Governing equations on integral form: Difference between revisions
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\dfrac{d}{dt}\iiint_{\Omega}\rho\left(e+\dfrac{V^2}{2}\right)dV+\iint_{\partial \Omega}\left[\rho\left(e+\dfrac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math> | \dfrac{d}{dt}\iiint_{\Omega}\rho\left(e+\dfrac{V^2}{2}\right)dV+\iint_{\partial \Omega}\left[\rho\left(e+\dfrac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math> | ||
\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV | \iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV | ||
</math>|align= | </math>|align=center}} | ||
The surface integral in the energy equation may be rewritten as | The surface integral in the energy equation may be rewritten as | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
\iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS | \iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS | ||
</math>|align= | </math>|align=center}} | ||
and with the definition of enthalpy <math>h=e+p/\rho</math>, we get | and with the definition of enthalpy <math>h=e+p/\rho</math>, we get | ||
