Dimension reduction for the full Navier-Stokes-Fourier system

TL;DR

This paper rigorously justifies reducing the three-dimensional Navier-Stokes-Fourier system to a one-dimensional model for flow in thin pipes, facilitating numerical analysis by showing convergence of solutions as pipe thickness diminishes.

Contribution

It provides a rigorous mathematical proof that weak solutions of the 3D NSF system converge to strong 1D solutions in the thin pipe limit, validating the dimension reduction approach.

Findings

Weak solutions tend to strong 1D solutions as pipe thickness approaches zero.

The dimension reduction is mathematically justified for the NSF system in thin geometries.

The approach simplifies complex 3D flow modeling in practical applications.

Abstract

It is well known that the full Navier-Stokes-Fourier system does not possess a strong solution in three dimensions which causes problems in applications. However, when modeling the flow of a fluid in a thin long pipe, the influence of the cross section can be neglected and the flow is basically one-dimensional. This allows us to deal with strong solutions which are more convenient for numerical computations. The goal of this paper is to provide a rigorous justification of this approach. Namely, we prove that any suitable weak solution to the three-dimensional NSF system tends to a strong solution to the one-dimensional system as the thickness of the pipe tends to zero.