Self-sustained deformable rotating liquid He cylinders: The pure normal fluid $^3$He and superfluid $^4$He cases

We have studied self-sustained, deformable, rotating liquid He cylinders of infinite length. In the normal fluid $^3$He case, we have employed a classical model where only surface tension and centrifugal forces are taken into account, as well as the Density Functional Theory (DFT) approach in conjunction with a semi-classical Thomas-Fermi approximation for the kinetic energy. In both approaches, if the angular velocity is sufficiently large, it is energetically favorable for the $^3$He cylinder to undergo a shape transition, acquiring an elliptic-like cross section which eventually becomes two-lobed. In the $^4$He case, we have employed a DFT approach that takes into account its superfluid character, limiting the description to vortex-free configurations where angular momentum is exclusively stored in capillary waves on a deformed cross section cylinder. The calculations allow us to carry out a comparison between the rotational behavior of a normal, rotational fluid ($^3$He) and a superfluid, irrotational fluid ($^4$He).