Free-surface potential flow of an ideal fluid due to a singular sink

A two-dimensional problem of a potential free-surface flow of an ideal incompressible fluid caused by a singular sink is considered. The sink is placed at the horizontal bottom of the channel. By employing a conformal map, the problem is equivalently rewritten in the unit circle. After that, it is investigated by the Levi — Civita technique with the extraction of the singular part of the flow that corresponds to the sink. We derive a Nekrasov type equation that describes exactly the form of the free boundary. This equation is studied at first numerically and then by an exact mathematical technique. It is shown that for the Froude number greater than some particular value, there exists a unique solution of the problem such that the free surface decreases monotonically when moving from the infinity to the sink. At the point over the sink, the free surface has a cusp.