The first contribution of this thesis is the analysis and discretization of multiphase flow in a microfluidic framework. We propose a new scheme wich is second order accurate, based on mixed finite volume / finite element method for the two phases Stokes system with surface tension. We then present the comparison of the accuracy of this new scheme with the MAC discretization in 2D and 3D axi. The second contribution is related to the study of numerical schemes in time for viscoelastic fluids.We present the current limitations in this area by studying the case of flows of wormlike micelles, polymers having the ability to reorganize themselves according to the shear rate. We show that a condition of stability related to the ratio of viscosity of the polymer and solvent in time - very restrictive - exists. A new scheme is then proposed to overcome this limitation and stability studies are conducted to demonstrate our results.
