Existence of solutions for an interaction problem between a bubble and a compressible viscous fluid

TL;DR

This paper proves the existence of weak solutions for a complex fluid-bubble interaction model involving nonlinear PDEs and ODEs, accounting for bubble dynamics including radial expansion and contraction.

Contribution

It introduces a novel fluid-bubble interaction model with radial bubble motion and proves solution existence using penalization techniques.

Findings

Existence of weak solutions until bubble collision or collapse

Inclusion of radial bubble dynamics adds new nonlinear terms

Application of penalization methods from fluid-solid interaction

Abstract

In this paper, we study the dynamics of a finite number of spherical bubbles in a compressible fluid within a bounded open domain of R 3 . The fluid-bubble interaction is described by a system of nonlinear partial differential equations (PDEs) and ordinary differential equations (ODEs) coupling the fluid's density, velocity and pressure to the bubble's translational, rotational and radial velocities. We prove the existence of weak solutions for this model until the collision or collapse of the bubbles. The formulation of the fluid-bubble system, along with the techniques used for the existence proof, is inspired by penalization methods developed for fluid-solid interaction. The main contribution of this work is the addition of a radial expansion-contraction mode in the bubble motion, which introduces new nonlinear terms in the momentum equations that need to be treated carefully in the compactness arguments.