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In the numerical simulation of blood flow problems it might happen that the only available physical boundary conditions prescribe the flow rate incoming/outgoing the vascular district at hand. In order to have a well-posed Navier–Stokes (NS) problem, these conditions need to be completed. In the bioengineering community, this problem is usually faced by choosing a priori a velocity profile on the inflow/outflow sections, that should fit the assigned flow rates. This approach strongly influences the accuracy of the numerical solutions. A less perturbative strategy is based on the so-called ‘do-nothing’approach, advocated in Heywood et al.(Int. J. Num. Meth. Fluids 1996; 22: 325–352). An equivalent approach, however, easier from the numerical discretization viewpoint, has been proposed in Formaggia et al.(SIAM J. Numer. Anal. 2002; 40 (1): 376–401). It is based on an augmented formulation of the problem, in which …