Critical velocity of a two-dimensional superflow past a potential barrier of arbitrary penetrability

We theoretically investigate the critical velocity for dissipationless motion of a two-dimensional superfluid past a static potential barrier of large width. The circular-shaped barrier provides a comprehensive analytical framework for the critical speed, for which we derive closed-form expressions using the hydraulic approximation, the hodograph method, and Janzen-Rayleigh expansions of the velocity potential. These analytical estimates are shown to be in good agreement with the numerical results of an imaginary-time integration of the full wave equation. In contrast to most of the state of the art, our study is not restricted to an impenetrable potential barrier nor to a quartic interaction Hamiltonian, which enables realistic modeling of recent experiments with atomic Bose-Einstein condensates and paraxial superfluids of light in two dimensions.