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tel-00656013, version 1 - 3 Jan 2012
tel-00656013, version 1 - 3 Jan 2012 Chapitre 5 Asymptotic analysis of blood flow in stented arteries : time dependency This work aims to extend results recently obtained in [148]. We will focus on the possible extension of our results to the time dependent case. So we consider 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. 5.1 Introduction Rupture of aneurysm are common lethal pathologies in western countries. It is mainly due to a loss of elastic properties of tissues that constitutes the arterial walls on some branching. Recently emerged a new kind of stent: a metallic wired mutli-layered prosthesis (see fig. 5.1 right) that unlike the classical endograft stent need not to be sutured to the arterial wall. Their form-memory metallic structure allow self-expansion recovering the original form without a need of a balloon. A recent work of the first author establishes, thanks to asymptotic analysis tools, several advantages of this new device [148]. These can be summarized as follows : • the presence of a stent at the inlet of a collateral artery (see fig. 5.1 left) gives rise to a secondary flow explicitly computable : it depends on the pressure jump occurring at zero order (when the stent totally closes the inlet of the collateral artery) and on some periodic microscopic resistivity (computed independently of any kind of macroscopic flow). • the presence of a stent above a closed aneurysmal sac (see fig. 5.1 middle) imposes a 145
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