Governing equations on differential form: Difference between revisions

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Gauss's divergence theorem applied to the surface integral term in the energy equation (Eqn. \ref{eq:governing:energy:int}) gives
Gauss's divergence theorem applied to the surface integral term in the energy equation (Eqn. \ref{eq:governing:energy:int}) gives


\[\oiint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\nabla\cdot(\rho h_o\mathbf{v})d\mathscr{V}\]\\
<math display="block">
\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\nabla\cdot(\rho h_o\mathbf{v})dV
</math>


\noindent Fixed control volume \\
Fixed control volume


\[\frac{d}{dt}\iiint_{\Omega}\rho e_o d\mathscr{V}=\iiint_{\Omega}\frac{\partial}{\partial t}(\rho e_o) d\mathscr{V}\]\\
<math display="block">
\frac{d}{dt}\iiint_{\Omega}\rho e_o dV=\iiint_{\Omega}\frac{\partial}{\partial t}(\rho e_o) dV
</math>


\noindent The energy equation can now be written as\\
The energy equation can now be written as


\[\iiint_{\Omega}\left[\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) - \rho\mathbf{f}\cdot\mathbf{v} - \dot{q}\rho \right]d\mathscr{V}=0\]\\
<math display="block">
\iiint_{\Omega}\left[\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) - \rho\mathbf{f}\cdot\mathbf{v} - \dot{q}\rho \right]dV=0
</math>


\noindent $\Omega$ is an arbitrary control volume and thus\\
<math>\Omega</math> is an arbitrary control volume and thus


\begin{equation}
<math display="block">
\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho
\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho
\label{eq:governing:energy:pde}
</math>
\end{equation}\\


\noindent which is the energy equation on partial differential form\\
which is the energy equation on partial differential form


==== Summary ====
==== Summary ====
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