Periodically-driven flows occur in several important applications such as flows in the blood circulatory system, the respiratory system, etc. Our work is focused on the study of the stability of periodically-driven flows, i.e., which sequence of bifurcations transforms a periodic periodically-driven flow into a quasi-periodic flow and eventually in a turbulent flow. The method proposed consists in solving a transient problem starting from an initial flow (we use a fluid that is at rest or a steady flow here) and iterate in time over the transient flow until the periodic solution is completely set. The Re is gradually increased in order to identify those critical values which make the solution unstable. The experiments have been carried out for the case of a 2-D artery with a stenotic region in the middle with two types of inflow :i) steady and ii)time-periodic (with different periods). Another purpose of the simulations is to identify those mechanisms that control the convergence of the transient solution toward the periodic solution of a periodically-driven non-symmetric parabolic model problem. Some theoretical results will be presented for the advection-diffusion equation.The Navier-Stokes equations share many characteristics with the advection-diffusion equation, of course with the extra difficulties caused by their nonlinearity, so the stability analysis will be extended to this type of equations.