Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

Abstract

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.

Keywords:

• fluid-structure interaction
• viscous fluid
• axisymmetric flow
• periodic case
• asymptotic solution
• error estimate

AMS Subject Classifications::

• 76M45
• 74F10

Acknowledgements

The authors are grateful to Prof E. Sanchez-Palencia for his useful suggestions during the preparation of this article. G.P. Panesenko is supported by the SFR MODMAD of the University of Saint Etienne, by the grant no. 14.740.11.0875 ‘Multiscale Problems: Analysis and Methods’ of the Ministry of Research of Russian Federation, by the PICS CNRS project ‘Mathematical Modelling of blood diseases’ and by the ANR project MECAMERGE. The work was done during the stay of R. Stavre in Saint-Etienne University.