12 | | || Conservation of Mass || [[latex($\frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x} (\rho v) = 0$)]] || |
13 | | || Conservation of Momentum || [[latex($\frac{\partial}{\partial t} (\rho v) + \frac{\partial}{\partial x} (\rho v^2 + p) = 0$)]] || |
14 | | || Conservation of Energy || [[latex($\frac{\partial}{\partial t} E + \frac{\partial}{\partial x} (v(E+p)) = 0$)]] || |
15 | | Where [[latex($\rho$)]] is mass density, v is velocity (only in x-direction for 1D), p is pressure, and E is total energy per unit volume. E is further defined as [[latex($E = \rho (\frac{1}{2} v^2 + e)$)]] where e is the specific internal energy. The specific internal energy depends on the equation of state. For an ideal gas [[latex($e = \frac{p}{\rho(\gamma - 1)}$)]] where [[latex($\gamma$)]] is the ratio of specific heats. Basically, these laws state that in a given volume, the change in a conserved quantity must be equal to the flux through the boundaries of that volume. In other words, the conserved quantity is in front of the time derivative and its flux is in front of the spatial derivative. |
| 12 | || Conservation of Mass || [[latex(\frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x} (\rho v) = 0)]] || |
| 13 | || Conservation of Momentum || [[latex(\frac{\partial}{\partial t} (\rho v) + \frac{\partial}{\partial x} (\rho v^2 + p) = 0)]] || |
| 14 | || Conservation of Energy || [[latex(\frac{\partial}{\partial t} E + \frac{\partial}{\partial x} (v(E+p)) = 0)]] || |
| 15 | Where [[latex(\rho)]] is mass density, v is velocity (only in x-direction for 1D), p is pressure, and E is total energy per unit volume. E is further defined as [[latex(E = \rho (\frac{1}{2} v^2 + e))]] where e is the specific internal energy. The specific internal energy depends on the equation of state. For an ideal gas [[latex(e = \frac{p}{\rho(\gamma - 1)})]] where [[latex(\gamma)]] is the ratio of specific heats. Basically, these laws state that in a given volume, the change in a conserved quantity must be equal to the flux through the boundaries of that volume. In other words, the conserved quantity is in front of the time derivative and its flux is in front of the spatial derivative. |