Among the several methods used to solve the Navier-Stokes equations Hierarchical Expansion Method has demonstrated satisfactory results. This work aimed to apply the expansion of the variables in hierarchical functions for the solution of the Navier-Stokes equations for incompressible fluids in two dimensions in laminar flow. This method is based on the finite element method. The expansion functions in this study were based on Legendre polynomials, adjusted in the rectangular elements in such a way that corner, side and area functions were defined. The order of the expansion functions associated with the sides and with the area of the elements is adjusted to the necessary or desired degree. This method is denominated by Hierarchical Expansion Method. In order to validate the proposed numeric method three well-known problems of the literature in two dimensions were analyzed; however, for this paper only one problem was presented. The results demonstrated that method was able to provide precise results. From the results obtained in this paper it is possible to conclude that the hierarchical expansion method can be effective for the solution of fluid dynamic problems that involve incompressible fluids.