Wolfgang Pauli Institute (WPI), UMI CNRS
2 ICJ, UMR 5208, Lyon I, FRANCE
3 Université Pierre et Marie Curie (UPMC), UMR 7598, Paris, FRANCE
This work aims to extend in two distinct directions results recently obtained in . 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.
The authors would like to thank Cardiatis (www.cardiatis.com), the industrial partner of the Cemracs 09 project Rugosity for the financial support
© EDP Sciences, ESAIM 2010