  In this paper we developed a numerical methodology to study some
incompressible fluid flows without free surface, using the curvilinear
coordinate system and whose edge geometry is constructed via parametrized
spline. First, we discussed the representation of the Navier-Stokes and
continuity equations on the curvilinear coordinate system, along with the
auxiliary conditions. Then, we presented the numerical method -- a simplified
version of MAC (\textit{Marker and Cell}) method -- along with the
discretization of the governing equations, which is carried out using the
finite differences method and the implementation of the FOU (\textit{First
Order Upwind}) scheme. Finally, we applied the numerical methodology to the
parallel plates problem, lid-driven cavity problem and atherosclerosis problem,
and then we compare the results obtained with those presented in the
literature.
  Keywords: finite differences, simplified MAC, curvilinear coordinates,
parallel plates, did-driven cavity, atherosclerosis.
