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Line 9: |
Line 9: |
| </math><br> | | </math><br> |
| <math> | | <math> |
| \frac{\partial \rho }{\partial t} + \frac{\partial}{\partial x} \left( u(E+p) \right) + \frac{\partial}{\partial y} \left( v(E+p) \right) =0 \, . | | \frac{\partial E }{\partial t} + \frac{\partial}{\partial x} \left( u(E+p) \right) + \frac{\partial}{\partial y} \left( v(E+p) \right) =0 \, . |
| </math><br> | | </math><br> |
| A suitable equation of state used to close the system is given by <br> | | A suitable equation of state used to close the system is given by <br> |
Latest revision as of 16:00, 17 May 2011
The compressible Euler equations for gas dynamics are (mass, momentum, internal energy)
![{\displaystyle {\frac {\partial (\rho u)}{\partial t}}+{\frac {\partial }{\partial x}}\left(\rho u^{2}+p\right)+{\frac {\partial }{\partial y}}\left(\rho uv\right)=0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cfe8107e9648f4d6fc4447da93c9f4b1675fc61)
![{\displaystyle {\frac {\partial (\rho v)}{\partial t}}+{\frac {\partial }{\partial x}}\left(\rho uv\right)+{\frac {\partial }{\partial y}}\left(\rho v^{2}+p\right)=0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59370cd852d2a9183e2d655ec9404293d1f6de22)
![{\displaystyle {\frac {\partial E}{\partial t}}+{\frac {\partial }{\partial x}}\left(u(E+p)\right)+{\frac {\partial }{\partial y}}\left(v(E+p)\right)=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87a271827fe14773f87754f96322b06789f86b2f)
A suitable equation of state used to close the system is given by
![{\displaystyle E={\frac {p}{\gamma -1}}+{\frac {\rho }{2}}\left(u^{2}+v^{2}\right)\;,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca645491db78d8c28b8db593e4f9c9f259a88fa)
where typically
for a monoatomic gas.