Fluid dynamics equations, number of variables and number of equations

The missing equation is energy conservation $$ \frac{\partial}{\partial t} {\cal E} + \vec\nabla \cdot\vec\jmath^{\,\cal E}=0 $$ where ${\cal E}$ is the energy density and $\vec\jmath^{\,\cal E}$ is the energy current $$ \vec\jmath^{\,\cal E} = \vec{u}\left[ {\cal E}+P \right] -\eta u\cdot\sigma-\kappa\vec\nabla T\, . $$ Now the equations close if you have an equation of state, $P=P({\cal E}^0,\rho)$, where ${\cal E}^0={\cal E}-\frac{1}{2}\rho u^2$ is the internal energy density. Note that the equation of state also fixes $T({\cal E}^0,\rho)$ using thermodynamic identities (although this is tedious in practice; for a non-interacting gas things are simple, $T=mP/\rho$). The energy equation can be rewritten in various ways, for example as an equation for entropy production.