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In this video, around 26 minutes and 30 seconds, you can see that a fluid (whose velocity is made visible) in a Couette cell shows nice patterns at certain rotation velocities, while between these velocities chaotic behavior takes over. I searched the net how these patterns come about, but couldn't find an answer. I'm sure the Reynold number is somehow involved, but I can't figure out the cause of what is shown. So my question is obvious: what explains the behavior shown?

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The flow is the response of the fluid to the imposed boundary conditions. The basic pattern is called Taylor-Couette flow. As for the changes in the observed patterns, there are called bifurcations, where, as a parameter is varied, the dynamical behavior (corresponding to a given pattern) becomes unstable and is substituted by a different one.

For many of those experiments, the control parameters were indeed the Reynolds numbers associated with the rotation speed of the cylinders. For details you could check some of the relevant papers from Harry Swinney, for example Bifurcation to periodic, quasiperiodic, and chaotic regimes in rotation and convecting fluids, the book chapter Instabilities and transition in flow between concentric rotating cylinders, or the less old Flow regimes in a circular Couette system with independently rotating cylinders (e-print).

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