Editing Euler equations (fluid dynamics) (section)

===Conservation form===
{{See also|Conservation equation|}}
The 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 \\
    \mathbf j \\
    E^t
  \end{pmatrix}; \qquad {\mathbf F} = \begin{pmatrix}
    \mathbf j \\
    \frac 1 \rho \mathbf j \otimes \mathbf j + p \mathbf I \\
    \left(E^t + p\right) \frac{1}{\rho}\mathbf{j}
  \end{pmatrix},
</math>

where:
* <math>\mathbf j = \rho \mathbf u</math> is the [[momentum]] density, a conservation variable.
* <math display="inline">E^t = \rho e + \frac{1}{2} \rho u^2</math> is the [[total energy]] density (total energy per unit volume).

Here <math>\mathbf y</math> has length N + 2 and <math>\mathbf F</math> has size N(N + 2).{{efn|In 3D for example y has length 5, I has size 3×3 and F has size 3×5, so the explicit forms are:
<math display="block">
  {\mathbf y} = \begin{pmatrix} j_1 \\ j_2  \\ j_3 \end{pmatrix}; \quad
  {\mathbf F} = \begin{pmatrix}
                       j_1 &                    j_2 &                    j_3 \\
    \frac{j_1^2}{\rho} + p & \frac{j_1j_2}{\rho}    & \frac{j_1j_3}{\rho}    \\
    \frac{j_1j_2}{\rho}    & \frac{j_2^2}{\rho} + p & \frac{j_2j_3}{\rho}    \\
    \frac{j_3j_1}{\rho}    & \frac{j_3j_2}{\rho}    & \frac{j_3^2}{\rho} + p \\
    \left(E^t + p\right) \frac{j_1}{\rho} &
                            \left(E^t + p\right) \frac{j_2}{\rho} &
                                                     \left(E^t + p\right) \frac{j_3}{\rho}
  \end{pmatrix}.
</math>
}} In general (not only in the Froude limit) Euler equations are expressible as:

{{Equation box 1
 |indent=:
 |title='''Euler equation(s)'''<br/>(''original conservation or Eulerian form'')
 |equation=<math>\frac{\partial}{\partial t}\begin{pmatrix}
    \rho \\
    \mathbf{j} \\
    E^t
  \end{pmatrix} + \nabla \cdot \begin{pmatrix}
    \mathbf{j} \\
    \frac{1}{\rho}\mathbf{j} \otimes \mathbf{j} + p \mathbf{I} \\
    \left(E^t + p\right) \frac{1}{\rho}\mathbf{j}
  \end{pmatrix} = \begin{pmatrix}
    0 \\
    \mathbf f \\
    \frac{1}{\rho}\mathbf{j} \cdot \mathbf{f}
  \end{pmatrix}
</math>
 |cellpadding
 |border
 |border colour = #FF0000
 |background colour = #ECFCF4
}}

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

We remark that also the Euler equation even when conservative (no external field, Froude limit) have '''no [[Riemann invariant]]s''' in general.{{sfn|Chorin|Marsden|2013|p=118|loc=par. 3.2 Shocks}} Some further assumptions are required

However, we already mentioned that for a thermodynamic fluid the equation for the total energy density is equivalent to the conservation equation:
<math display="block">{\partial \over \partial t} (\rho s) + \nabla \cdot (\rho s \mathbf u) = 0.</math>

Then the conservation equations in the case of a thermodynamic fluid are more simply expressed as:

{{Equation box 1
|indent=:
|title='''Euler equation(s)'''<br/>(''conservation form, for thermodynamic fluids'')
|equation=<math>
  \frac{\partial}{\partial t}\begin{pmatrix}\rho  \\  \mathbf{j} \\S \end{pmatrix} + \nabla \cdot \begin{pmatrix}\mathbf{j} \\ \frac{1}{\rho}\mathbf{j} \otimes \mathbf{j} + p\mathbf{I} \\ S\frac{\mathbf{j}}{\rho}\end{pmatrix} =
  \begin{pmatrix}0 \\ \mathbf{f} \\ 0 \end{pmatrix}
</math>
|cellpadding
|border
|border colour = #FFFF00
|background colour = #ECFCF4
}}

where <math>S = \rho s</math> is the entropy density, a thermodynamic conservation variable.

Another possible form for the energy equation, being particularly useful for [[isobaric process|isobarics]], is:
<math display="block">
  \frac{\partial H^t}{\partial t} + \nabla \cdot \left(H^t \mathbf u\right) =
  \mathbf u \cdot \mathbf f - \frac{\partial p}{\partial t},
</math>
where <math display="inline">H^t = E^t + p = \rho e + p + \frac{1}{2} \rho u^2</math> is the total [[enthalpy]] density.