Editing ThreeDimensionalConfigurations/ChallengesPt6 (section)

==Riemann Flow==

<font color="red"><b>STEP #5</b></font>

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<tr><th align="center" colspan="2">Example Type I<br />Ellipsoid<br />([[ThreeDimensionalConfigurations/RiemannTypeI#Example_b1.25c0.470|see also]])</th></tr>
<tr>
  <td align="center"><math>~\frac{b}{a} = \frac{a_2}{a_1}</math></td>
  <td align="center">1.25</td>
</tr>
<tr>
  <td align="center"><math>~\frac{c}{a} = \frac{a_3}{a_1}</math></td>
  <td align="center">0.4703</td>
</tr>
<tr>
  <td align="center"><math>~\Omega_2</math></td>
  <td align="center">0.3639</td>
</tr>
<tr>
  <td align="center"><math>~\Omega_3</math></td>
  <td align="center">0.6633</td>
</tr>
<tr>
  <td align="center"><math>~\tan^{-1} \biggl[ \frac{\Omega_3}{\Omega_2} \biggr]</math></td>
  <td align="center">61.25&deg;</td>
</tr>
<tr>
  <td align="center"><math>~\zeta_2</math></td>
  <td align="center">-2.2794</td>
</tr>
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  <td align="center"><math>~\zeta_3</math></td>
  <td align="center">-1.9637</td>
</tr>
<tr>
  <td align="center"><math>~\tan^{-1} \biggl[ \frac{\zeta_3}{\zeta_2} \biggr]</math></td>
  <td align="center">40.74&deg;</td>
</tr>
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  <td align="center"><math>~\beta_+</math></td>
  <td align="center">1.13449 (1.13332)</td>
</tr>
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  <td align="center"><math>~\gamma_+</math></td>
  <td align="center">1.8052</td>
</tr>
</table>
</td>
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</table>

As we have summarized in an [[ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities|accompanying discussion]] of Riemann Type 1 ellipsoids &#8212; see also [[ThreeDimensionalConfigurations/ChallengesPt3#Riemann-Derived_Expressions|our separate discussion]] &#8212; [[Appendix/References#EFE|[<font color="red">EFE</font>] ]]  provides an expression for the velocity vector of each fluid element, given its  instantaneous ''body''-coordinate position (x, y, z) = (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) &#8212; see his Eq. (154), Chapter 7, &sect;51 (p. 156).  As viewed from the rotating ''body'' coordinate frame, the three component expressions are,

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<math>~\dot{x} = u_1 = \boldsymbol{\hat\imath} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 y - \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\dot{y} = u_2 = \boldsymbol{\hat\jmath} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma \Omega_3 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\dot{z} = u_3 = \boldsymbol{\hat{k}} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \beta \Omega_2 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x \, ,</math>
  </td>
</tr>
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<span  id="betagamma">where,</span>

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  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} \, .
</math>
  </td>
</tr>
</table>


In order to transform Riemann's velocity vector from the ''body'' frame (unprimed) to the "tipped orbit" frame (primed coordinates), we use the following mappings of the three unit vectors:
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<td align="left">

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<math>~\boldsymbol{\hat\imath}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\imath'} \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\boldsymbol{\hat\jmath}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath'}\cos\theta - \boldsymbol{\hat{k}'}\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{k}}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath'}\sin\theta + \boldsymbol{\hat{k}'}\cos\theta \, .</math>
  </td>
</tr>
</table>
</td>
<td align="left">

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  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~x' \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~y' \cos\theta - z' \sin\theta \, ,</math>
  </td>
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  <td align="right">
<math>~z - z_0</math>
  </td>
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<math>~\rightarrow</math>
  </td>
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<math>~y' \sin\theta + z'\cos\theta \, .</math>
  </td>
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</table>

</td></tr></table>

In the ''tipped'' frame, we find,
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<math>~\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 (  y'\cos\theta - z'\sin\theta ) 
- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( z_0 + y'\sin\theta + z'\cos\theta  ) \biggr]
+
[\boldsymbol{\hat{k}'}\sin\theta -\boldsymbol{\hat\jmath'}\cos\theta  ] \biggl[ \gamma \Omega_3 x' \biggr]
+
[ \boldsymbol{\hat\jmath'}\sin\theta + \boldsymbol{\hat{k}'}\cos\theta] \biggl[ \beta \Omega_2 x' \biggr]
</math>
  </td>
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&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 (  y'\cos\theta - z'\sin\theta ) 
- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 ( z_0 + y'\sin\theta + z'\cos\theta  ) \biggr]
+
\boldsymbol{\hat\jmath'}  \biggl[\beta \Omega_2 x' \cdot \sin\theta 
- \gamma \Omega_3 x' \cdot \cos\theta \biggr]
+
\boldsymbol{\hat{k}'}\biggl[ \beta \Omega_2 x' \cdot \cos\theta
+
\gamma \Omega_3 x' \cdot \sin\theta \biggr] \, .
</math>
  </td>
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<span id="ThetaDef">In order</span> for the <math>~\boldsymbol{k}'</math> component to be zero in the tipped plane, we must choose the tipping angle such that,
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<math>~\tan\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~
-\frac{\beta\Omega_2}{\gamma \Omega_3} = -0.344793 ~~~\Rightarrow~~~ \theta = -19.0238^\circ \, .
</math>
  </td>
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<div align="center"><font color="red"><b>NO!!!</b> This is wrong!</font></div>

And if we examine the flow only in the tipped x'-y' plane, then we should set <math>~z' = -z_0/\cos\theta</math>.  These two constraints lead to the velocity expression,
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  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 (  y'\cos\theta + z_0\tan\theta ) 
- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 (  y'\sin\theta ) \biggr]
+
\boldsymbol{\hat\jmath'}  \biggl[\beta \Omega_2  \sin\theta 
- \gamma \Omega_3  \cos\theta \biggr]x' \, .
</math>
  </td>
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</table>

</td></tr></table>

And if we examine the flow only in the tipped x'-y' plane, then we should set <math>~z' = 0</math>.  These two constraints lead to the velocity expression,
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<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 (  y'\cos\theta ) 
- \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 (z_0 +  y'\sin\theta ) \biggr]
+
\boldsymbol{\hat\jmath'}  \biggl[\beta \Omega_2  \sin\theta 
- \gamma \Omega_3  \cos\theta \biggr]x' \, .
</math>
  </td>
</tr>
</table>