Viscous entrainment from a nozzle: Singular liquid spouts

We analyze a long-wavelength model of viscous entrainment from a nozzle and show that, when appropriate large-scale boundary conditions are imposed, the entrainment transition can become continuous. As the entrainment threshold is approached from above, the entrained steady-state spout becomes vanishingly thin compared to the nozzle radius and takes the form of a thin tendril emerging from a base profile whose tip is perfectly conical.