  Supersonic flow of a superfluid past a slender impenetrable macroscopic
obstacle is studied in the framework of the two-dimensional defocusing
nonlinear Schr\"odinger (NLS) equation. This problem is of fundamental
importance as a dispersive analogue of the corresponding classical gas-dynamics
problem. Assuming the oncoming flow speed sufficiently high, we asymptotically
reduce the original boundary-value problem for a steady flow past a slender
body to the one-dimensional dispersive piston problem described by the
nonstationary NLS equation, in which the role of time is played by the
stretched $x$-coordinate and the piston motion curve is defined by the spatial
body profile. Two steady oblique spatial dispersive shock waves (DSWs)
spreading from the pointed ends of the body are generated in both half-planes.
These are described analytically by constructing appropriate exact solutions of
the Whitham modulation equations for the front DSW and by using a generalized
Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear
DSW. We propose an extension of the traditional modulation description of DSWs
to include the linear "ship wave" pattern forming outside the nonlinear
modulation region of the front DSW. Our analytic results are supported by
direct 2D unsteady numerical simulations and are relevant to recent experiments
on Bose-Einstein condensates freely expanding past obstacles.
