User:Tohline/PGE - Revision history

Tohline: /* Principal Governing Equations */

2018-05-05T03:45:53Z

Principal Governing Equations

← Older revision		Revision as of 03:45, 5 May 2018
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	{{User:Tohline/Math/EQ_FirstLaw02}} .		{{User:Tohline/Math/EQ_FirstLaw02}} .

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

Tohline: /* Principal Governing Equations */

2017-07-05T21:44:32Z

Principal Governing Equations

← Older revision		Revision as of 21:44, 5 July 2017
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	{{User:Tohline/Math/EQ_Poisson01}}		{{User:Tohline/Math/EQ_Poisson01}}
	[[Image:LSU_OriginButton.jpg\|~~150px~~\|link=http://www.vistrails.org/index.php/User:Tohline/SR/PoissonOrigin]]		[[Image:LSU_OriginButton.jpg\|125px\|link=http://www.vistrails.org/index.php/User:Tohline/SR/PoissonOrigin]]
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Tohline: /* Principal Governing Equations */

2017-07-05T21:44:05Z

Principal Governing Equations

← Older revision		Revision as of 21:44, 5 July 2017
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	{{User:Tohline/Math/EQ_Poisson01}}		{{User:Tohline/Math/EQ_Poisson01}}
			[[Image:LSU_OriginButton.jpg\|150px\|link=http://www.vistrails.org/index.php/User:Tohline/SR/PoissonOrigin]]
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Tohline: /* Principal Governing Equations */

2016-11-18T19:07:52Z

Principal Governing Equations

← Older revision		Revision as of 19:07, 18 November 2016
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	To complete the description of any specific astrophysical system, this set of differential equations must be [[User:Tohline/SR\|supplemented by additional relations]] — for example, a relationship between {{User:Tohline/Math/VAR_SpecificInternalEnergy01}} and the other two state variables, {{User:Tohline/Math/VAR_Pressure01}} & {{User:Tohline/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 [[User:Tohline/SR#Supplemental_Relations\|supplemented by additional relations]] — for example, a relationship between {{User:Tohline/Math/VAR_SpecificInternalEnergy01}} and the other two state variables, {{User:Tohline/Math/VAR_Pressure01}} & {{User:Tohline/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.

	==Related Wikipedia Discussions==		==Related Wikipedia Discussions==

Tohline: /* Principal Governing Equations */ get rid of "nabla" symbols within the line of text

2015-07-12T22:20:22Z

Principal Governing Equations: get rid of "nabla" symbols within the line of text

← Older revision		Revision as of 22:20, 12 July 2015
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	In the 1<sup>st</sup> Law of Thermodynamics, {{User:Tohline/Math/VAR_Temperature01}} is the temperature, {{User:Tohline/Math/VAR_SpecificEntropy01}} is the specific entropy, {{User:Tohline/Math/VAR_SpecificInternalEnergy01}} is the specific internal energy of the gas; and in the Poisson equation, {{User:Tohline/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, {{User:Tohline/Math/VAR_Time01}}, and variations in space, {{User:Tohline/Math/VAR_PositionVector01}}, are accounted for through the differential operators: gradient ~~(<math>\nabla</math>)~~, divergence ~~(<math>\nabla\cdot</math>)~~, and Laplacian ~~(<math>\nabla^2</math>)~~.		In the 1<sup>st</sup> Law of Thermodynamics, {{User:Tohline/Math/VAR_Temperature01}} is the temperature, {{User:Tohline/Math/VAR_SpecificEntropy01}} is the specific entropy, {{User:Tohline/Math/VAR_SpecificInternalEnergy01}} is the specific internal energy of the gas; and in the Poisson equation, {{User:Tohline/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, {{User:Tohline/Math/VAR_Time01}}, and variations in space, {{User:Tohline/Math/VAR_PositionVector01}}, are accounted for through the differential operators: the gradient, divergence, and Laplacian.

Tohline: /* Principal Governing Equations */

2015-07-10T21:50:03Z

Principal Governing Equations

← Older revision		Revision as of 21:50, 10 July 2015
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	By restricting our discussions to physical systems that are governed by this set of equations, for the most part '''we will be considering the structure, stability, and dynamical behavior of compressible, inviscid fluid systems that are self-gravitating.''' We will assume that no electromagnetic forces act on the fluid (''e.g.,'' the effects of magnetic fields on an ionized plasma fluid will not be considered). Also, in the absence of dynamically generated shocks, we will usually assume that all compressions and rarefactions occur adiabatically (<i>i.e.</i>, ~~d{{User:Tohline~~/~~Math/VAR_SpecificEntropy01}}/d{{User:Tohline/Math/VAR_Time01}}~~ = 0), in which case the third governing equation identified above will be replaced by the		By restricting our discussions to physical systems that are governed by this set of equations, for the most part '''we will be considering the structure, stability, and dynamical behavior of compressible, inviscid fluid systems that are self-gravitating.''' We will assume that no electromagnetic forces act on the fluid (''e.g.,'' the effects of magnetic fields on an ionized plasma fluid will not be considered). Also, in the absence of dynamically generated shocks, we will usually assume that all compressions and rarefactions occur adiabatically (<i>i.e.</i>, <math>~ds/dt = 0</math>), in which case the third governing equation identified above will be replaced by the

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	To complete the description of any specific astrophysical system, this set of differential equations must be [~~http://www.vistrails.org/index.php/~~User:Tohline/SR supplemented by additional relations] — for example, a relationship between {{User:Tohline/Math/VAR_SpecificInternalEnergy01}} and the other two state variables, {{User:Tohline/Math/VAR_Pressure01}} & {{User:Tohline/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 [[User:Tohline/SR\|supplemented by additional relations]] — for example, a relationship between {{User:Tohline/Math/VAR_SpecificInternalEnergy01}} and the other two state variables, {{User:Tohline/Math/VAR_Pressure01}} & {{User:Tohline/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.

	==Related Wikipedia Discussions==		==Related Wikipedia Discussions==

Tohline: /* Principal Governing Equations */

2015-07-10T21:38:05Z

Principal Governing Equations

← Older revision		Revision as of 21:38, 10 July 2015
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	{{User:Tohline/Math/EQ_Euler01}}		{{User:Tohline/Math/EQ_Euler01}}
	[http://www.vistrails.org/index.php/User:Tohline/PGE/Euler ~~{{User:Tohline/Buttons/OtherForms}}~~]		[[Image:LSU_OtherFormsButton.jpg\|150px\|link=http://www.vistrails.org/index.php/User:Tohline/PGE/Euler#Euler_Equation]]


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

Tohline: /* Principal Governing Equations */

2015-07-10T21:27:41Z

Principal Governing Equations

← Older revision		Revision as of 21:27, 10 July 2015
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	{{User:Tohline/Math/EQ_Continuity01}}		{{User:Tohline/Math/EQ_Continuity01}}
	[http://www.vistrails.org/index.php/User:Tohline/PGE/ConservingMass ~~{{User:Tohline/Buttons/OtherForms}}~~]		[[Image:LSU_OtherFormsButton.jpg\|150px\|link=http://www.vistrails.org/index.php/User:Tohline/PGE/ConservingMass#Continuity_Equation]]

Tohline: Move title image

2012-11-06T17:40:27Z

Move title image

← Older revision		Revision as of 17:40, 6 November 2012
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			=Principal Governing Equations=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	~~=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> {{User:Tohline/Math/VAR_VelocityVector01}} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{User:Tohline/Math/VAR_Pressure01}} <FONT COLOR="#007700">and the density</FONT> {{User:Tohline/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 {{User:Tohline/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> {{User:Tohline/Math/VAR_VelocityVector01}} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{User:Tohline/Math/VAR_Pressure01}} <FONT COLOR="#007700">and the density</FONT> {{User:Tohline/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 {{User:Tohline/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>

Tohline: /* Related Wikipedia Discussions */

2012-11-03T21:19:05Z

Related Wikipedia Discussions

← Older revision		Revision as of 21:19, 3 November 2012
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	==Related Wikipedia Discussions==		==Related Wikipedia Discussions==
	* [http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics) Euler equations (fluid dynamics)]		* [http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics) Euler equations (fluid dynamics)]


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}