Editing Euler equations (fluid dynamics) (section)

==Incompressible Euler equations==
In convective form the incompressible Euler equations in case of density variable in space are:{{sfn|Hunter|2006|p=}}

{{Equation box 1
|indent=:
|title='''Incompressible Euler equations'''<br/>(''convective or Lagrangian form'')
|equation=<math>\begin{align} 
         {D\rho \over Dt} &= 0\\
   {D\mathbf{u} \over Dt} &= -\frac{\nabla p}{\rho} + \mathbf{g} \\
  \nabla \cdot \mathbf{u} &= 0
\end{align}</math>
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|border colour = #0073CF
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}}

where the additional variables are:
*<math>\rho</math> is the fluid [[mass density]],
*<math>p</math> is the [[pressure]], <math>p = \rho w</math>.

The first equation, which is the new one, is the incompressible [[continuity equation]]. In fact the general continuity equation would be:
<math display="block">{\partial \rho \over\partial t} + \mathbf u \cdot \nabla \rho +  \rho \nabla \cdot \mathbf u  = 0,</math>

but here the last term is identically zero for the incompressibility constraint.

===Conservation form===
{{See also|conservation equation|}}
The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:
<math display="block">
\mathbf y = \begin{pmatrix}\rho  \\  \rho \mathbf u  \\0\end{pmatrix}; \qquad {\mathbf F} = \begin{pmatrix}\rho \mathbf u \\ \rho \mathbf u \otimes \mathbf u + p \mathbf I\\\mathbf u\end{pmatrix}.
</math>

Here <math>\mathbf y</math> has length <math>N+2</math> and <math>\mathbf F</math> has size <math>(N+2)N</math>.{{efn|In 3D for example <math>\mathbf y</math> has length 5, <math>\mathbf I</math> has size 3×3 and <math>\mathbf F</math> has size 5×3, so the explicit forms are:
<math display="block">
{\mathbf y}=\begin{pmatrix}\rho \\  \rho u_1 \\ \rho u_2  \\ \rho u_3  \\0\end{pmatrix}; \quad
{\mathbf F}=\begin{pmatrix}\rho u_1 & \rho u_2 & \rho u_3 \\
\rho u_1^2 + p &  \rho u_1u_2 &  \rho u_1u_3
\\  \rho u_1 u_2 & \rho u_2^2 + p &  \rho u_2u_3
\\ \rho u_3 u_1 &  \rho u_3 u_2 &  \rho u_3^2 + p
\\ u_1 & u_2 & u_3 \end{pmatrix}.
</math>
}}
In general (not only in the Froude limit) Euler equations are expressible as:
<math display="block">
\frac {\partial}{\partial t}\begin{pmatrix}\rho  \\  \rho \mathbf u  \\0\end{pmatrix}+ \nabla \cdot \begin{pmatrix}\rho \mathbf u\\\rho \mathbf u \otimes \mathbf u + p \mathbf I\\ \mathbf u\end{pmatrix} = \begin{pmatrix}0 \\  \rho \mathbf g \\ 0 \end{pmatrix}.
</math>

===Conservation variables===
The variables for the equations in conservation form are not yet optimised. In fact we could define:
<math display="block">
{\mathbf y}=\begin{pmatrix}\rho  \\ \mathbf j  \\0\end{pmatrix}; \qquad {\mathbf F}=\begin{pmatrix} \mathbf j \\ \mathbf j \otimes \frac {1} \rho \, \mathbf j+ p \mathbf I\\ \frac \mathbf j \rho \end{pmatrix},
</math>
where <math>\mathbf j = \rho \mathbf u</math> is the [[momentum]] density, a conservation variable.

{{Equation box 1
|indent=:
|title='''Incompressible Euler equation(s)'''<br/>(''conservation or Eulerian form'')
|equation=<math>
\frac {\partial}{\partial t}\begin{pmatrix}\rho  \\  \mathbf j  \\0\end{pmatrix}+ \nabla \cdot \begin{pmatrix}\mathbf j \\ \mathbf j \otimes \frac 1 \rho \, \mathbf j + p \mathbf I\\ \frac \mathbf j \rho\end{pmatrix} = \begin{pmatrix}0 \\  \mathbf f \\ 0 \end{pmatrix}
</math>
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|border colour = #0073CF
|background colour=#F5FFFA
}}

where <math>\mathbf f = \rho \mathbf g</math> is the [[force density]], a conservation variable.