Asymptotic Analysis of a Viscous Fluid in a Curved Pipe with Elastic Walls

TL;DR

This paper presents an asymptotic analysis of viscous fluid flow in elastic-walled, curved pipes, deriving a one-dimensional model from Navier-Stokes equations as the cross-sectional radius shrinks, applicable to blood flow in arteries.

Contribution

It introduces a new asymptotic approach for modeling viscous flow in elastic curved pipes, coupling fluid dynamics with wall elasticity, and derives a simplified one-dimensional model from Navier-Stokes equations.

Findings

Derived a one-dimensional model for viscous flow in elastic pipes.

Established asymptotic expansions for velocity and pressure.

Provided a framework for coupling fluid flow with wall elasticity.

Abstract

This communication is devoted to the presentation of our recent results regarding the asymptotic analysis of a viscous flow in a tube with elastic walls. This study can be applied, for example, to the blood flow in an artery. With this aim, we consider the dynamic problem of the incompressible flow of a viscous fluid through a curved pipe with a smooth central curve. Our analysis leads to obtain an one dimensional model via singular perturbation of the Navier-Stokes system as $\varepsilon$, a non dimensional parameter related to the radius of cross-section of the tube, tends to zero. We allow the radius depend on tangential direction and time, so a coupling with an elastic or viscoelastic law on the wall of the pipe is possible. To perform the asymptotic analysis, we do a change of variable to a reference domain where we assume the existence of asymptotic expansions on $\varepsilon$ for both velocity and pressure which, upon substitution on Navier-Stokes equations, leads to the characterization of various terms of the expansion. This allows us to obtain an approximation of the solution of the Navier-Stokes equations.