Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure

TL;DR

This paper develops an asymptotic expansion for the steady Stokes equation with variable viscosity in a thin, two-dimensional tube structure, including boundary layer corrections near bifurcations, and provides error estimates.

Contribution

It introduces a new asymptotic expansion method for the Stokes equation with variable viscosity in thin structures, accounting for boundary layers at bifurcations.

Findings

Constructed an asymptotic expansion with Poiseuille flows and boundary layer correctors.

Proved estimates for the difference between exact solution and approximation.

Validated the asymptotic method for thin tube structures with variable viscosity.

Abstract

The Stokes equation with the varying viscosity is considered in a thin tube structure, i.e. in a connected union of thin rectangles with heights of order $\varepsilon<<1 $ and with bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed: it contains some Poiseuille type flows in the channels (rectangles) with some boundary layers correctors in the neighborhoods of the bifurcations of the channels. The estimates for the difference of the exact solution and its asymptotic approximation are proved.