Editing PGE/Euler (section)

=Euler Equation=

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<font size="-1">[[H_BookTiledMenu#Context|<b>Euler</b>]]</font>
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==Lagrangian Representation==

===in terms of velocity:===
Among the [[PGE#Principal_Governing_Equations|principal governing equations]] we have included the
&nbsp;<br />
&nbsp;<br />

<div align="center">
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''Lagrangian Representation'''</font></span><br />
of the Euler Equation,

{{Template:Math/EQ_Euler01}}

[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 13, Eq. (1.55)
</div>

===in terms of momentum density:===
<!--
Multiplying this equation through by the mass density {{Template:Math/VAR_Density01}} produces the relation,
<div align="center">
<math>\rho\frac{d\vec{v}}{dt} = - \nabla P - \rho\nabla \Phi</math> ,
</div>
which may be rewritten as,
<div align="center">
<math>\frac{d(\rho\vec{v})}{dt}- \vec{v}\frac{d\rho}{dt} = - \nabla P - \rho\nabla \Phi</math> .
</div>
Combining this with the [[PGE/ConservingMass#ConservingMass:Lagrangian|Standard Lagrangian Representation of the Continuity Equation]], we derive,
<div align="center">
<math>\frac{d(\rho\vec{v})}{dt}+ (\rho\vec{v})\nabla\cdot\vec{v} = - \nabla P - \rho\nabla \Phi</math> .
</div>
-->

Multiplying this equation through by the mass density {{Template:Math/VAR_Density01}} produces the relation,
<table border="0" align="center" cellpadding="5"><tr><td align="left">
<math>\rho\frac{d\vec{v}}{dt} = - \nabla P - \rho\nabla \Phi</math> ,
</td></tr></table>

which may be rewritten as,

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{d(\rho\vec{v})}{dt}- \vec{v}\frac{d\rho}{dt} </math>
  </td>
  <td align="center">
<math>=</math> 
  </td>
  <td align="left">
<math>- \nabla P - \rho\nabla \Phi</math> .
  </td>
</tr>
</table>

Combining this with the [[PGE/ConservingMass#ConservingMass:Lagrangian|Standard Lagrangian Representation of the Continuity Equation]], we derive,
<table border="0" align="center" cellpadding="5"><tr><td align="left">
<math>\frac{d(\rho\vec{v})}{dt}+ (\rho\vec{v})\nabla\cdot\vec{v} = - \nabla P - \rho\nabla \Phi</math> .
</td></tr></table>

==Eulerian Representation==

===in terms of velocity:===

By replacing the so-called Lagrangian (or "material") time derivative <math>d\vec{v}/dt</math> in the Lagrangian representation of the Euler equation by its Eulerian counterpart (see, for example, the wikipedia discussion titled, "[https://en.wikipedia.org/wiki/Material_derivative Material_derivative]", to understand how the Lagrangian and Eulerian descriptions of fluid motion differ from one another conceptually as well as how to mathematically transform from one description to the other), we directly obtain the

<div align="center">
<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,

{{Template:Math/EQ_Euler02}}
</div>

===in terms of momentum density:===

As was done above in the context of the Lagrangian representation of the Euler equation, we can multiply this expression through by {{Template:Math/VAR_Density01}} and combine it with the continuity equation to derive what is commonly referred to as the,

<div align="center">
<span id="ConservingMomentum:Conservative"><font color="#770000">'''Conservative Form'''</font></span><br />
of the Euler Equation,

{{Template:Math/EQ_Euler03}}

[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 8, Eq. (1.31)
</div>

The second term on the left-hand-side of this last expression represents the divergence of the "[https://en.wikipedia.org/wiki/Dyadics dyadic product]" or "[https://en.wikipedia.org/wiki/Outer_product outer product]" of the vector momentum density and the velocity vector, and is sometimes written as, <math>~\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>.

===in terms of the vorticity:===
Drawing on one of the standard [https://en.wikipedia.org/wiki/Vector_calculus_identities#Dot_product_rule dot product rule vector identities], the nonlinear term on the left-hand-side of the Eulerian representation of the Euler equation can be rewritten as,
<div align="center">
<math>
(\vec{v}\cdot\nabla)\vec{v} = \frac{1}{2}\nabla(\vec{v}\cdot\vec{v}) - \vec{v}\times(\nabla\times\vec{v})
= \frac{1}{2}\nabla(v^2) + \vec{\zeta}\times \vec{v} ,
</math>
</div>
where,
<div align="center">
<math>
\vec\zeta \equiv \nabla\times\vec{v}
</math>
</div>
is commonly referred to as the [https://en.wikipedia.org/wiki/Vorticity vorticity].  Making this substitution leads to an expression for the,

<div align="center">
Euler Equation<br />
<span id="ConservingMomentum:Vorticity"><font color="#770000">'''in terms of the Vorticity'''</font></span>,

{{Template:Math/EQ_Euler04}}
</div>

==Double Check Vector Identities==

In a subsection of an accompanying chapter titled, [[AxisymmetricConfigurations/SolutionStrategies#Double_Check_Vector_Identities|''Double Check Vector Identities,'']] we explicitly demonstrate for four separate "simple rotation profiles" that these two separate terms involving a nonlinear velocity expression do indeed generate identical mathematical relations, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\vec\zeta \times \vec{v} + \frac{1}{2}\nabla (v^2) \, ;</math>
  </td>
</tr>
</table>
and we explicitly demonstrate that they are among the set of velocity profiles that can also be expressed in terms of the gradient of a "centrifugal potential," <math>~\nabla\Psi</math>.