Solutions are presented to the problem of steady, axisymmetric flow of an inviscid fluid into a point sink. The fluid is of finite depth and has a free surface. Two numerical schemes, a spectral method and an integral equation approach, are implemented to confirm results for the maximum-flow-rate steady solution for each configuration. The effects of surface tension and sink depth are included and constitute the new component of the work. Surface tension has the effect of increasing the maximum flow rate at which steady-state solutions can exist.