Editing ThreeDimensionalConfigurations/ChallengesPt6 (section)

==Body Frame of Riemann Type I Ellipsoid==
We recognize that, in the ''body frame'' of a Riemann ellipsoid, the surface of the configuration is defined by the following expression:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{x}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 + \biggl( \frac{z}{c}\biggr)^2 </math>
  </td>
</tr>
</table>

===Blue (x = 0) Ellipse===
By setting <math>x = z = 0</math>, we find the point where the y-axis intersects the surface of the ellipsoid, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{y}{b}\biggr)^2 ~~\Rightarrow ~~ y = y_\mathrm{max} = b \, .</math>
  </td>
</tr>
</table>

Similarly, by setting <math>x = y = 0</math>, we find the point where the z-axis intersects the surface of the ellipsoid, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{z}{c}\biggr)^2 ~~\Rightarrow ~~ z = z_\mathrm{max} = c \, .</math>
  </td>
</tr>
</table>

If we only set, <math>x = 0</math>, this expression generates an ellipse in the y-z plane whose semi-axes are <math>(y_\mathrm{max}, z_\mathrm{max}) = (1.25, 0.4703)</math>.  The <math>(y, z)</math> coordinates of individual points along the ellipse can be determined by choosing values of <math>y</math> in the range,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>- y_\mathrm{max} \le y \le + y_\mathrm{max} \, ,</math>
  </td>
</tr>
</table>
then determining the corresponding pair of values of <math>z</math> via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>z_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\pm ~z_\mathrm{max} \biggl[1 - \frac{y^2}{y_\mathrm{max}^2}  \biggr]^{1 / 2}
\, .</math>
  </td>
</tr>
</table>

This ellipse is identified in Figure 2 by the dotted-blue curve.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="2">Figure 2:  &nbsp; Y-Z plane(s) of Riemann Type I Ellipsoid[[File:DataFileButton02.png|right|60px|file = Dropbox/3Dviewers/RiemannModels/RiemannCalculations.xlsx --- worksheet = Feb22]]</td>
</tr>
<tr>
  <td align="center">[[File:YZplane3.png|400px|x = +0.70]]</td>
  <td align="center">[[File:YZplaneXm085.png|400px|x = -0.85]]
</tr>

<tr>
  <td align="center">Blue ellipse (x/a = 0.0); Green ellipse (x/a = + 0.70)</td>
  <td align="center">Blue ellipse (x/a = 0.0); Green ellipse (x/a = - 0.85)</td>
</tr>
</table>

===Green (x/a = 0.7) Ellipse===

Next, let's examine the surface-intersection-ellipse that results from a y-z plane that slices through the ellipsoid at <math>x/a = 0.7</math>.  By setting <math>z = 0</math>, we find the point where the y-axis intersects the surface of the ellipsoid, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1 - (0.7)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{y}{b}\biggr)^2 ~~\Rightarrow ~~ y = y_\mathrm{max} = b [1 - (0.7)^2]^{1 / 2} = 0.89268\, .</math>
  </td>
</tr>
</table>

Similarly, by setting <math>y = 0</math>, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1 - (0.7)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{z}{c}\biggr)^2 ~~\Rightarrow ~~ z = z_\mathrm{max} = c [1 - (0.7)^2]^{1 / 2} = 0.33586\, .</math>
  </td>
</tr>
</table>
The <math>(y, z)</math> coordinates of individual points along this ellipse can be determined, as before, by choosing values of <math>y</math> in the range,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>- y_\mathrm{max} \le y \le + y_\mathrm{max} \, ,</math>
  </td>
</tr>
</table>
then determining the corresponding pair of values of <math>z</math> via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>z_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\pm ~z_\mathrm{max} \biggl[1 - (0.7)^2 - \frac{y^2}{y_\mathrm{max}^2}  \biggr]^{1 / 2}
\, .</math>
  </td>
</tr>
</table>

This ellipse is identified in Figure 2 by the dotted-green curve.  All four of the red arrows (velocity vectors, as  explained below) that are displayed in Figure 2 are anchored on this dotted-green curve; the <math>(x, y, z)_\mathrm{base}</math> coordinates of these anchor positions are listed in the yellow-colored elements of the following Table titled, "Red Arrows."  (There is nothing special about these four chosen anchor positions other than they lie on the dotted-green ellipse.)

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="12">'''Red Arrows'''<br />(Velocity Components in an y-z Plane)</td>
</tr>
<tr>
  <td align="center" rowspan="2">Number</td>
  <td align="center" colspan="4">Base of each Arrow</td>
  <td align="center" colspan="1" rowspan="6" bgcolor="lightgray">&nbsp;</td>
  <td align="center" colspan="2" rowspan="1">Velocity</td>
  <td align="center" colspan="1" rowspan="6" bgcolor="lightgray">&nbsp;</td>
  <td align="center" colspan="3">Arrow Tips</td>
</tr>
<tr>
  <td align="center"><math>x</math></td>
  <td align="center"><math>y</math></td>
  <td align="center"><math>z</math></td>
  <td align="center"><math>\biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2</math></td>
  <td align="center" colspan="1" rowspan="1"><math>\dot{y} = u_2</math></td>
  <td align="center" colspan="1" rowspan="1"><math>\dot{z} = u_3</math></td>
  <td align="center"><math>x</math></td>
  <td align="center"><math>y</math></td>
  <td align="center"><math>z</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="right" bgcolor="yellow">0.7</td>
  <td align="right" bgcolor="yellow">-0.89268</td>
  <td align="right" bgcolor="yellow">0.00000</td>
  <td align="right">1.00000</td>
  <td align="right" rowspan="4">-1.19738 x</td>
  <td align="right" rowspan="4">+0.41285 x</td>
  <td align="right">0.7</td>
  <td align="right">-1.10222</td>
  <td align="right">+0.07225</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="right" bgcolor="yellow">0.7</td>
  <td align="right" bgcolor="yellow">-0.16396</td>
  <td align="right" bgcolor="yellow">+0.33015</td>
  <td align="right">1.00001</td>
  <td align="right">0.7</td>
  <td align="right">-0.37350</td>
  <td align="right">+0.40240</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="right" bgcolor="yellow">0.7</td>
  <td align="right" bgcolor="yellow">+0.81981</td>
  <td align="right" bgcolor="yellow">+0.13291</td>
  <td align="right">1.00000</td>
  <td align="right">0.7</td>
  <td align="right">+0.61027</td>
  <td align="right">+0.20516</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="right" bgcolor="yellow">0.7</td>
  <td align="right" bgcolor="yellow">+0.38258</td>
  <td align="right" bgcolor="yellow">-0.30345</td>
  <td align="right">0.99999</td>
  <td align="right">0.7</td>
  <td align="right">+0.17304</td>
  <td align="right">-0.23121</td>
</tr>
</table>
As a check, we have also included in the "Red Arrows" table a column that tallies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl[ \biggl(\frac{x}{a}\biggr)^2 + 
\biggl(\frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2
\biggr]_\mathrm{base} \, ,</math>
  </td>
</tr>
</table>
which in every case totals 1.000, as it should.

===Velocity Components===

In steady-state, the velocity that is associated with each coordinate location can be ascertained from our [[#Riemann_Flow|above <font color="red">STEP #5</font> discussion of the Riemann Flow]].  Here we are especially interested in the velocity components that are in an x = constant, y-z plane, that is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\dot{y} = u_2 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x 
=
+\biggl[ \frac{(1.25)^2}{1 + (1.25)^2} \biggr] (-1.9637) x
=
-1.19738 x
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\dot{z} = u_3 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x 
=
-\biggl[ \frac{(0.4703)^2}{1 + (0.4703)^2} \biggr] (-2.2794) x
=
+0.41285 x
\, .</math>
  </td>
</tr>
</table>
Given that all of the points along the black-dotted ellipse in our ''Red Arrows'' figure are positioned in the <math>x = 0</math>, y-z plane, we appreciate that <math>u_2 = u_3 = 0</math> at all points along this black ellipse.  But along the green-dotted ellipse, for which <math>x/a = 0.7</math>, each fluid element exhibits a nonzero component of motion in the relevant y-z plane; specifically, for all points along the green ellipse,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>u_2\biggr|_{x=0.7} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-0.83816
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>u_3\biggr|_{x=0.7} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
+0.28899
\, .</math>
  </td>
</tr>
</table>

The pointed tip of each of the four red arrows in our figure is located at a coordinate position, <math>(y_\mathrm{tip}, z_\mathrm{tip})</math>, determined from the following pair of expressions: 

<table border="0" align="center" cellpadding="5">
<tr>
   <td align="center"><math>y_\mathrm{tip}</math></td>
   <td align="center"><math>=</math></td>
   <td align="center"><math>y_\mathrm{base} + \tfrac{1}{4} u_2\biggr|_{x=0.7} = y_\mathrm{base} - \tfrac{1}{4}(0.83817) \, ,</math></td>
</tr>
<tr>
   <td align="center"><math>z_\mathrm{tip}</math></td>
   <td align="center"><math>=</math></td>
   <td align="center"><math>z_\mathrm{base} + \tfrac{1}{4} u_2\biggr|_{x=0.7} = z_\mathrm{base} + \tfrac{1}{4}(0.28899) \, .</math></td>
</tr>
</table>
That is, each arrow illustrates how far a fluid element would travel away from its ''base'' location if it moved at the prescribed velocity for a time, <math>\Delta t = \tfrac{1}{4} \times [\pi G \rho]^{-1 / 2}</math>.  The four red arrows serve to illustrate that, at every point along the dotted-green ellipse, the component of the velocity that lies in the y-z plane is precisely the same, in both magnitude and direction.

[[File:PrimedCoordinates3.png|250px|right|Primed Coordinates]]Now, if we were to examine in a similar manner the component of the fluid motion in any other x = constant, y-z plane, we would find that the red velocity vectors arising from every ''base point'' along the relevant ellipse in this new y-z plane would be the same &#8212; in both magnitude and direction &#8212; around the entire ellipse.  Relative to the (dotted green) ellipse that lies in the x = 0.7, y-z plane, the magnitudes would be different &#8212; larger for larger values of <math>x</math> and smaller for smaller values of <math>x</math> &#8212; however, all of the ''red arrows'' in the new y-z plane would point in the same ''direction'' as the red arrows displayed in Figure 2.

All of the "red arrow" flow-components are tipped up, out of the x-y plane by an angle that is given by the ratio of the pair of velocity components, <math>(u_2, u_3)</math>.  Specifically,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\tan\theta</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{u_3}{u_2} = \frac{+0.41285 x}{- 1.19738 x} = -0.34479 \, ,</math></td>
</tr>
<tr>
  <td align="right"><math>\Rightarrow ~~~ \theta</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>-0.33203</math> radians <math>= -19.024^\circ \, .</math></td>
</tr>
</table>
Borrowing from Figure 1, above, and appreciating that <math>\theta</math> is ''negative'' in the example being used here, we would find that all of the red arrows, in all of the x = constant, y-z planes would lie parallel to the <math>y'</math> axis.  In our Figure 2, above, the black, dashed line serves to illustrate one such <math>y'</math> axis; it has been drawn using the expression,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>z</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>y \tan\theta + z_0 \, ,</math></td>
</tr>
</table>
where we have set <math>\tan\theta = -0.34479</math> and <math>z_0 = 0.12758</math> for <math>-1.25 \le y \le + 1.25</math>.