Governing equations on integral form: Difference between revisions

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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">
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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>
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