Euler equations describe a smooth flow. The equations represent conservation of mass (continuity equation), momentum and energy in the fluid. The set of equations are given by:
where is the total energy density. So far we have three equations and four variables, so we need a forth equation. In the case of a compressible fluid we add an equation of state (EOS) of the type . For inccompressible fluids we need to assume that the divergence of the fluid velocity is zero (which is equivalent to consider constant density).
Euler equations can be rewritten in a vectorial form, which is useful to use when solving them numerically:
A final equation is usually included in numerical codes to fully characterized a fluid. Although this is not a fluid equation strictly speaking, it is very useful when computing forces and acceleration in a system. Both quantities can be calculated from the potential field, which thanks to this equation is related to the density of the fluid. It is known as the Poisson equation: