Editing Euler equations (fluid dynamics) (section)

===Ideal polytropic gas===
{{See also|ideal gas}}
For an ideal polytropic gas the fundamental [[equation of state]] is:{{sfn|Quartapelle|Auteri|2013|p=A-61|loc=Appendix E}}

<math display="block">e(v, s) = e_0 e^{(\gamma-1)m\left(s-s_0\right)} \left({v_0 \over v}\right)^{\gamma-1},</math>

where <math>e</math> is the specific energy, <math>v</math> is the specific volume, <math>s</math> is the specific entropy, <math>m</math> is the molecular mass, <math>\gamma</math> here is considered a constant ([[polytropic process]]), and can be shown to correspond to the [[heat capacity ratio]]. This equation can be shown to be consistent with the usual equations of state employed by thermodynamics.

{{hidden
|Demonstration of consistency with the thermodynamics of an ideal gas
|By the thermodynamic definition of temperature:

<math display="block">T(e) \equiv {\partial e \over \partial s} = (\gamma - 1) m e</math>

Where the temperature is measured in energy units. At first, note that by combining these two equations one can deduce the '''[[ideal gas law]]''':

<math display="block">p v = m T,</math>

or, in the usual form:

<math display="block">p = n T,</math>

where: <math>n \equiv \frac m v</math> is the number density of the material. On the other hand the ideal gas law is less strict than the original fundamental equation of state considered.

Now consider the molar heat capacity associated to a process ''x'':

<math display="block">c_x = \left(m T {\partial s \over \partial T}\right)_x</math>

according to the first law of thermodynamics:

<math display="block">d e(v,s)=-p dv + T \, ds</math>

it can be simply expressed as:

<math display="block">c_x \equiv m \left({\partial e \over \partial T}\right)_x + m p \left({\partial v \over \partial T}\right)_x</math>

Now inverting the equation for temperature T(e) we deduce that for an ideal polytropic gas the isochoric heat capacity is a constant:

<math display="block">c_v \equiv m \left({\partial e \over \partial T}\right)_v = m {d e \over dT} = \frac {1}{(\gamma -1)}</math>

and similarly for an ideal polytropic gas the isobaric heat capacity results constant:

<math display="block">c_p \equiv m \left({\partial e \over \partial T}\right)_p + m p  \left({\partial v \over \partial T}\right)_p = m {d e \over dT} + p \left({\partial v \over \partial T}\right)_p = \frac {1}{(\gamma -1)} + 1</math>

This brings to two important [[relations between heat capacities]]: the constant gamma actually represents the '''[[heat capacity ratio]]''' in the ideal polytropic gas:

<math display="block">\frac {c_p}{c_v}= \gamma</math>

and one also arrives to the '''Meyer's relation''':

<math display="block">c_p = c_v+1</math>

The specific energy is then, by inverting the relation T(e):

<math display="block">e(T) = \frac {mT} {\gamma - 1} = c_v m T</math>

The specific enthalpy results by substitution of the latter and of the ideal gas law:

<math display="block">h(T) \equiv e(T) + (p v)(T) = c_v m T + m T = c_p m T</math>

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From this equation one can derive the equation for pressure by its thermodynamic definition:

<math display="block">p(v,e) \equiv - {\partial e \over \partial v} = (\gamma - 1) \frac e v.</math>

By inverting it one arrives to the mechanical equation of state:

<math display="block">e(v,p) = \frac {pv}{\gamma - 1}.</math>

Then for an ideal gas the compressible Euler equations can be simply expressed in the ''mechanical'' or ''primitive variables'' specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in convective form they result:

{{Equation box 1
|indent=:
|title='''Euler equations for an ideal polytropic gas'''<br/>(''convective form''){{sfn|Toro|1999|p= 91|loc=par 3.1.2 Nonconservative formulations}}
|equation=<math>\begin{align} 
           {Dv \over Dt} &= v\nabla \cdot \mathbf{u} \\[1.2ex]
  \frac{D\mathbf{u}}{Dt} &= v\nabla p + \mathbf{g} \\[1.2ex]
           {Dp \over Dt} &= -\gamma p\nabla \cdot \mathbf{u}
\end{align}</math>
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and in one-dimensional quasilinear form they results:

<math display="block"> \frac{\partial \mathbf y}{\partial t} + \mathbf A \frac{\partial \mathbf y}{\partial x}  = {\mathbf 0}. </math>

where the conservative vector variable is:

<math display="block">{\mathbf y}=\begin{pmatrix}v\\ u  \\p \end{pmatrix},</math>

and the corresponding jacobian matrix is:{{sfn|Zingale|2013|p=}}{{sfn|Toro|1999|p= 92}}

<math display="block">{\mathbf A}=\begin{pmatrix}u & -v & 0 \\ 0 & u & v \\ 0 & \gamma p & u \end{pmatrix}.</math>