Euler equations

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This page discusses classical compressible fluid flow. For other uses, see Euler function (disambiguation).

In fluid dynamics, the Euler equations govern the motion of a compressible, inviscid fluid. 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.

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:

{\partial\rho\over\partial t}+ \nabla\cdot(\rho\bold u)=0
{\partial\rho{\bold u}\over\partial t}+ \nabla\cdot(\rho \bold u)\bold u+\nabla p=0
{\partial E\over\partial t}+ \nabla\cdot(\bold u(E+p))=0

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:

{\partial\rho u_j\over\partial t}+ {\partial\rho u_i u_j\over\partial x_i}+ {\partial p\over\partial x_j} =0

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:

\rho\left( \frac{\partial}{\partial t}+{\bold u}\cdot\nabla \right){\bold u}+\nabla p=0

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 \left(\partial/{\partial t}+{\bold u}\cdot\nabla\right)(\rho{\bold u})+\nabla p=0 is incorrect). In conservation vector form, Euler equations become

\frac{\partial U}{\partial t}+ \frac{\partial F}{\partial x}+ \frac{\partial G}{\partial y}+ \frac{\partial H}{\partial z}=0

where

U=\begin{pmatrix}\rho \\ \rho u \\ \rho v \\ \rho w \\E\end{pmatrix}\qquad F=\begin{pmatrix}\rho u\\p+\rho u^2\\ \rho uv \\ \rho uw\\u(E+p)\end{pmatrix}\qquad G=\begin{pmatrix}\rho v\\ \rho uv \\p+\rho v^2\\ \rho vw \\v(E+p)\end{pmatrix}\qquad H=\begin{pmatrix}\rho w\\ \rho uw \\ \rho vw \\p+\rho w^2\\w(E+p)\end{pmatrix}.\qquad

This form makes it clear that F,G,H 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 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).

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 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.

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