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Because the arms of a spiral galaxy have a cylindrical-linear center, perhaps they have a cyilindrical rotation around their center same as a vortex of smoke? Have similar movements within galaxies been measured?

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Sort of. This is demonstrated clearest in barred spiral galaxies, which make up about 1/2 to 2/3 of all spiral galaxies. A dramatic example is NGC 1365:


Image courtesy of Wikipedia.

Others, such as M95, have spiral arms that wrap even further around while still retaining the central bar:


Original image courtesy of Wikipedia; color added by me in Paint.

In both cases, you can see the bar clearly dominating the central structure of the galaxy.

The thing is, though, that the spiral arms (and central areas) of a galaxy don't rotate as much as the stuff inside them. According to density wave theory, these structures are just regions of greater density, where there is more gas, dust, and stars. These objects can move in and out of the spiral arms and bars. The spiral arms are just the places where there are more stars, gas, and dust than in other places. Same goes for bars.

The arms do rotate, to a certain extend, at something called the pattern speed. This is greater than the speed of some of the stars and less than the speed of other stars - it depends on just where in the arm the stars are.

I have yet to see "pattern speed" applied to bars, but the same phenomena may be at work.

Have similar movements within galaxies been measured?

I suppose that I've answered this already, but I did find a good paper, Gordon (1977). He mentions that some popular techniques involve measuring radio emissions from neutral hydrogen, although he measures carbon monoxide abundances. Both techniques are used to find rotation curves and determine the pattern speed. The paper's 38 years old, though.

A good overview of bar formation and evolution can be found in Bournaud & Combes (2002). You can find another good essay on spiral arms here.

Some other related and helpful questions on Physics:

Here's one from Astronomy:

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  • $\begingroup$ The OP just edited, and I think (though it's rather unclear) the question is about whether the arms are rolling around their long axis (independent of their traveling around the galaxy). $\endgroup$
    – user10851
    Jan 9, 2016 at 23:40
  • $\begingroup$ @ChrisWhite Ah, that would be interesting (thanks for the update notification). I kind of doubt that that's been studied, or if a mechanism for that has been identified (if it's been observed), if one exists. But I'm not familiar with the literature. $\endgroup$
    – HDE 226868
    Jan 9, 2016 at 23:49
  • $\begingroup$ Since density wave theory works in 2D, it's not really concerned with this sort of thing. I think the fact that these are density waves rather than matter-transporting waves severs the connection to vortex rings. There might be nonzero coupling to vertical frequency, especially in warped disks, but we've gone pretty far on this tangent already. $\endgroup$
    – user10851
    Jan 9, 2016 at 23:53

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