# The compressible Navier-Stokes equations¶

In two dimensions the compressible Navier-Stokes equations can be written as

$$$\vec{Q}_t + \frac{\partial}{\partial x}(\vec{f}(\vec{Q})+\vec{F}(\vec{Q})) + \frac{\partial}{\partial y} (\vec{g}(\vec{Q})+\vec{G}(\vec{Q})) = 0.$$$

Where the conservative state variables are $$\vec{Q} = [\rho, \rho u, \rho v, E]^T$$ and the hyperbolic fluxes are

$$$\vec{f}(\vec{Q}) = \left( \begin{array}{c} \rho u\\ \rho u^2 + p \\ \rho u v \\ u (E+p) \end{array} \right), \ \ \ \vec{g}(\vec{Q}) = \left( \begin{array}{c} \rho v\\ \rho u v \\ \rho v^2 + p \\ v (E+p) \end{array} \right)$$$

Here we assume an equation of state $$p = (\gamma-1) \rho e$$ and note that we can relate $$E$$ and $$p$$ through $$\rho e = E - \rho (u^2+v^2)/2$$.

The viscous fluxes are:

$$$\vec{F}(\vec{Q}) = - \left( \begin{array}{c} 0\\ \frac{4 \mu}{3} u_x+(\frac{\mu}{3}+\mu_b) v_y \\ \mu (u_y+v_x) \\ u F_2 + v F_3 + \frac{\mu \gamma}{Pr} T_x \end{array} \right), \ \ \ \vec{G}(\vec{Q}) = - \left( \begin{array}{c} 0\\ \mu (u_y+v_x) \\ \frac{4 \mu}{3} v_y + (\frac{mu}{3} + \mu_b) u_x\\ u G_2 + v G_3 + \frac{\mu \gamma}{Pr} T_y \end{array} \right).$$$

Here $$\rho$$ is the density, $$(u,v)$$ is the velocity vector, $$E$$ is the total energy, $$e$$ is the specific internal energy, $$\mu$$ is the dynamic shear viscosity, $$\mu_b$$ is the bulk viscosity, $$\gamma$$ is the ratio of specific heats, $$Pr$$ is the Prandtl number, $$T = (\gamma-1)e/R$$ is the temperature and $$R$$ is the gas constant.