The entropy equation - Revision history

Nian at 05:18, 1 April 2026

2026-04-01T05:18:54Z

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	From the second law of thermodynamics		From the second law of thermodynamics

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)		\frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)
	</math>		</math>}}

	From the energy equation on differential non-conservation form internal energy formulation		From the energy equation on differential non-conservation form internal energy formulation

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v})		\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v})
	</math>		</math>}}

	The continuity equation on differential non-conservation form		The continuity equation on differential non-conservation form

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 \Rightarrow \nabla\cdot\mathbf{v}=-\frac{1}{\rho}\frac{D\rho}{Dt}		\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 \Rightarrow \nabla\cdot\mathbf{v}=-\frac{1}{\rho}\frac{D\rho}{Dt}
	</math>		</math>}}

	and thus		and thus

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{De}{Dt} = \dot{q} +\frac{p}{\rho^2}\frac{D\rho}{Dt}		\frac{De}{Dt} = \dot{q} +\frac{p}{\rho^2}\frac{D\rho}{Dt}
	</math>		</math>}}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{D\rho}{Dt}=-\frac{1}{\nu^2}\frac{D\nu}{Dt}		\frac{D\rho}{Dt}=-\frac{1}{\nu^2}\frac{D\nu}{Dt}
	</math>		</math>}}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\rho\frac{De}{Dt} = \rho\dot{q} -\frac{p}{\rho\nu^2}\frac{D\nu}{Dt} = \rho\dot{q} -\rho p\frac{D\nu}{Dt}		\rho\frac{De}{Dt} = \rho\dot{q} -\frac{p}{\rho\nu^2}\frac{D\nu}{Dt} = \rho\dot{q} -\rho p\frac{D\nu}{Dt}
	</math>		</math>}}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\rho\left[\frac{De}{Dt}+p\frac{D\nu}{Dt}-\dot{q}\right]=0\Rightarrow\frac{De}{Dt}=\dot{q}-p\frac{D\nu}{Dt}		\rho\left[\frac{De}{Dt}+p\frac{D\nu}{Dt}-\dot{q}\right]=0\Rightarrow\frac{De}{Dt}=\dot{q}-p\frac{D\nu}{Dt}
	</math>		</math>}}

	Insert <math>De/Dt</math> in Eqn. \ref{eq:second:law}		Insert <math>De/Dt</math> in Eqn. \ref{eq:second:law}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\dot{q}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\Rightarrow		\dot{q}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\Rightarrow
	</math>		</math>}}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	T\frac{Ds}{Dt}=-\dot{q}		T\frac{Ds}{Dt}=-\dot{q}
	</math>		</math>}}

	Adiabatic flow:		Adiabatic flow:

	<math ~~display="block"~~>		{{NumEqn\|<math>
	T\frac{Ds}{Dt}=0		T\frac{Ds}{Dt}=0
	</math>		</math>}}

	In an adiabatic, steady-state, inviscid flow, entropy is constant along a streamline.		In an adiabatic, steady-state, inviscid flow, entropy is constant along a streamline.

Nian at 19:28, 20 March 2026

2026-03-20T19:28:06Z

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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>

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Nian at 18:56, 20 March 2026

2026-03-20T18:56:34Z

← Older revision		Revision as of 18:56, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
	[[Category:Topic]]		[[Category:Compressible flow:Topic]]
	[[Category:Governing equations]]		[[Category:Governing equations]]

Nian at 15:47, 20 March 2026

2026-03-20T15:47:22Z

← Older revision		Revision as of 15:47, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Topic]]
	[[Category:Governing equations]]		[[Category:Governing equations]]

Nian at 06:36, 17 March 2026

2026-03-17T06:36:37Z

← Older revision		Revision as of 06:36, 17 March 2026
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	__TOC__		__TOC__

	~~\section{The Entropy Equation}~~		From the second law of thermodynamics

	~~\noindent From the second law of thermodynamics\\~~		<math display="block">

	~~\begin{equation}~~
	\frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)		\frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)
	~~\label{eq:second:law}~~		</math>
	~~\end{equation}\\~~

	~~\noindent~~ From the energy equation on differential non-conservation form internal energy formulation\\		From the energy equation on differential non-conservation form internal energy formulation

	\[\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v})~~\]\\~~		<math display="block">
			\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v})
			</math>

	~~\noindent~~ The continuity equation on differential non-conservation form		The continuity equation on differential non-conservation form

	\[\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 \Rightarrow \nabla\cdot\mathbf{v}=-\frac{1}{\rho}\frac{D\rho}{Dt}\]		<math display="block">
			\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 \Rightarrow \nabla\cdot\mathbf{v}=-\frac{1}{\rho}\frac{D\rho}{Dt}
			</math>

	~~\noindent~~ and thus\\		and thus

	\[\frac{De}{Dt} = \dot{q} +\frac{p}{\rho^2}\frac{D\rho}{Dt}~~\]\\~~		<math display="block">
			\frac{De}{Dt} = \dot{q} +\frac{p}{\rho^2}\frac{D\rho}{Dt}
			</math>

	\[\frac{D\rho}{Dt}=-\frac{1}{\nu^2}\frac{D\nu}{Dt}~~\]\\~~		<math display="block">
			\frac{D\rho}{Dt}=-\frac{1}{\nu^2}\frac{D\nu}{Dt}
			</math>

	\[\rho\frac{De}{Dt} = \rho\dot{q} -\frac{p}{\rho\nu^2}\frac{D\nu}{Dt} = \rho\dot{q} -\rho p\frac{D\nu}{Dt} ~~\]\\~~		<math display="block">
			\rho\frac{De}{Dt} = \rho\dot{q} -\frac{p}{\rho\nu^2}\frac{D\nu}{Dt} = \rho\dot{q} -\rho p\frac{D\nu}{Dt}
			</math>

	\[\rho\left[\frac{De}{Dt}+p\frac{D\nu}{Dt}-\dot{q}\right]=0\Rightarrow\frac{De}{Dt}=\dot{q}-p\frac{D\nu}{Dt}~~\]\\~~		<math display="block">
			\rho\left[\frac{De}{Dt}+p\frac{D\nu}{Dt}-\dot{q}\right]=0\Rightarrow\frac{De}{Dt}=\dot{q}-p\frac{D\nu}{Dt}
			</math>

	~~\noindent~~ Insert $De/Dt$ in Eqn. \ref{eq:second:law} \\		Insert <math>De/Dt</math> in Eqn. \ref{eq:second:law}

	\[\dot{q}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\Rightarrow~~\]\\~~		<math display="block">
			\dot{q}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\Rightarrow
			</math>

	~~\begin{equation}~~		<math display="block">
	T\frac{Ds}{Dt}=-\dot{q}		T\frac{Ds}{Dt}=-\dot{q}
	~~\label{eq:second:law:b}~~		</math>
	~~\end{equation}\\~~

	~~\noindent~~ Adiabatic flow:\\		Adiabatic flow:

	~~\begin{equation}~~		<math display="block">
	T\frac{Ds}{Dt}=0		T\frac{Ds}{Dt}=0
	~~\label{eq:second:law:b}~~		</math>
	~~\end{equation}\\~~

	~~\noindent~~ In an adiabatic, steady-state, inviscid flow, entropy is constant along a streamline.		In an adiabatic, steady-state, inviscid flow, entropy is constant along a streamline.

Nian: Created page with "Category:Compressible flow Category:Governing equations TOC \section{The Entropy Equation} \noindent From the second law of thermodynamics\\ \begin{equation} \frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right) \label{eq:second:law} \end{equation}\\ \noindent From the energy equation on differential non-conservation form internal energy formulation\\ \[\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v})\]\\ \noindent The co..."

2026-03-16T13:42:44Z

Created page with "Category:Compressible flow Category:Governing equations __TOC__ \section{The Entropy Equation} \noindent From the second law of thermodynamics\\ \begin{equation} \frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right) \label{eq:second:law} \end{equation}\\ \noindent From the energy equation on differential non-conservation form internal energy formulation\\ \[\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v})\]\\ \noindent The co..."

New page

[[Category:Compressible flow]]
[[Category:Governing equations]]

__TOC__

\section{The Entropy Equation}

\noindent From the second law of thermodynamics\\

\begin{equation}
\frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)
\label{eq:second:law}
\end{equation}\\

\noindent From the energy equation on differential non-conservation form internal energy formulation\\

\[\frac{De}{Dt} = \dot{q} - \frac{p}{\rho}(\nabla\cdot\mathbf{v})\]\\

\noindent The continuity equation on differential non-conservation form

\[\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 \Rightarrow \nabla\cdot\mathbf{v}=-\frac{1}{\rho}\frac{D\rho}{Dt}\]

\noindent and thus\\

\[\frac{De}{Dt} = \dot{q} +\frac{p}{\rho^2}\frac{D\rho}{Dt}\]\\

\[\frac{D\rho}{Dt}=-\frac{1}{\nu^2}\frac{D\nu}{Dt}\]\\

\[\rho\frac{De}{Dt} = \rho\dot{q} -\frac{p}{\rho\nu^2}\frac{D\nu}{Dt} = \rho\dot{q} -\rho p\frac{D\nu}{Dt} \]\\

\[\rho\left[\frac{De}{Dt}+p\frac{D\nu}{Dt}-\dot{q}\right]=0\Rightarrow\frac{De}{Dt}=\dot{q}-p\frac{D\nu}{Dt}\]\\

\noindent Insert $De/Dt$ in Eqn. \ref{eq:second:law} \\

\[\dot{q}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\Rightarrow\]\\

\begin{equation}
T\frac{Ds}{Dt}=-\dot{q}
\label{eq:second:law:b}
\end{equation}\\

\noindent Adiabatic flow:\\

\begin{equation}
T\frac{Ds}{Dt}=0
\label{eq:second:law:b}
\end{equation}\\

\noindent In an adiabatic, steady-state, inviscid flow, entropy is constant along a streamline.