We consider models which are classically used in the simulation of airflows and blood flows andinvestigate the numerical stability of some discretization strategies. The geometrical complexity of the networksin which air/blood flows leads to a classical decomposition of two areas: a truncated 3D geometry correspondingto the largest contribution of the domain and a 0D part connected to the 3D part, modelling air/blood flowsin smaller airways/vessels. The resulting Navier-Stokes system in the 3D truncated part may involve non-local boundary conditions, deriving from a mechanical model. For various 3D/0D coupled models, differentdiscretization processes are presented and analyzed in terms of numerical stability, highlighting strong differencesaccording to the regimes that are considered. In particular, two main stability issues are investigated: first thecoupling between the 3D and the 0D part for which implicit or explicit strategies are studied and, second, thequestion of estimating the amount of kinetic energy entering the 3D domain because of the artificial boundaries.In particular, we prove new estimates in appropriate norms for the discretized-in-time Navier-Stokes system.These estimates are derived under conditions on the smallness of the data, enlighting the intrinsic difficultyencountered with such systems to perform realistic simulations. We illustrate some of the theoretical resultswith numerical simulations, firstly in a single tube, then in a bifurcation geometry and finally in real geometries.Finally, we discuss the difference between airflows and blood flows in terms of numerical stability related tothe magnitude of the physiological and physical parameters. Stokes and Navier-Stokes equations; coupling ofmodels; numerical stability; numerical computations; finite element method