A finite difference method is described which solves steady free boundary problems arising in classical hydrodynamics. The vena contracta model for both plane and axially symmetric geometries are considered. The approach is to establish a fixed rectangle R in a parameter domain and seek the stream function and the physical plane in terms of the parameter variables. The physical plane is determined as a conformal map of R. Well‐known variational principles connected with conformal mapping and free boundary problems are used, combined with the method of steepest descent, to produce an iterative scheme which yields the desired parametric form of the solution.