Given rotational symmetry around X axis, how can XY possibly differ from XZ?
 

I have a 3D model of a sphere in a cylindrical fluid channel. The sphere is at the dead center of the channel. Fluid flow is along the X axis, and this is Stokes flow, so no turbulence. The model is not set up using an actual symmetry function, but it is inherently symmetric about X.

It would seem to me that in this case, the XY plane and the XZ plane are exactly the same -- rotate the model 90 degrees in the proper direction, replace "Y" with "Z", and theta with phi, in the equations, and you literally should not be able to tell the difference.

But, it doesn't work. I can solve for translational velocity in X, Y and Z, and I can solve for rotational velocity in XY. But, if I try to solve for rotation in XZ (which should essentially be 0, and is essentially 0 in XY), I get crazy solutions (sometimes > 10,000 rad/sec). Additionally, the XZ solutions bounce around randomly if I change the mesh settings, whereas the XY solutions do not.

How is this possible?? Is there some formalism, perhaps of vector direction or sign, that makes Z different than Y?

Here is a screenshot of the geometry, most of the variables that go into it, and the results at the bottom when solving for translation in all axis, and rotation in XY:

https://www.dropbox.com/s/l8zmmrx2wo...hot-1.png?dl=0

https://www.dropbox.com/s/l8zmmrx2wo...hot-1.png?dl=0

Here is the result when I also try to solve for rotation in XZ (note that it is not the combination that is the problem -- solving for only XZ doesn't work either, just showing a combined version to illustrate the equations I am using):

https://www.dropbox.com/s/601uhx38xr...hot-2.png?dl=0

http://www.dropbox.com/s/601uhx38xrz...hot-2.png?dl=0

I fixed this, but although my equations were wrong, they were consistent in XY versus XZ, so I still wonder why the two planes of rotation gave totally different answers.