What is it about?
Investigations of various physical and biological processes in channels, junctions and networks are urgent for numerous fields of natural sciences. Especial interest is the investigation of the influence of a local geometrical heterogeneity in vessels on the blood flow. This is both an aneurysm (a pathological extension of an artery like bulge) and a stenosis (a pathological restriction of an artery). The understanding of the impact of a local geometric irregularity on properties of solutions to boundary-value problems in such domains is made in our paper.
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Why is it important?
Till now the question how to model different processes in thin aneurysm-type domain with sufficient accuracy is still open. This question was the main motivation for us to develop a new approach (asymptotic one) for the study of boundary-value problems in domains of such type, since numerical methods do not give good approximations through the presence of a local geometric irregularity.
Perspectives
We hope that this asymptotic approach can be applied to the study of the blood flow in vessels with a local geometric heterogeneity what we are going to do in our further studies. Nevertheless, the results obtained in this article can be considered as the first steps in this direction, since it is known that for the incompressible flow it is possible in some cases to couple pressure and velocity through the Poisson equation for pressure. Also the pressure Poisson equation with Neumann boundary conditions is encountered in the time-discretization of the incompressible Navier-Stokes equations.
Prof. Dr. Taras Mel'nyk
Universitat Stuttgart
Read the Original
This page is a summary of: Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain, Open Mathematics, November 2017, De Gruyter,
DOI: 10.1515/math-2017-0114.
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