In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\rho$ and velocity field $\mathbf{v}$. So we need 5 equations. Landau's "Hydrodynamics" states that the equations are:
The mass continuity equation $\frac{\partial \rho}{\partial t}+\nabla (\rho \mathbf{v}) = 0$
The Euler equation (3 components) $\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p$
A statement of the fact, that there is no dissipation of energy $\frac{d s}{d t} = 0$ ($s$ is entropy per unit mass)
My question: how to express the last equation in terms of the actual variables we are using? We have to assume some form of the equation of state for the fluid in order to do it right? Landau in his typical fashion glance over it, assuming the readers' perfect understanding of thermodynamics, which is not my case unfortunately.
PS. The question is a little related to a more general one I posted yesterday: Explicit form of the entropy production in hydrodynamics