The goals of this project are to study and numerically solve a reduced fluid-structure interaction problem for cardiovascular applications. The first section of this project presents two test cases in order to understand the finite element method for two dimensional space problems. Then, we focus our attention on the study of the Navier-Stokes equations in a fixed domain. The last section is dedicated to the structure model and the coupling conditions. To avoid the use of the ALE framework and all the numerical issues related to, we use the transpiration conditions of order zero. Finally we point out that thanks to the simplifications the structure equation is embedded in the fluid equations as a generalized Robin boundary condition. The original problem is then reduced to a fluid problem in a moving domain.