Governing equations on integral form: Difference between revisions
From Flowpedia
No edit summary |
No edit summary |
||
| (4 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
[[Category:Compressible flow]] | <!-- | ||
[[Category:Governing equations]] | -->[[Category:Compressible flow]]<!-- | ||
<noinclude><!-- | -->[[Category:Governing equations]]<!-- | ||
--><noinclude><!-- | |||
-->[[Category:Compressible flow:Topic]]<!-- | -->[[Category:Compressible flow:Topic]]<!-- | ||
--></noinclude><!-- | --></noinclude><!-- | ||
| Line 28: | Line 29: | ||
==== The Continuity Equation ==== | ==== The Continuity Equation ==== | ||
{{ | {{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> | ||
| Line 64: | Line 65: | ||
==== The Momentum Equation ==== | ==== The Momentum Equation ==== | ||
{{ | {{QuoteBox|The time rate of change of momentum of a body equals the net force exerted on it}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
| Line 131: | Line 132: | ||
==== The Energy Equation ==== | ==== The Energy Equation ==== | ||
{{ | {{QuoteBox|Energy can be neither created nor destroyed; it can only change in form}} | ||
<math display="block"> | <math display="block"> | ||
| Line 230: | Line 231: | ||
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;"> | |||
{{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>}} | </math>|description=Continuity:}} | ||
{{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>}} | </math>|description=Momentum:}} | ||
{{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>}} | </math>|description=Energy:}} | ||
</div> | |||
