Athough the Euler equations formally reduce to potential flow in the limit of vanishing Mach number, this is not helpful in practice, essentially because the approximation of incompressibility is almost invariably very close. They are:

the fluid density. The second equation includes the divergence of a dyadic tensor, and may be clearer in subscript notation:

Note that the above equations are expressed in conservation form, as this format emphasises their physical origins (and is by far the most convenient form for computational fluid dynamics simulations). The momentum component of the Euler equations is usually expressed as follows:

The equations above thus represent conservation of mass, three components of momentum, and energy. There are thus five equations and six unknowns. Closing the system requires an equation of state; the most commonly used one would be the ideal gas law (ie ).

Note the odd form for the energy equation; see Rankine-Hugoniot equation. The extra terms involving *p* may be interpreted as the mechanical work done on a fluid element by nearby fluid elements moving around. These terms sum to zero in an incompressible fluid.

Apart from shocks, there appear to be no non-trivial analytical solutions to the Euler equations. Indeed, it is not clear to this author that a shock is a solution: the Euler equations as stated above do not admit discontinuities; resolving the shock's structure requires the bulk viscosity, but Knudsen number effects clearly dominate at those lengthscales.

The more well known Bernoulli's equation can be derived by integrating Euler's equation along a streamline under the assumption of constant density and a sufficiently stiff equation of state.