Governing equations on differential form: Difference between revisions
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==== Conservation of Mass ==== | ==== Conservation of Mass ==== | ||
Apply Gauss's divergence theorem on the surface integral | The continuity equation on integral form reads | ||
{{NumEqn|<math> | |||
\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0 | |||
</math>|nonumber=1}} | |||
Apply Gauss's divergence theorem on the surface integral gives | |||
{{NumEqn|<math> | {{NumEqn|<math> | ||
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==== Conservation of Momentum ==== | ==== Conservation of Momentum ==== | ||
The momentum equation on integral form reads | |||
{{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 | |||
</math>|nonumber=1}} | |||
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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==== Conservation of Energy ==== | ==== Conservation of Energy ==== | ||
Gauss's divergence theorem applied to the surface integral term in the energy equation | The energy equation on integral form reads | ||
{{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 | |||
</math>|nonumber=1}} | |||
Gauss's divergence theorem applied to the surface integral term in the energy equation gives | |||
{{NumEqn|<math> | {{NumEqn|<math> | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 | \frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 | ||
</math>}} | </math>|label=eq-pde-noncons-cont}} | ||
{{EquationNote|label=eq-pde-noncons-cont|nopar=1}} says that the mass of a fluid element with a fixed set of fluid particles is constant as the element moves in space. | |||
==== Conservation of Momentum ==== | ==== Conservation of Momentum ==== | ||
