Weak solutions of a viscous model for fluid-bubbles interaction

TL;DR

This paper introduces a Navier-Stokes based model for multiple gas bubbles in a liquid, relaxing stress continuity to maintain bubble sphericity, and constructs weak solutions up to bubble collision, addressing complex fluid-bubble interactions.

Contribution

It develops a novel mathematical framework for fluid-bubble interaction with relaxed stress conditions and constructs weak solutions that handle bubble collisions.

Findings

Successful construction of weak solutions up to bubble collision

Addressed mathematical challenges from bubble compression/dilation

Extended weak solution concepts to fluid-bubble systems

Abstract

We present a system of Navier-Stokes type that describes the dynamics of several spherical bubbles of gas in a liquid. It is derived from a more complete model, where the bubbles are seen as inclusions of gas of homogeneous barotropic pressure with free surfaces. The usual condition of continuity of the stress is relaxed in order to preserve the sphericity of the bubbles through time. We construct weak solutions \`a la Leray for this relaxed system, up to collision between the bubbles. Although these solutions are reminiscent of weak solutions for fluid-solid interaction systems, accounting for the compression/dilation of the bubbles creates new and significant mathematical difficulties.