Asymptotic-numerical derivation of the Robin type coupling conditions for the macroscopic pressure at a reservoir–capillaries interface

Abstract

In this article the Stokes equations are considered in a domain simulating a capillary bed system. The capillaries are supposed to be thin, parallel and periodic. An asymptotic approximation is constructed. The macroscopic pressure satisfies a Robin interface condition whose coefficients are calculated numerically through a finite element approximation of a boundary layer problem, which is inspired to a domain decomposition technique.

Keywords:

• asymptotic expansion
• Robin interface conditions
• Stokes equations
• porous medium
• domain decomposition

AMS Subject Classifications::

• 35B27
• 35J25
• 65N30
• 76D07
• 76D05
• 76M55

Acknowledgements

The first and third authors gratefully acknowledge the ERC Advanced Grant No. 227058 MATHCARD – Mathematical Modelling and Simulation of the Cardiovascular System. The second author gratefully acknowledges the Grant No. 14.740.11.0875 ‘Multiscale problems: analysis and methods’ of Russian Ministry of Education and Research, the joint French-Russian PICS CNRS grant ‘Mathematical modelling of blood diseases’ and the SFR MODMAD grant of the University of Saint-Etienne/ENISE.