Family of self-similar solutions with a free boundary in a fluid wedge

Plane unsteady potential flows of an ideal incompressible fluid with a free boundary are considered in the absence of external forces and surface tension. At the initial time, the flow occupy a wedge with an angle at the apex. For different initial flow velocities and values of the angle at the vertex, a family of exact solutions is found. A method for finding solutions based on reducing the boundary-value problems to systems of ordinary differential equations.