Governing equations on differential form: Difference between revisions
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The continuity equation on integral form reads | The continuity equation on integral form reads | ||
{{ | {{InfoBox|<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>}} | ||
Apply Gauss's divergence theorem on the surface integral gives | Apply Gauss's divergence theorem on the surface integral gives | ||
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The momentum equation on integral form reads | The momentum equation on integral form reads | ||
{{ | {{InfoBox|<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>}} | ||
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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The energy equation on integral form reads | The energy equation on integral form reads | ||
{{ | {{InfoBox|<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>}} | ||
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 | ||
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<div style="border: solid 1px;"> | <div style="border: solid 1px;"> | ||
{{ | {{OpenInfoBox|<math> | ||
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0 | \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0 | ||
</math> | </math>|description=Continuity:}} | ||
{{ | {{OpenInfoBox|<math> | ||
\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} | ||
</math> | </math>|description=Momentum:}} | ||
{{ | {{OpenInfoBox|<math> | ||
\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 | ||
</math> | </math>|description=Energy:}} | ||
</div> | </div> | ||
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<div style="border: solid 1px;"> | <div style="border: solid 1px;"> | ||
{{ | {{OpenInfoBox|<math> | ||
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 | \frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 | ||
</math> | </math>|description=Continuity:}} | ||
{{ | {{OpenInfoBox|<math> | ||
\frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f} | \frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f} | ||
</math> | </math>|description=Momentum:}} | ||
{{ | {{OpenInfoBox|<math> | ||
\rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | \rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | ||
</math> | </math>|description=Energy:}} | ||
</div> | </div> | ||
