Analysis of compressible bubbly flows. Part II: Derivation of a macroscopic model

TL;DR

This paper derives a macroscopic averaged model for compressible bubbly flows from a microscopic description, establishing existence of solutions and connecting to the Williams-Boltzmann equation, advancing the theoretical understanding of bubbly flow dynamics.

Contribution

It introduces a rigorous derivation of a 1D macroscopic model from microscopic interactions, including existence proofs and the connection to kinetic equations.

Findings

Existence of solutions independent of the number of bubbles.

Construction of macroscopic variables from microscopic data.

Derivation of the Williams-Boltzmann equation for bubbly flows.

Abstract

This paper is the second of the series of two papers, which focuses on the derivation of an averaged 1D model for compressible bubbly flows. For this, we start from a microscopic description of the interactions between a large but finite number of small bubbles with a surrounding compressible fluid. This microscopic model has been derived and analysed in the first paper. In the present one, provided physical parameters scale according to the number of bubbles, we prove that solutions to the microscopic model exist on a timespan independent of the number of bubbles. Considering then that we have a large number of bubbles, we propose a construction of the macroscopic variables and derive the averaged system satisfied by these quantities. Our method is based on a compactness approach in a strong-solution setting. In the last section, we propose the derivation of the Williams-Boltzmann equation corresponding to our setting.