In fluid dynamics, the Euler equations govern the motion of a compressible, inviscid fluid. 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 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 subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances like liquids and gases. ... The pitch drop experiment at the University of Queensland. ... It has been suggested that Leonhard Euler/EB1911 biography be merged into this article or section. ... 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: In fluid dynamics, potential flow, also known as irrotational flow (of incompressible fluids) is steady flow defined by the equations (zero rotation) (zero divergence = volume conservation) Equivalently, where: v is the vector fluid velocity Φ is the fluid flow potential, scalar × is curl · is divergence. ... Mach number (Ma) (pronounced mack in British English and mock in American English) is defined as a ratio of speed to the speed of sound in the medium in case. ...

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), 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 a generalized quantity or a certain kind of geometrical entity that includes all the ideas of scalars, vectors, matrices and linear operators. ...

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: Computational fluid dynamics (CFD) is the use of computers to analyze problems in fluid dynamics. ...

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. In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks. ...

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. ... The ideal gas law or equation is the equation of state of an ideal gas. ...

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. ...

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. ... In fluid dynamics, a streamline is a line which is everywhere tangent to the velocity of the flow. ...