流体力学守恒形式Euler方程（笛卡尔坐标、柱坐标、球坐标）
- 笛卡尔坐标
\[\begin{align}
& \frac{\partial \rho }{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{1}} \right)}{\partial {{x}_{1}}}+\frac{\partial \left( \rho {{v}_{2}} \right)}{\partial {{x}_{2}}}+\frac{\partial \left( \rho {{v}_{3}} \right)}{\partial {{x}_{3}}} \right]=0 \\
& \frac{\partial \left( \rho {{v}_{1}} \right)}{\partial t}+\frac{\partial \left( \rho {{v}_{1}}{{v}_{1}} \right)}{\partial {{x}_{1}}}+\frac{\partial \left( \rho {{v}_{1}}{{v}_{2}} \right)}{\partial {{x}_{2}}}+\frac{\partial \left( \rho {{v}_{1}}{{v}_{3}} \right)}{\partial {{x}_{3}}}=\frac{\partial \left( {{\sigma }_{11}} \right)}{\partial {{x}_{1}}}+\frac{\partial \left( {{\sigma }_{12}} \right)}{\partial {{x}_{2}}}+\frac{\partial \left( {{\sigma }_{13}} \right)}{\partial {{x}_{3}}} \\
& \frac{\partial \left( \rho {{v}_{2}} \right)}{\partial t}+\frac{\partial \left( \rho {{v}_{2}}{{v}_{1}} \right)}{\partial {{x}_{1}}}+\frac{\partial \left( \rho {{v}_{2}}{{v}_{2}} \right)}{\partial {{x}_{2}}}+\frac{\partial \left( \rho {{v}_{2}}{{v}_{3}} \right)}{\partial {{x}_{3}}}=\frac{\partial \left( {{\sigma }_{12}} \right)}{\partial {{x}_{1}}}+\frac{\partial \left( {{\sigma }_{22}} \right)}{\partial {{x}_{2}}}+\frac{\partial \left( {{\sigma }_{23}} \right)}{\partial {{x}_{3}}} \\
& \frac{\partial \left( \rho {{v}_{3}} \right)}{\partial t}+\frac{\partial \left( \rho {{v}_{3}}{{v}_{1}} \right)}{\partial {{x}_{1}}}+\frac{\partial \left( \rho {{v}_{3}}{{v}_{2}} \right)}{\partial {{x}_{2}}}+\frac{\partial \left( \rho {{v}_{3}}{{v}_{3}} \right)}{\partial {{x}_{3}}}=\frac{\partial \left( {{\sigma }_{13}} \right)}{\partial {{x}_{1}}}+\frac{\partial \left( {{\sigma }_{23}} \right)}{\partial {{x}_{2}}}+\frac{\partial \left( {{\sigma }_{33}} \right)}{\partial {{x}_{3}}} \\
& \frac{\partial \left( \rho \mathcal{E} \right)}{\partial t}+\left[ \frac{\partial \left( \rho \mathcal{E}{{v}_{1}} \right)}{\partial {{x}_{1}}}+\frac{\partial \left( \rho \mathcal{E}{{v}_{2}} \right)}{\partial {{x}_{2}}}+\frac{\partial \left( \rho \mathcal{E}{{v}_{3}} \right)}{\partial {{x}_{3}}} \right]=\left[ \frac{\partial \left( \left( {{\sigma }_{11}}{{v}_{1}}+{{\sigma }_{12}}{{v}_{2}}+{{\sigma }_{13}}{{v}_{3}} \right) \right)}{\partial {{x}_{1}}}+\frac{\partial \left( \left( {{\sigma }_{12}}{{v}_{1}}+{{\sigma }_{22}}{{v}_{2}}+{{\sigma }_{23}}{{v}_{3}} \right) \right)}{\partial {{x}_{2}}}+\frac{\partial \left( \left( {{\sigma }_{13}}{{v}_{1}}+{{\sigma }_{23}}{{v}_{2}}+{{\sigma }_{33}}{{v}_{3}} \right) \right)}{\partial {{x}_{3}}} \right] \\
\end{align}
\]
- 柱坐标
\[\begin{align}
& \frac{\partial \left( r\rho \right)}{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{1}}r \right)}{\partial r}+\frac{\partial \left( \rho {{v}_{2}} \right)}{\partial \theta }+\frac{\partial \left( \rho {{v}_{3}}r \right)}{\partial z} \right]=0 \\
& \frac{\partial \left( r\rho {{v}_{1}} \right)}{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{1}}{{v}_{1}}r \right)}{\partial r}+\frac{\partial \left( \rho {{v}_{1}}{{v}_{2}} \right)}{\partial \theta }+\frac{\partial \left( \rho {{v}_{1}}{{v}_{3}}r \right)}{\partial z} \right]-\rho {{v}_{2}}{{v}_{2}}=\left[ \frac{\partial \left( {{\sigma }_{11}}r \right)}{\partial r}+\frac{\partial \left( {{\sigma }_{12}} \right)}{\partial \theta }+\frac{\partial \left( {{\sigma }_{13}}r \right)}{\partial z} \right]-{{\sigma }_{22}} \\
& \frac{\partial \left( r\rho {{v}_{2}} \right)}{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{2}}{{v}_{1}}r \right)}{\partial r}+\frac{\partial \left( \rho {{v}_{2}}{{v}_{2}} \right)}{\partial \theta }+\frac{\partial \left( \rho {{v}_{2}}{{v}_{3}}r \right)}{\partial z} \right]+\rho {{v}_{1}}{{v}_{2}}=\left[ \frac{\partial \left( {{\sigma }_{12}}r \right)}{\partial r}+\frac{\partial \left( {{\sigma }_{22}} \right)}{\partial \theta }+\frac{\partial \left( {{\sigma }_{23}}r \right)}{\partial z} \right]+{{\sigma }_{12}} \\
& \frac{\partial \left( r\rho {{v}_{3}} \right)}{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{1}}{{v}_{3}}r \right)}{\partial r}+\frac{\partial \left( \rho {{v}_{2}}{{v}_{3}} \right)}{\partial \theta }+\frac{\partial \left( \rho {{v}_{3}}{{v}_{3}}r \right)}{\partial z} \right]=\left[ \frac{\partial \left( {{\sigma }_{13}}r \right)}{\partial r}+\frac{\partial \left( {{\sigma }_{23}} \right)}{\partial \theta }+\frac{\partial \left( {{\sigma }_{33}}r \right)}{\partial z} \right] \\
& \frac{\partial \left( r\rho \mathcal{E} \right)}{\partial t}+\left[ \frac{\partial \left( \rho \mathcal{E}{{v}_{1}}r \right)}{\partial r}+\frac{\partial \left( \rho \mathcal{E}{{v}_{2}} \right)}{\partial \theta }+\frac{\partial \left( \rho \mathcal{E}{{v}_{3}}r \right)}{\partial z} \right]=\left[ \frac{\partial \left( \left( {{\sigma }_{11}}{{v}_{1}}+{{\sigma }_{12}}{{v}_{2}}+{{\sigma }_{13}}{{v}_{3}} \right)r \right)}{\partial r}+\frac{\partial \left( \left( {{\sigma }_{12}}{{v}_{1}}+{{\sigma }_{22}}{{v}_{2}}+{{\sigma }_{23}}{{v}_{3}} \right) \right)}{\partial \theta }+\frac{\partial \left( \left( {{\sigma }_{13}}{{v}_{1}}+{{\sigma }_{23}}{{v}_{2}}+{{\sigma }_{33}}{{v}_{3}} \right)r \right)}{\partial z} \right] \\
\end{align}
\]
- 球坐标
\[\begin{align}
& \frac{\partial \left( {{r}^{2}}\sin \theta \rho \right)}{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{1}}{{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( \rho {{v}_{2}}r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( \rho {{v}_{3}}r \right)}{\partial \varphi } \right]=0 \\
& \frac{\partial \left( {{r}^{2}}\sin \theta \rho {{v}_{1}} \right)}{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{1}}{{v}_{1}}{{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( \rho {{v}_{1}}{{v}_{2}}r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( \rho {{v}_{1}}{{v}_{3}}r \right)}{\partial \varphi } \right]-\rho {{v}_{2}}{{v}_{2}}r\sin \theta -\rho {{v}_{3}}{{v}_{3}}r\sin \theta =\left[ \frac{\partial \left( {{\sigma }_{11}}{{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( {{\sigma }_{12}}r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( {{\sigma }_{13}}r \right)}{\partial \varphi } \right]-{{\sigma }_{22}}r\sin \theta -{{\sigma }_{33}}r\sin \theta \\
& \frac{\partial \left( {{r}^{2}}\sin \theta \rho {{v}_{2}} \right)}{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{1}}{{v}_{2}}{{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( \rho {{v}_{2}}{{v}_{2}}r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( \rho {{v}_{2}}{{v}_{3}}r \right)}{\partial \varphi } \right]+\rho {{v}_{1}}{{v}_{2}}r\sin \theta -\rho {{v}_{3}}{{v}_{3}}r\cos \theta =\left[ \frac{\partial \left( {{\sigma }_{12}}{{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( {{\sigma }_{22}}r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( {{\sigma }_{23}}r \right)}{\partial \varphi } \right]+{{\sigma }_{12}}r\sin \theta -{{\sigma }_{33}}r\cos \theta \\
& \frac{\partial \left( {{r}^{2}}\sin \theta \rho {{v}_{3}} \right)}{\partial t}+\left[ \frac{\partial \left( \rho {{v}_{1}}{{v}_{3}}{{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( \rho {{v}_{2}}{{v}_{3}}r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( \rho {{v}_{3}}{{v}_{3}}r \right)}{\partial \varphi } \right]+\rho {{v}_{1}}{{v}_{3}}r\sin \theta +\rho {{v}_{2}}{{v}_{3}}r\cos \theta =\left[ \frac{\partial \left( {{\sigma }_{13}}{{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( {{\sigma }_{23}}r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( {{\sigma }_{33}}r \right)}{\partial \varphi } \right]+{{\sigma }_{13}}r\sin \theta +{{\sigma }_{23}}r\cos \theta \\
& \frac{\partial \left( {{r}^{2}}\sin \theta \rho \mathcal{E} \right)}{\partial t}+\left[ \frac{\partial \left( \rho \mathcal{E}{{v}_{1}}{{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( \rho \mathcal{E}{{v}_{2}}r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( \rho \mathcal{E}{{v}_{3}}r \right)}{\partial \varphi } \right]=\left[ \frac{\partial \left( \left( {{\sigma }_{11}}{{v}_{1}}+{{\sigma }_{12}}{{v}_{2}}+{{\sigma }_{13}}{{v}_{3}} \right){{r}^{2}}\sin \theta \right)}{\partial r}+\frac{\partial \left( \left( {{\sigma }_{12}}{{v}_{1}}+{{\sigma }_{22}}{{v}_{2}}+{{\sigma }_{23}}{{v}_{3}} \right)r\sin \theta \right)}{\partial \theta }+\frac{\partial \left( \left( {{\sigma }_{13}}{{v}_{1}}+{{\sigma }_{23}}{{v}_{2}}+{{\sigma }_{33}}{{v}_{3}} \right)r \right)}{\partial \varphi } \right] \\
\end{align}
\]
