Homogenized Equations for Isentropic Gas in a Pipe with Periodically-Varying Cross-Section

TL;DR

This paper derives effective homogenized equations for isentropic gas flow in pipes with periodically varying cross-sections, revealing solitary wave solutions instead of shocks.

Contribution

It introduces a homogenization approach using multiple-scale perturbation theory to simplify complex variable-cross-section gas flow equations.

Findings

Homogenized equations include dispersive higher-order derivatives.

Numerical solutions show solitary wave behavior.

Homogenized model accurately approximates original system.

Abstract

We analyze the behavior of an isentropic gas in a narrow pipe with periodically-varying cross-sectional area. Using multiple-scale perturbation theory, we derive homogenized effective equations, which take the form of a constant-coefficient system of evolution equations, including dispersive higher-order derivative terms. We provide an approximate Riemann solver for the variable-cross-section isentropic gas equations, and compare numerical solutions of the original system with those of the homogenized system. We observe that the resulting solutions take the form of solitary waves, rather than shock waves, under fairly general conditions.