On the shape of the free-surface problem of an ideal incompressible fluid flow with a singular sink at the top of a triangular ledge at the bottom

A two-dimensional stationary problem of a potential free-surface flow of an ideal incompressible fluid caused by a singular sink is considered. The sink is located at the top of a triangular ledge at the bottom. The problem is to determine the shape of the free boundary and the velocity field of the fluid. By employing a conformal map and the Levi-Civita technique, the problem is rewritten as an operator equation in a Hilbert space. It is proved that, for the Froude number greater than some particular value, there is a solution of the problem. It is established that the free boundary has a cusp at the point over the sink. It is shown that the inclination angle of the free surface is less than $\pi/2$ everywhere except at the cusp point, where is it equal to $\pi/2$.