Refinement of a criterion for superfluidity of a classical liquid in a nanotube

The problem has a large parameter, the number $N$ of particles (outside the condensate), and a small parameter, Planck's constant $\hbar$ (more precisely, a dimensionless parameter containing $\hbar$ in the numerator). They turn out to be interrelated for Bose particles. Using the exact solution of the variational equation for a bosonic system in a capillary, we can obtain precise criteria restricting the superfluidity of a classical liquid in the capillary.