Three Dimensional Asymptotic Analysis of an Axisymmetric Flow in a Thin Tube with Thin Stiff Elastic Wall

Abstract

An interaction between a thin cylindrical elastic tube and a viscous fluid filling the thin interior of the elastic tube is considered when the thickness of the elastic medium ($\varepsilon$) and that of the fluid domain ($\varepsilon _1$) are small parameters, with $\varepsilon<<\varepsilon _1<<1$ while the scale of the longitudinal characteristic size is of order one. At the same time the magnitude of the stiffness and density of the elastic tube may be big or finite parameters with respect to the viscosity and density of the fluid when the characteristic time is of order one. A three dimensional asymptotic analysis is provided for an axisymmetric flow. The combination between the small parameters $\varepsilon , \varepsilon _1$ on one hand and the Young’s modulus of the elastic medium and its density (presented as some powers of $\varepsilon$) on the other hand leads to the analysis of 10 different cases. Several examples of real world applications of the considered mathematical model are presented in the introduction. We show that the error between the exact solution and the truncated asymptotic solution of order K is small, so these estimates justify the asymptotic expansion. The leading term of the expansion is computed and compared to the Poiseuille flow. It gives a Poiseuille type solution for the fluid-structure interaction problem.