Editing Apps/RiemannEllipsoidsCompressible (section)

====<font color="darkblue">Step #2 (simplified):</font>====
Use the continuity equation and the curl of the Euler equation to numerically derive the <i>structure</i> but not the overall magnitude of the velocity flow-field throughout the 3D configuration, assuming <math>\vec{\omega}=\hat{k}\omega</math> and the 3D density distribution is known.  Here are the relevant equations:
<div align="center">
<math>
\frac{\partial}{\partial x}\biggl[ \rho v_x \biggr] + \frac{\partial}{\partial y}\biggl[ \rho v_y \biggr] = 0 .
</math>
</div>
<div align="left">
<math>
~~~~~\hat{i}:~~~~~ v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y} +2\omega v_x  = C_{z1}(x,y);
</math><br />
<math>
~~~~~\hat{j}:~~~~~ v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} -2\omega v_y = C_{z2}(x,y) ;
</math><br />
<math>
~~~~~\hat{k}:~~~~~ \frac{\partial}{\partial x} \biggl[ C_{z1}(x,y) \biggr] = \frac{\partial}{\partial y} \biggl[C_{z2}(x,y)  \biggr] ,
</math>
</div>

Now, for the adopted flow-field of the Riemann ellipsoids, <math>C_{z1}(x,y)</math> is only a function of <math>y</math> and <math>C_{z2}(x,y)</math> is only a function of <math>x</math> &#8212; that is, <math>C_{z1}(x,y) \rightarrow C_{z1}(y)</math> and <math>C_{z2}(x,y) \rightarrow C_{z2}(x)</math>.  Hence, the third (<math>\hat{k}</math>) condition is automatically satisfied.  I don't know whether or not we will find that our more general velocity fields exhibit the same nice character.