Hydrodynamics

Euler equations

Euler equations describe a smooth flow. The equations represent conservation of mass (continuity equation), momentum and energy in the fluid. The set of equations are given by:

{\partial\rho\over\partial t}+\nabla\cdot(\rho\vec{v})=0
{\partial(\rho\vec{v})\over\partial t}+\nabla\cdot(\rho \vec{v}\cdot\vec{v})+\nabla p=0
{\partial \rho e\over\partial t}+\nabla\cdot(\vec{v}(\rho e+p))=0

where e=u+\vec{v}^2/2 is the total energy density. So far we have three equations and four variables, so we need a forth equation. In the case of a compressible fluid we add an equation of state (EOS) of the type p=(\gamma -1)\rho u. For inccompressible fluids we need to assume that the divergence of the fluid velocity is zero \nabla u =0 (which is equivalent to consider constant density).

Euler equations can be rewritten in a vectorial form, which is useful to use when solving them numerically:

{\partial \over \partial t} \begin{pmatrix} \rho \\ \rho\vec{v} \\ \rho e \end{pmatrix}  =\nabla \begin{pmatrix} \rho\vec{v} \\ \rho \vec{v}\cdot\vec{v} + p \\ (\rho e+p)\vec{v} \end{pmatrix}

A final equation is usually included in numerical codes to fully characterized a fluid. Although this is not a fluid equation strictly speaking, it is very useful when computing forces and acceleration in a system. Both quantities can be calculated from the potential field, which thanks to this equation is related to the density of the fluid. It is known as the Poisson equation:

\nabla^2\phi=-4\pi G\rho

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