Critical Velocities for Roton and Super-Flow Quantum Turbulence in Liquid ${}^4$He

Two different types of transitions of the superfluid ${}^4$He to quantum turbulence regimes are studied for $1\mathrm{D}$ geometry in the case when the influence of the normal fluid on superfluid flow is suppressed. It is shown that the roton mechanism of transition to quantum turbulence leads to a critical velocity satisfying the relation $v_c\propto d^{-1\mathrm{/}4}$. In the super-flow mechanism, the transition to quantum turbulence arises when the “quantum Reynolds number” is about ${10}^3$ and the critical velocity depends on channel size $d$ as $v_c\propto d^{-1}$ in agreement with the equations of motion for a superfluid component of the liquid ${}^4$He being disturbed by small fluctuations of the normal fluid.