Governing equations on integral form - Revision history

Nian at 08:03, 2 April 2026

2026-04-02T08:03:19Z

← Older revision		Revision as of 08:03, 2 April 2026
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	==== The Continuity Equation ====		==== The Continuity Equation ====

	{{~~quote~~\|Mass can be neither created nor destroyed, which implies that mass is conserved}}		{{QuoteBox\|Mass can be neither created nor destroyed, which implies that mass is conserved}}

	The net massflow into the control volume <math>\Omega</math> in Fig. \ref{fig:generic:cv} is obtained by integrating mass flux over the control volume surface <math>\partial \Omega</math>		The net massflow into the control volume <math>\Omega</math> in Fig. \ref{fig:generic:cv} is obtained by integrating mass flux over the control volume surface <math>\partial \Omega</math>
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	==== The Momentum Equation ====		==== The Momentum Equation ====

	{{~~quote~~\|The time rate of change of momentum of a body equals the net force exerted on it}}		{{QuoteBox\|The time rate of change of momentum of a body equals the net force exerted on it}}

	{{NumEqn\|<math>		{{NumEqn\|<math>
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	==== The Energy Equation ====		==== The Energy Equation ====

	{{~~quote~~\|Energy can be neither created nor destroyed; it can only change in form}}		{{QuoteBox\|Energy can be neither created nor destroyed; it can only change in form}}

	<math display="block">		<math display="block">

Nian: /* Summary */

2026-04-01T18:28:01Z

Summary

← Older revision		Revision as of 18:28, 1 April 2026
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	<div style="border: solid 1px;">		<div style="border: solid 1px;">
	{{~~NumEqn~~\|<math>		{{OpenInfoBox\|<math>
	\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0		\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0
	</math>~~\|background-color=whitesmoke\|color=steelblue~~\|description=Continuity:~~\|nonumber=1\|padding=2em~~}}		</math>\|description=Continuity:}}

	{{~~NumEqn~~\|<math>		{{OpenInfoBox\|<math>
	\frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV+\iint_{\partial \Omega} \left[(\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}+p\mathbf{n}\right]dS=\iiint_{\Omega}\rho \mathbf{f}dV		\frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV+\iint_{\partial \Omega} \left[(\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}+p\mathbf{n}\right]dS=\iiint_{\Omega}\rho \mathbf{f}dV
	</math>~~\|background-color=whitesmoke\|color=steelblue~~\|description=Momentum:~~\|nonumber=1\|padding=2em~~}}		</math>\|description=Momentum:}}

	{{~~NumEqn~~\|<math>		{{OpenInfoBox\|<math>
	\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV		\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
	</math>~~\|background-color=whitesmoke\|color=steelblue~~\|description=Energy:~~\|nonumber=1\|padding=2em~~}}		</math>\|description=Energy:}}
	</div>		</div>

Nian at 19:32, 31 March 2026

2026-03-31T19:32:09Z

← Older revision		Revision as of 19:32, 31 March 2026
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	The integral form of the governing equations for inviscid compressible flow has been derived		The integral form of the governing equations for inviscid compressible flow has been derived

			<div style="border: solid 1px;">
	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0		\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0
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	\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV		\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
	</math>\|background-color=whitesmoke\|color=steelblue\|description=Energy:\|nonumber=1\|padding=2em}}		</math>\|background-color=whitesmoke\|color=steelblue\|description=Energy:\|nonumber=1\|padding=2em}}
			</div>

Nian at 19:28, 31 March 2026

2026-03-31T19:28:43Z

← Older revision		Revision as of 19:28, 31 March 2026
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	The integral form of the governing equations for inviscid compressible flow has been derived		The integral form of the governing equations for inviscid compressible flow has been derived

	~~Continuity:~~

	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0		\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0
	</math>}}		</math>\|background-color=whitesmoke\|color=steelblue\|description=Continuity:\|nonumber=1\|padding=2em}}

	~~Momentum:~~

	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV+\iint_{\partial \Omega} \left[(\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}+p\mathbf{n}\right]dS=\iiint_{\Omega}\rho \mathbf{f}dV		\frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV+\iint_{\partial \Omega} \left[(\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}+p\mathbf{n}\right]dS=\iiint_{\Omega}\rho \mathbf{f}dV
	</math>}}		</math>\|background-color=whitesmoke\|color=steelblue\|description=Momentum:\|nonumber=1\|padding=2em}}

	~~Energy:~~

	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV		\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
	</math>}}		</math>\|background-color=whitesmoke\|color=steelblue\|description=Energy:\|nonumber=1\|padding=2em}}

Nian at 07:35, 31 March 2026

2026-03-31T07:35:58Z

← Older revision		Revision as of 07:35, 31 March 2026
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	[[Category:Compressible flow]]		<!--
	[[Category:Governing equations]]		-->[[Category:Compressible flow]]<!--
	<noinclude><!--		-->[[Category:Governing equations]]<!--
			--><noinclude><!--
	-->[[Category:Compressible flow:Topic]]<!--		-->[[Category:Compressible flow:Topic]]<!--
	--></noinclude><!--		--></noinclude><!--

Nian at 13:35, 30 March 2026

2026-03-30T13:35:39Z

← Older revision		Revision as of 13:35, 30 March 2026
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	--></nomobile><!--		--></nomobile><!--

			--><noinclude><!--
	-->{{#vardefine:secno\|2}}<!--		-->{{#vardefine:secno\|2}}<!--
	~~--><noinclude><!--~~
	-->{{#vardefine:eqno\|0}}<!--		-->{{#vardefine:eqno\|0}}<!--
	--></noinclude><!--		--></noinclude><!--

Nian at 12:21, 30 March 2026

2026-03-30T12:21:45Z

← Older revision		Revision as of 12:21, 30 March 2026
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	\dfrac{d}{dt}\iiint_{\Omega}\rho\left(e+\dfrac{V^2}{2}\right)dV+\iint_{\partial \Omega}\left[\rho\left(e+\dfrac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>		\dfrac{d}{dt}\iiint_{\Omega}\rho\left(e+\dfrac{V^2}{2}\right)dV+\iint_{\partial \Omega}\left[\rho\left(e+\dfrac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>
	\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV		\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
	</math>\|align=~~left~~}}		</math>\|align=center}}

	The surface integral in the energy equation may be rewritten as		The surface integral in the energy equation may be rewritten as
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	\iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS		\iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS
	</math>\|align=~~left~~}}		</math>\|align=center}}

	and with the definition of enthalpy <math>h=e+p/\rho</math>, we get		and with the definition of enthalpy <math>h=e+p/\rho</math>, we get

Nian at 12:20, 30 March 2026

2026-03-30T12:20:31Z

← Older revision		Revision as of 12:20, 30 March 2026
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	~~\begin{aligned}~~		\dfrac{d}{dt}\iiint_{\Omega}\rho\left(e+\dfrac{V^2}{2}\right)dV+\iint_{\partial \Omega}\left[\rho\left(e+\dfrac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>
	&\dfrac{d}{dt}\iiint_{\Omega}\rho\left(e+\dfrac{V^2}{2}\right)dV+\iint_{\partial \Omega}\left[\rho\left(e+\dfrac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=\\		\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
	&\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV		</math>\|align=left}}
	~~\end{aligned}~~
	</math>}}

	The surface integral in the energy equation may be rewritten as		The surface integral in the energy equation may be rewritten as

	{{NumEqn\|<math>		{{NumEqn\|<math>
	\iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS		\iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS
	</math>}}		</math>\|align=left}}

	and with the definition of enthalpy <math>h=e+p/\rho</math>, we get		and with the definition of enthalpy <math>h=e+p/\rho</math>, we get
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV		\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
	</math>}}		</math>}}

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	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV		\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
	</math>}}		</math>}}

Nian at 11:28, 30 March 2026

2026-03-30T11:28:33Z

Show changes

Nian at 19:30, 20 March 2026

2026-03-20T19:30:21Z

← Older revision		Revision as of 19:30, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Governing equations]]
			<noinclude>
	[[Category:Compressible flow:Topic]]		[[Category:Compressible flow:Topic]]
	~~[[Category:Governing equations]]~~		</noinclude>

	__TOC__		__TOC__