A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions

TL;DR

This paper develops a rigorous mathematical framework to estimate the convergence of boundary layer approximations for a simplified model of blood flow in arteries with rough stent surfaces, using multi-scale analysis and decay estimates.

Contribution

It introduces a vertical boundary layer correction and provides error estimates for boundary layer approximations in a rough Poisson-Dirichlet problem with Neumann boundary conditions.

Findings

Decay estimates for the vertical boundary layer.

Error bounds for boundary layer and wall law approximations.

Construction of boundary layers respecting macroscopic conditions.

Abstract

Stents are medical devices designed to modify blood flow in aneurysm sacs, in order to prevent their rupture. Some of them can be considered as a locally periodic rough boundary. In order to approximate blood flow in arteries and vessels of the cardio-vascular system containing stents, we use multi-scale techniques to construct boundary layers and wall laws. Simplifying the flow we turn to consider a 2-dimensional Poisson problem that conserves essential features related to the rough boundary. Then, we investigate convergence of boundary layer approximations and the corresponding wall laws in the case of Neumann type boundary conditions at the inlet and outlet parts of the domain. The difficulty comes from the fact that correctors, for the boundary layers near the rough surface, may introduce error terms on the other portions of the boundary. In order to correct these spurious oscillations, we introduce a vertical boundary layer. Trough a careful study of its behavior, we prove rigorously decay estimates. We then construct complete boundary layers that respect the macroscopic boundary conditions. We also derive error estimates in terms of the roughness size epsilon either for the full boundary layer approximation and for the corresponding averaged wall law.