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This work aims to extend in two distinct directions results recently obtained in [10]. In a first step we focus on the possible extension of our results to the time  dependent case. Whereas in the second part some preliminary numerical simulations aim to  give orders of magnitudes in terms of numerical costs of direct 3D simulations.  We consider, in the first part, the time dependent rough problem for a simplified heat  equation in a straight channel that mimics the axial velocity under an oscillating  pressure gradient. We derive first order approximations with respect to  ϵ, the size of the roughness. In order to understand the problem and set  up correct boundary layer approximations, we perform a time periodic fourier analysis and  check that no frequency can interact with the roughness. We show rigorously on this toy  problem that the boundary layers remain stationary in time (independent on the frequency  number). Finally we perform numerical tests validating our theoretical approach.  In the second part, we determine actual limits, when running three-dimensional blood flow  simulations of the non-homogenized stented arteries. We solve the stationary Stokes  equations for an artery containing a saccular aneurysm. Consecutive levels of uniform mesh  refinement, serve to relate spatial resolution, problem scale, and required computation  time. Test computations are presented for femoral side aneurysm, where a simplified  ten-wire stent model was placed across the aneurysm throat. We advocate the proposed stent  homogenization model, by concluding that an actual computation power is not sufficient to  run accurate, direct simulations of a pulsatile flow in stented vessels

Asymptotic analysis of blood flow in stented arteries: time dependency and direct simulations