Eulerekvationerna

Eulerekvationerna are a set of fundamental conservation laws in fluid dynamics that describe the motion of an inviscid (zero viscosity) and compressible fluid. Named after Leonhard Euler, they form the inviscid limit of the more general Navier–Stokes equations and are widely used to model high-speed or externally dominated flows where viscous and conductive effects are small.

In three spatial dimensions, the Euler equations can be written in conservative form for the density ρ, velocity u, pressure p, and total energy per unit volume E. The equations are: continuity: ∂ρ/∂t + ∇·(ρu) = 0; momentum: ∂(ρu)/∂t + ∇·(ρu⊗u) + ∇p = 0; energy: ∂E/∂t + ∇·((E+p)u) = 0, with E = ρe + 1/2 ρ|u|^2, where e is the specific internal energy. Closure of the system requires an equation of state relating p, ρ, and e, such as for an ideal gas p = (γ−1)ρe or p = ρRT, where γ is the heat capacity ratio.

The Euler equations are hyperbolic, supporting waves such as pressure (sound) waves, contact discontinuities, and shock waves. They are frequently solved numerically in computational fluid dynamics (CFD) using Riemann solvers and finite-volume methods. While powerful for many problems, they neglect viscous effects and heat conduction, so they are often used in conjunction with or as the basis for more complete models like the Navier–Stokes equations.

In one dimension, the equations simplify to a reduced set commonly employed in gas dynamics to study shock waves, expansion fans, and other compressible-flow phenomena.