Editing Euler equations (fluid dynamics) (section)

===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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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.