Governing equations on differential form: Difference between revisions
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{{NumEqn|<math> | {{NumEqn|<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>|nonumber=1}} | </math>|nonumber=1|border=1|background-color=whitesmoke|padding=2em}} | ||
Apply Gauss's divergence theorem on the surface integral gives | Apply Gauss's divergence theorem on the surface integral gives | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=\iiint_{\Omega}\nabla\cdot(\rho\mathbf{v})dV | \iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=\iiint_{\Omega}\nabla\cdot(\rho\mathbf{v})dV | ||
</math> | </math>}} | ||
Also, if <math>\Omega</math> is a fixed control volume | Also, if <math>\Omega</math> is a fixed control volume | ||
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{{NumEqn|<math> | {{NumEqn|<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>|nonumber=1}} | </math>|nonumber=1|border=1|background-color=whitesmoke|padding=2em}} | ||
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. | ||
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{{NumEqn|<math> | {{NumEqn|<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>|nonumber=1}} | </math>|nonumber=1|border=1|background-color=whitesmoke|padding=2em}} | ||
Gauss's divergence theorem applied to the surface integral term in the energy equation gives | Gauss's divergence theorem applied to the surface integral term in the energy equation gives | ||
