Critical velocity for wake vortex generation behind a plate in a superflow

We study theoretically the critical velocity ${U}_{c}$ for quantum vortex generation by a thin plate-shaped obstacle moving through a uniform two-dimensional Bose-Einstein condensate. Our results based on the Gross-Pitaevskii theory reveal that the critical velocity monotonically decreases with increasing the plate size $L$. In the limit of large $L$, the critical velocity is asymptotic to ${L}^{\ensuremath{-}1/2}$ predicted by the potential flow theory for an incompressible ideal fluid with a phenomenological length correction. As $L$ decreases, however, the incompressible analysis breaks down quantitatively. By performing a perturbative analysis to incorporate compressibility into the potential flow theory, we successfully reproduced the numerical results analytically over a wide parameter range.