Conservation and Vector Form
In vector and conservation form, the Euler equations become:
where
This form makes it clear that fx, fy and fz 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 is 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 its neighbor fluid elements. These terms sum to zero in an incompressible fluid.
The well-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.
Read more about this topic: Euler Equations (fluid Dynamics)
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