In fluid dynamics, the Euler equations govern the compressible, Inviscid flow. They correspond to the Navier-Stokes equations with zero viscosity and heat conduction terms, 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 after Leonhard Euler. This page assumes that classical mechanics applies; see relativistic Euler equations for a discussion of compressible fluid flow when velocities approach the speed of light. Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... In thermodynamics and fluid mechanics, compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. ... A fluid flow where viscous (friction) forces are small in comparison to inertial forces is said to be inviscid. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ... Viscosity is a measure of the resistance of a fluid to deform under shear stress. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity. ...

Although 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. In differential form, the equations are: A potential flow is characterized by an irrotational velocity field. ... An F/A-18 Hornet breaking the sound barrier. ...

where E = ρe + ρ(u² + v² + w²) / 2 is the total energy per unit volume (e is the internal energy per unit mass for the fluid), u, v and w are the velocity components, p is the pressure, u the fluid velocity and ρ the fluid density. The second equation includes the divergence of a dyadic tensor, and may be clearer in subscript notation: In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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: Conservation form refers to an arrangement of an equation or system of equations, usually representing a physical system, that show that a property represented is conserved by making overall change equal to zero. ... A computer simulation of high velocity air flow around the Space Shuttle during re-entry. ...

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 F,G,H are fluxes. flux in science and mathematics. ...

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 is the ideal gas law (i.e. p = ρ(γ − 1)e, where ρ is the density, γ the adiabatic index, and e the internal energy). In physics and thermodynamics, an equation of state is a constitutive equation describing the state of matter under a given set of physical conditions. ... Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ...

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. The Rankine-Hugoniot equation governs the behaviour of shock waves normal to the oncoming flow. ...

The better 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. In fluid dynamics, Bernoullis equation, derived by Daniel Bernoulli, describes the behavior of a fluid moving along a streamline. ... Solid blue lines and broken grey lines represent the streamlines. ...