Governing equations on differential form: Difference between revisions

From Flowpedia
← Older editNewer edit →
Nian (talk | contribs)
Bureaucrats, Interface administrators, Administrators
580 edits
VisualWikitext
No edit summary
Line 4: Line 4:
__TOC__
__TOC__


\section{Governing Equations on Differential Form}
=== Conservation of Mass ===


\subsection{Conservation of Mass}
Apply Gauss's divergence theorem on the surface integral in Eqn. \ref{eq:governing:cont:int} gives


\noindent Apply Gauss's divergence theorem on the surface integral in Eqn. \ref{eq:governing:cont:int} gives\\
<math display="block">
\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=\iiint_{\Omega}\nabla\cdot(\rho\mathbf{v})dV
</math>


\[\oiint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=\iiint_{\Omega}\nabla\cdot(\rho\mathbf{v})d\mathscr{V}\]\\
Also, if <math>\Omega</math> is a fixed control volume


\noindent Also, if $\Omega$ is a fixed control volume\\
<math display="block">
\frac{d}{dt}\iiint_{\Omega} \rho dV=\iiint_{\Omega} \frac{\partial \rho}{\partial t} dV
</math>


\[\frac{d}{dt}\iiint_{\Omega} \rho d\mathscr{V}=\iiint_{\Omega} \frac{\partial \rho}{\partial t} d\mathscr{V}\]\\
The continuity equation can now be written as a single volume integral.


\noindent The continuity equation can now be written as a single volume integral.\\
<math display="block">
\iiint_{\Omega} \left[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})\right]dV=0
</math>


\[\iiint_{\Omega} \left[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})\right]d\mathscr{V}=0\]\\
<math>\Omega</math> is an arbitrary control volume and thus


\noindent $\Omega$ is an arbitrary control volume and thus\\
<math display="block">
 
\begin{equation}
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0
\label{eq:governing:cont:pde}
</math>
\end{equation}\\


\noindent which is the continuity equation on partial differential form.\\
which is the continuity equation on partial differential form.


\subsection*{Conservation of Momentum}
=== Conservation of Momentum ===


As for the continuity equation, the surface integral terms are rewritten as volume integrals using Gauss's divergence theorem.\\
As for the continuity equation, the surface integral terms are rewritten as volume integrals using Gauss's divergence theorem.


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


\[\oiint_{\partial \Omega} p\mathbf{n}dS=\iiint_{\Omega} \nabla pd\mathscr{V}\]\\
<math display="block">
\iint_{\partial \Omega} p\mathbf{n}dS=\iiint_{\Omega} \nabla pdV
</math>


Also, if $\Omega$ is a fixed control volume\\
Also, if <math>\Omega</math> is a fixed control volume


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


\noindent The momentum equation can now be written as one single volume integral\\
The momentum equation can now be written as one single volume integral


\[\iiint_{\Omega}\left[\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p - \rho \mathbf{f}\right]d\mathscr{V}=0\]\\
<math display="block">
\iiint_{\Omega}\left[\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p - \rho \mathbf{f}\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 \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f}  
\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f}  
\label{eq:governing:mom:pde}
</math>
\end{equation}\\


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


\subsection{Conservation of Energy}
=== Conservation of Energy ===


\noindent Gauss's divergence theorem applied to the surface integral term in the energy equation (Eqn. \ref{eq:governing:energy:int}) gives\\
\noindent Gauss's divergence theorem applied to the surface integral term in the energy equation (Eqn. \ref{eq:governing:energy:int}) gives\\
Retrieved from "https://wiki.g3dflow.com/index.php?title=Governing_equations_on_differential_form"