Editing ThreeDimensionalConfigurations/Challenges (section)

===Part II===

Consider a steady-state configuration that is the compressible analog of a Riemann S-type ellipsoid; even better, give the configuration a "peanut" shape rather than the shape of an ellipsoid.  As viewed from a frame that is spinning with the configuration's overall angular velocity, <math>~\vec\Omega_f = \boldsymbol{\hat{k}} \Omega_f</math>, generally we expect the configuration's internal (and surface) flow to be represented by a set of nested stream-lines and at every <math>~(x, y)</math> location the fluid's velocity (and its momentum-density vector) will be tangent to the stream-line that runs through that point.  It is customary to represent the stream-function by a scalar quantity, <math>~\Psi(x, y)</math>, appreciating that each stream-line will be defined by a curve, <math>~\Psi = \mathrm{constant}</math>; also, the local spatial gradient of <math>~\Psi(x,y)</math> will be perpendicular to the local stream-line, hence it will be perpendicular to the local velocity vector as well.  If we specifically introduce the stream-function via the relation,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\rho\bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\nabla \times (\boldsymbol{\hat{k}} \Psi) 
=
\boldsymbol{\hat\imath} \biggl[ \frac{\partial \Psi}{\partial y} \biggr]
- \boldsymbol{\hat\jmath} \biggl[  \frac{\partial \Psi}{\partial x}\biggr] \, ,</math>
  </td>
</tr>
</table>

it will display all of the just-described attributes and we are also guaranteed that the steady-state continuity equation will be satisfied everywhere, because the divergence of a curl is always zero.

We also have demonstrated that the vector, <math>~\bold{A}</math>, has the right properties if,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\bold{u} \cdot \nabla \ln \biggl[ \frac{(2\Omega_f + \zeta_z)}{\rho} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>

This means that, at every location in the plane of the fluid flow, the gradient of the ''vortensity'' also must be perpendicular to the velocity vector.  This constraint can be immediately satisfied if we simply demand that the vortensity be a function of the stream-function, <math>~\Psi</math>, that is, we need,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{(2\Omega_f + \zeta_z)}{\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~g(\Psi) \, .</math>
  </td>
</tr>
</table>
In other words, the scalar vortensity is constant along each stream-line.  And, once we have determined the mathematical expression for this function, we will know that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ \boldsymbol{\hat{k}} g(\Psi) ] \times \rho\bold{u} \, ;</math>
  </td>
</tr>
</table>
Furthermore, we should be able to determine the mathematical expression for <math>~F_B(x,y)</math> because,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dF_B}{d\Psi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- g(\Psi) \, .</math>
  </td>
</tr>
</table>
As a check, we should find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla F_B + \bold{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>