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Superconductivity has limits of currents and magnetic fields they can endure before dropping the superconductivity phase

What are the corresponding limits of superfluidity? what are the limit velocities of superfluid substances relative to the container, above which the fluid stops being superfluid?

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I believe the term is "critical velocity". For liquid He-4, the dispersion relation can be found here: Elementary excitations of superfluid 4He

The critical velocity is usually the lowest slope which intersects with the dispersion relation, since then at that speed one can create excitations that will damp the motion. Notice that for He-4 in particular, because of the roton part of the spectrum, this is lower than the linearised theory would predict.

For rotating fluids, it is also possible to get localised breakdown as in type II superconductors, by threading quanta of flux through the bulk.

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the second point you make is important, as the critical velocity given by the slope condition is too fast by orders of magnitude. –  Ron Maimon Oct 26 '12 at 21:02

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