />
			<br />
	<div align="center">		<div align="center">
	<span id="PGE:Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />		<span id="PGE:Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

Jet53man: /* Principal Governing Equations */

2021-07-28T02:59:51Z

Principal Governing Equations

← Older revision		Revision as of 22:59, 27 July 2021
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	=Principal Governing Equations=		=Principal Governing Equations=

			{\| class="PGEclass" style="float:right; margin-left: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 150px; width: 150px; background-color:lightgreen;" \|[[H_BookTiledMenu#Context\|<b>Principle<br />Governing<br />Equations</b>]]<br />(PGEs)
			\|}
	According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">"the state of a moving fluid is determined by five quantities: the three components of the velocity</FONT> {{ Math/VAR_VelocityVector01 }} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{ Math/VAR_Pressure01 }} <FONT COLOR="#007700">and the density</FONT> {{ Math/VAR_Density01 }}<FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{ Math/VAR_NewtonianPotential01 }}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>		According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">"the state of a moving fluid is determined by five quantities: the three components of the velocity</FONT> {{ Math/VAR_VelocityVector01 }} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{ Math/VAR_Pressure01 }} <FONT COLOR="#007700">and the density</FONT> {{ Math/VAR_Density01 }}<FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{ Math/VAR_NewtonianPotential01 }}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>

Jet53man: /* Principal Governing Equations */

2021-07-17T21:26:36Z

Principal Governing Equations

← Older revision		Revision as of 17:26, 17 July 2021
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	=Principal Governing Equations=		=Principal Governing Equations=
	According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">"the state of a moving fluid is determined by five quantities: the three components of the velocity</FONT> {{~~Template:~~Math/VAR_VelocityVector01}} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{~~Template:~~Math/VAR_Pressure01}} <FONT COLOR="#007700">and the density</FONT> {{~~Template:~~Math/VAR_Density01}}<FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{~~Template:~~Math/VAR_NewtonianPotential01}}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>		According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">"the state of a moving fluid is determined by five quantities: the three components of the velocity</FONT> {{ Math/VAR_VelocityVector01 }} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{ Math/VAR_Pressure01 }} <FONT COLOR="#007700">and the density</FONT> {{ Math/VAR_Density01 }}<FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{ Math/VAR_NewtonianPotential01 }}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>

	<div align="center">		<div align="center">
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	('''Mass Conservation''')		('''Mass Conservation''')

	{{~~Template:~~Math/EQ_Continuity01}}		{{ Math/EQ_Continuity01 }}
	[[File:OtherFormsButton.jpg\|150px\|link=PGE/ConservingMass]]		[[File:OtherFormsButton.jpg\|150px\|link=PGE/ConservingMass]]

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	('''Momentum Conservation''')		('''Momentum Conservation''')

	{{~~Template:~~Math/EQ_Euler01}}		{{ Math/EQ_Euler01 }}
	[[File:OtherFormsButton.jpg\|150px\|link=PGE/Euler#Euler_Equation]]		[[File:OtherFormsButton.jpg\|150px\|link=PGE/Euler#Euler_Equation]]

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	<span id="PGE:FirstLaw"><font color="#770000">'''First Law of Thermodynamics'''</font></span>		<span id="PGE:FirstLaw"><font color="#770000">'''First Law of Thermodynamics'''</font></span>

	{{~~Template:~~Math/EQ_FirstLaw01}}		{{ Math/EQ_FirstLaw01 }}
	<!-- [[Image:LSU_OtherFormsButton.jpg\|150px\|link=http://www.vistrails.org/index.php/User:Tohline/PGE/ConservingEntropy]] -->		<!-- [[Image:LSU_OtherFormsButton.jpg\|150px\|link=http://www.vistrails.org/index.php/User:Tohline/PGE/ConservingEntropy]] -->

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	<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />		<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

	{{~~Template:~~Math/EQ_Poisson01}}		{{ Math/EQ_Poisson01 }}
	[[File:OriginButton.jpg\|125px\|link=PGE/PoissonOrigin]]		[[File:OriginButton.jpg\|125px\|link=PGE/PoissonOrigin]]
	</div>		</div>


	In the 1<sup>st</sup> Law of Thermodynamics, {{~~Template:~~Math/VAR_Temperature01}} is the temperature, {{~~Template:~~Math/VAR_SpecificEntropy01}} is the specific entropy, {{~~Template:~~Math/VAR_SpecificInternalEnergy01}} is the specific internal energy of the gas; and in the Poisson equation, {{~~Template:~~Math/C_GravitationalConstant}} is the Newtonian gravitational constant. These differential equations relate the spatial and temporal variations of the principal variables to one another in a physically consistent fashion. Temporal variations are accounted for through first-order derivatives with respect to time, {{~~Template:~~Math/VAR_Time01}}, and variations in space, {{~~Template:~~Math/VAR_PositionVector01}}, are accounted for through the differential operators: the gradient, divergence, and Laplacian.		In the 1<sup>st</sup> Law of Thermodynamics, {{ Math/VAR_Temperature01 }} is the temperature, {{ Math/VAR_SpecificEntropy01 }} is the specific entropy, {{ Math/VAR_SpecificInternalEnergy01 }} is the specific internal energy of the gas; and in the Poisson equation, {{ Math/C_GravitationalConstant }} is the Newtonian gravitational constant. These differential equations relate the spatial and temporal variations of the principal variables to one another in a physically consistent fashion. Temporal variations are accounted for through first-order derivatives with respect to time, {{ Math/VAR_Time01 }}, and variations in space, {{ Math/VAR_PositionVector01 }}, are accounted for through the differential operators: the gradient, divergence, and Laplacian.


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	('''Specific Entropy Conservation''')		('''Specific Entropy Conservation''')

	{{~~Template:~~Math/EQ_FirstLaw02}} .		{{ Math/EQ_FirstLaw02 }} .

	[<b>[[Appendix/References#BLRY07\|<font color="red">BLRY07</font>]]</b>], p. 14, Eq. (1.60)		[<b>[[Appendix/References#BLRY07\|<font color="red">BLRY07</font>]]</b>], p. 14, Eq. (1.60)
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	To complete the description of any specific astrophysical system, this set of differential equations must be [[SR#Supplemental_Relations\|supplemented by additional relations]] — for example, a relationship between {{~~Template:~~Math/VAR_SpecificInternalEnergy01}} and the other two state variables, {{~~Template:~~Math/VAR_Pressure01}} & {{~~Template:~~Math/VAR_Density01}} — which (at least in the context of this H_Book) usually will be algebraic expressions motivated by the specific physics that is relevant to the chosen system.		To complete the description of any specific astrophysical system, this set of differential equations must be [[SR#Supplemental_Relations\|supplemented by additional relations]] — for example, a relationship between {{ Math/VAR_SpecificInternalEnergy01 }} and the other two state variables, {{ Math/VAR_Pressure01 }} & {{ Math/VAR_Density01 }} — which (at least in the context of this H_Book) usually will be algebraic expressions motivated by the specific physics that is relevant to the chosen system.

	=See Also=		=See Also=

Jet53man: /* Principal Governing Equations */

2021-07-03T22:28:13Z

Principal Governing Equations

← Older revision		Revision as of 18:28, 3 July 2021
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	{{Template:Math/EQ_Euler01}}		{{Template:Math/EQ_Euler01}}
	[[File:OtherFormsButton.jpg\|150px\|link=~~http://www.vistrails.org/index.php/User:Tohline/~~PGE/Euler#Euler_Equation]]		[[File:OtherFormsButton.jpg\|150px\|link=PGE/Euler#Euler_Equation]]

Jet53man: /* Principal Governing Equations */

2021-07-03T22:26:33Z

Principal Governing Equations

← Older revision		Revision as of 18:26, 3 July 2021
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	To complete the description of any specific astrophysical system, this set of differential equations must be [[~~User:Tohline/~~SR#Supplemental_Relations\|supplemented by additional relations]] — for example, a relationship between {{Template:Math/VAR_SpecificInternalEnergy01}} and the other two state variables, {{Template:Math/VAR_Pressure01}} & {{Template:Math/VAR_Density01}} — which (at least in the context of this H_Book) usually will be algebraic expressions motivated by the specific physics that is relevant to the chosen system.		To complete the description of any specific astrophysical system, this set of differential equations must be [[SR#Supplemental_Relations\|supplemented by additional relations]] — for example, a relationship between {{Template:Math/VAR_SpecificInternalEnergy01}} and the other two state variables, {{Template:Math/VAR_Pressure01}} & {{Template:Math/VAR_Density01}} — which (at least in the context of this H_Book) usually will be algebraic expressions motivated by the specific physics that is relevant to the chosen system.

	=See Also=		=See Also=

Jet53man: /* Principal Governing Equations */

2021-07-02T19:03:14Z

Principal Governing Equations

← Older revision		Revision as of 15:03, 2 July 2021
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	{{Template:Math/EQ_Poisson01}}		{{Template:Math/EQ_Poisson01}}
	[[File:OriginButton.jpg\|125px\|link=~~http://www.vistrails.org/index.php/User:Tohline/SR~~/PoissonOrigin]]		[[File:OriginButton.jpg\|125px\|link=PGE/PoissonOrigin]]
	</div>		</div>

Jet53man: /* Principal Governing Equations */

2021-07-01T19:19:04Z

Principal Governing Equations

← Older revision		Revision as of 15:19, 1 July 2021
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	{{Template:Math/EQ_FirstLaw02}} .		{{Template:Math/EQ_FirstLaw02}} .

	[<b>[[~~User:Tohline/~~Appendix/References#BLRY07\|<font color="red">BLRY07</font>]]</b>], p. 14, Eq. (1.60)		[<b>[[Appendix/References#BLRY07\|<font color="red">BLRY07</font>]]</b>], p. 14, Eq. (1.60)
	</div>		</div>

Jet53man: /* Principal Governing Equations */

2021-06-30T23:04:09Z

Principal Governing Equations

← Older revision		Revision as of 19:04, 30 June 2021
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	{{Template:Math/EQ_Continuity01}}		{{Template:Math/EQ_Continuity01}}
	[[File:OtherFormsButton.jpg\|150px\|link=~~http://www.vistrails.org/index.php/User:Tohline/~~PGE/ConservingMass~~#Continuity_Equation~~]]		[[File:OtherFormsButton.jpg\|150px\|link=PGE/ConservingMass]]

← Older revision		Revision as of 11:33, 21 September 2021
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	=See Also=		=See Also=
	* [https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics) Euler equations (fluid dynamics)]		* [https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics) Euler equations (fluid dynamics)]
			* [[Cylindrical3D\|PGEs Cast in Cylindrical Coordinates]]


	{{ SGFfooter }}		{{ SGFfooter }}