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

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This work aims to extend in two distinct directions results recently obtained in (10). In a rst 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 rst part, the time dependent rough problem for a simplied heat equation in a straight channel that mimics the axial velocity under an oscillating pressure gradient. We derive rst order approximations with respect to , 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 ow simula- tions of the non-homogenized stented arteries. We solve the stationary Stokes equations for an artery containing a saccular aneurysm. Consecutive levels of uniform mesh renement, serve to relate spatial resolution, problem scale, and required computation time. Test computations are presented for femoral side aneurysm, where a simplied 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 sucient to run accurate, direct simulations of a pulsatile ow in stented vessels.