Euler equations

In fluid dynamics, the Euler equations govern the motion of a compressible, inviscid fluid. They are chiefly interesting when the Mach number exceeds unity and are useful in the investigation of shock waves. They correspond to the Navier-Stokes equations with zero viscosity, although they are usually written in the form shown here because this emphasises the fact that they directly represent conservation of mass, momentum, and energy. The equations are named for Leonard Euler.

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:

where is the total energy per unit volume ( is the internal energy per unit mass for the fluid), is the pressure, the fluid velocity and

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:

but this form obscures the direct connection between the Euler equations and Newton's second law of motion (in particular, it is not intuitively clear why this equation is correct and is incorrect). In conservation vector form, Euler equations become

where

This form makes it clear that are fluxes.

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.