Editing Euler equations (fluid dynamics) (section)

{{Short description|Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow}}
{{About|Euler equations in classical fluid flow||List of topics named after Leonhard Euler}}
{{Hatnote|This page assumes that [[classical mechanics]] applies; For a discussion of compressible fluid flow when velocities approach the [[speed of light]] see [[relativistic Euler equations]].}}

[[File:Flow around a wing.gif|thumb|Flow around a wing. This incompressible flow satisfies the Euler equations.]]
In [[fluid dynamics]], the '''Euler equations''' are a set of [[partial differential equation]]s governing [[adiabatic process|adiabatic]] and [[inviscid flow]]. They are named after [[Leonhard Euler]]. In particular, they correspond to the [[Navier–Stokes equations]] with zero [[viscosity]] and zero [[thermal conductivity]].{{sfn|Toro|1999|p= 24}}

The Euler equations can be applied to [[incompressible flow|incompressible]] and [[compressible flow]]s. The incompressible Euler equations consist of [[Cauchy momentum equation|Cauchy equations]] for conservation of mass and balance of momentum, together with the incompressibility condition that the [[flow velocity]] is  [[solenoidal field|divergence-free]]. The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable [[constitutive equation]] for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. However, fluid dynamics literature often refers to the full set of the compressible Euler equations – including the energy equation – as "the compressible Euler equations".{{sfn|Anderson|1995|p=}}

The mathematical characters of the incompressible and compressible Euler equations are rather different. For constant fluid density, the incompressible equations can be written as a quasilinear [[advection]] equation for the fluid velocity together with an elliptic [[Poisson's equation]] for the pressure. On the other hand, the compressible Euler equations form a quasilinear [[hyperbolic partial differential equation|hyperbolic]] system of [[conservation equation]]s.

The Euler equations can be formulated in a "convective form" (also called the "[[Lagrangian and Eulerian specification of the flow field|Lagrangian form]]") or a "conservation form" (also called the "[[Lagrangian and Eulerian specification of the flow field|Eulerian form]]"). The convective form emphasizes changes to the state in a frame of reference moving with the fluid. The conservation form emphasizes the mathematical interpretation of the equations as conservation equations for a control volume fixed in space (which is useful
from a numerical point of view).