  We study the evolution of an initially confined atom condensate which is
progressively outcoupled by gradually lowering the confining barrier on one
side. The goal is to identify protocols that best lead to a quasi-stationary
sonic black hole separating regions of subsonic and supersonic flow. An optical
lattice is found to be more efficient than a single barrier in yielding a
long-time stationary flow. This is best achieved if the final conduction band
is broad and its minimum not much lower than the initial chemical potential. An
optical lattice with a realistic Gaussian envelope yields similar results. We
analytically prove and numerically check that, within a spatially
coarse-grained description, the sonic horizon is bound to lie right at the
envelope maximum. We derive an analytical formula for the Hawking temperature
in that setup.
