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

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