Pressure driven flows typically occur in hydraulic networks, e.g. oil ducts, water supply, biological flows, microfluidic channels etc. However, Stokes and Navier-Stokes problems are most often studied in a framework where Dirichlet type boundary conditions on the velocity field are imposed, thanks to the simpler settings from the theoretical and numerical points of view. In this work, we propose a novel formulation of the Stokes system with pressure boundary condition, together with no tangential flow, on a part of the boundary in a standard Stokes functional framework using Lagrange multipliers to enforce the latter constraint on velocity. More precisely, we carry out (i) a complete analysis of the formulation from the continuous to discrete level in two and three dimensions (ii) the description of our solution strategy, (iii) a verification of the convergence properties with an analytic solution and finally (iv) three-dimensional simulations of blood ow in the cerebral venous network that are in line with in-vivo measurements and the presentation of some performance metrics with respect to our solution strategy.
