Editing Appendix/Ramblings/HybridSchemeOld (section)

===Alternative Reference Frames===

Now, we might want to examine the time-dependent behavior of a fluid system while viewing the flow from a reference frame that is more or less moving along with the fluid.  This new frame of reference need not be an inertial frame; for example, when studying a rotating fluid, we may want to view the system's evolution from a rotating frame of reference.  This will be accomplished mathematically by adjusting the dynamical equations so that the velocity that appears in the divergence term accounts for the new "frame" velocity field; specifically, we want to replace <math>~\vec{v}</math> with,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\vec{u} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\vec{v} - \vec{v}_\mathrm{frame} \, .
</math>
  </td>
</tr>
</table>
</div>
(Here, we will consider only time-independent functional expressions for the frame velocity, <math>~\vec{v}_\mathrm{frame}</math>.)  Of course, switching to the rotating frame must be done in such a way that the resulting, new PDE describes exactly the same physical behavior of the system as was described by the original equation; that is, the new equation must be derivable from the original one.  

If <math>~\vec{v}_\mathrm{frame}</math> is a divergence-free velocity field, then the transformation is trivial.  For example, if we choose a frame of reference that is rotating uniformly with angular velocity, <math>~\Omega_0</math>, then,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\vec{v}_\mathrm{frame} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat{e}}_\phi (\varpi \Omega_0) \, ,
</math>
  </td>
</tr>
</table>
</div>
and, utilizing cylindrical coordinates,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\nabla\cdot\vec{v}_\mathrm{frame} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{\partial(0)}{\partial \varpi} + \frac{1}{\varpi}\frac{\partial(\varpi \Omega_0)}{\partial \phi}
+ \frac{\partial(0)}{\partial z} = 0 \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot \vec{u}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot [\vec{v} - \vec{v}_\mathrm{frame}]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot \vec{v} \, ,
</math>
  </td>
</tr>
</table>
</div>
so the new generic hyperbolic PDE becomes,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot \vec{u}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, ,
</math>
  </td>
</tr>
</table>
</div>
and we can be confident that this new PDE represents the physics of the system just as well as the original PDE.