Editing Euler equations (fluid dynamics) (section)

==Euler equations==
In differential convective form, the compressible (and most general) Euler equations can be written shortly with the [[material derivative]] notation:

{{Equation box 1
|indent=:
|title='''Euler equations'''<br/>(''convective form'')
|equation=<math>\begin{align} 
        {D\rho \over Dt} &= -\rho\nabla \cdot \mathbf{u} \\[1.2ex]
  \frac{D\mathbf{u}}{Dt} &= -\frac{\nabla p}{\rho} + \mathbf{g} \\[1.2ex]
           {De \over Dt} &= -\frac{p}{\rho}\nabla \cdot \mathbf{u}
\end{align}</math>
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}}

where the additional variables here is:
*<math>e</math> is the specific [[internal energy]] (internal energy per unit mass).

The equations above thus represent [[conservation of mass]], [[conservation of momentum|momentum]], and [[conservation of energy|energy]]: the energy equation expressed in the variable internal energy allows to understand the link with the incompressible case, but it is not in the simplest form.
Mass density, flow velocity and pressure are the so-called ''convective variables'' (or physical variables, or lagrangian variables), while mass density, momentum density and total energy density are the so-called ''[[conserved variable]]s'' (also called eulerian, or mathematical variables).{{sfn|Toro|1999|p= 24}}

If one expands the material derivative, the equations above become:
<math display="block">\begin{align} 
        {\partial\rho \over \partial t} + \mathbf{u} \cdot \nabla\rho + \rho\nabla \cdot \mathbf{u} &= 0,\\[1.2ex]
  \frac{\partial\mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla\mathbf{u} + \frac{\nabla p}{\rho} &= \mathbf{g}, \\[1.2ex]
  {\partial e \over \partial t} + \mathbf{u} \cdot \nabla e + \frac{p}{\rho}\nabla \cdot \mathbf{u} &= 0.
\end{align}</math>

===Incompressible constraint (revisited)===
Coming back to the incompressible case, it now becomes apparent that the ''incompressible constraint'' typical of the former cases actually is a particular form valid for incompressible flows of the ''energy equation'', and not of the mass equation. In particular, the incompressible constraint corresponds to the following very simple energy equation:
<math display="block">\frac{D e}{D t} = 0.</math>

Thus '''for an incompressible inviscid fluid the specific internal energy is constant along the flow lines''', also in a time-dependent flow. The pressure in an incompressible flow acts like a [[Lagrange multiplier]], being the multiplier of the incompressible constraint in the energy equation, and consequently in incompressible flows it has no thermodynamic meaning. In fact, thermodynamics is typical of compressible flows and degenerates in incompressible flows.{{sfn|Quartapelle|Auteri|2013|p=13|loc=Ch. 9}}

Basing on the mass conservation equation, one can put this equation in the conservation form:
<math display="block">{\partial \rho e \over \partial t} + \nabla \cdot (\rho e \mathbf u) = 0, </math>
meaning that for an incompressible inviscid nonconductive flow a continuity equation holds for the internal energy.

===Enthalpy conservation===
Since by definition the specific enthalpy is:
<math display="block">h = e + \frac p \rho.</math>

The material derivative of the specific internal energy can be expressed as:
<math display="block">{D e \over Dt} = {D h \over Dt} - \frac 1 \rho \left({D p \over Dt} - \frac p \rho {D \rho \over Dt} \right).</math>

Then by substituting the momentum equation in this expression, one obtains:
<math display="block">{D e \over Dt}= {D h \over Dt} - \frac 1 \rho \left(p \nabla \cdot \mathbf u + {D p \over Dt} \right).</math>

And by substituting the latter in the energy equation, one obtains that the enthalpy expression for the Euler energy equation:
<math display="block">{D h \over Dt} = \frac 1 \rho {D p \over Dt}. </math>
In a reference frame moving with an inviscid and nonconductive flow, the variation of enthalpy directly corresponds to a variation of pressure.

===Thermodynamics of ideal fluids===
In [[thermodynamics]] the independent variables are the [[specific volume]], and the [[specific entropy]], while the [[specific energy]] is a [[function of state]] of these two variables.

{{hidden
|Deduction of the form valid for thermodynamic systems
|Considering the first equation, variable must be changed from density to specific volume. By definition:
<math display="block"> v \equiv \frac 1 \rho </math>

Thus the following identities hold:
<math display="block"> \nabla \rho = \nabla \left(\frac{1}{v}\right) = -\frac{1}{v^2} \nabla v</math>
<math display="block"> \frac{\partial\rho}{\partial t} = \frac{\partial}{\partial t} \left(\frac{1}{v}\right) = -\frac{1}{v^2} \frac{\partial v}{\partial t} </math>

Then by substituting these expressions in the mass conservation equation:
<math display="block"> - \frac{\mathbf{u}}{v^2} \cdot \nabla v - \frac 1 {v^2} \frac {\partial v}{\partial t} = - \frac 1 v \nabla \cdot \mathbf{u} </math>

And by multiplication:
<math display="block"> {\partial v \over\partial t}+\mathbf u \cdot \nabla v = v \nabla \cdot \mathbf u </math>

This equation is the only belonging to general continuum equations, so only this equation have the same form for example also in Navier-Stokes equations.

On the other hand,  the pressure in thermodynamics is the opposite of the partial derivative of the specific internal energy with respect to the specific volume:
<math display="block">p(v, s) = - {\partial e(v, s) \over \partial v}</math>
since the internal energy in thermodynamics is a function of the two variables aforementioned, the pressure gradient contained into the momentum equation should be explicited as:
<math display="block">- \nabla p (v,s) = - \frac {\partial p}{\partial v} \nabla v - \frac {\partial p}{\partial s} \nabla s = \frac {\partial^2 e}{\partial v^2} \nabla v +  \frac {\partial^2 e}{\partial v \partial s}\nabla s </math>

It is convenient for brevity to switch the notation for the second order derivatives:
<math display="block"> - \nabla p (v,s) =  e_{vv} \nabla v + e_{vs} \nabla s </math>

Finally, the energy equation:
<math display="block">{D e \over Dt} = - p v \nabla \cdot \mathbf u </math>
can be further simplified in convective form by changing variable from specific energy to the specific entropy: in fact the [[first law of thermodynamics]] in local form can be written:
<math display="block">{D e \over Dt} = T {D s \over Dt} - p {D v \over Dt}</math>
by substituting the material derivative of the internal energy, the energy equation becomes:
<math display="block">T {D s \over Dt} + \frac p {\rho^2} \left( {D \rho \over Dt} + \rho \nabla \cdot \mathbf u \right) = 0</math>
now the term between parenthesis is identically zero according to the conservation of mass, then the Euler energy equation becomes simply:
<math display="block">{D s \over Dt} = 0</math>

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For a thermodynamic fluid, the compressible Euler equations are consequently best written as:

{{Equation box 1
|indent=:
|title='''Euler equations'''<br/>(''convective form, for a thermodynamic system'')
|equation=<math>\begin{align} 
           {Dv \over Dt} &= v \nabla \cdot \mathbf u\\[1.2ex]
  \frac{D\mathbf{u}}{Dt} &= ve_{vv}\nabla v + ve_{vs}\nabla s + \mathbf{g} \\[1.2ex]
           {Ds \over Dt} &= 0
\end{align}</math>
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|border colour = #FFFF00
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}}

where:
* <math>v</math> is the specific volume
* <math>\mathbf u</math> is the flow velocity vector
* <math>s</math> is the specific entropy

In the general case and not only in the incompressible case, the energy equation means that '''for an inviscid thermodynamic fluid the specific entropy is constant along the [[flow lines]]''', also in a time-dependent flow. Basing on the mass conservation equation, one can put this equation in the conservation form:{{sfn|Landau|Lifshitz|2013|p=4|loc= Eqs 2.6 and 2.7}}
<math display="block">{\partial \rho s \over \partial t} + \nabla \cdot (\rho s \mathbf u) = 0,</math>
meaning that for an inviscid nonconductive flow a continuity equation holds for the entropy.

On the other hand, the two second-order partial derivatives of the specific internal energy in the momentum equation require the specification of the [[fundamental equation of state]] of the material considered, i.e. of the specific internal energy as function of the two variables specific volume and specific entropy:
<math display="block">e = e(v, s).</math>

The ''fundamental'' equation of state contains all the thermodynamic information about the system (Callen, 1985),{{sfn|Henderson|2000|p=152|loc=2.6 Thermodynamic properties of materials}} exactly like the couple of a ''thermal'' equation of state together with a ''caloric'' equation of state.

===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>
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|border colour = #FFFF00
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}}

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.