A variational approach to the derivation of reduced models for bubbly flows

TL;DR

This paper derives simplified models for gas bubble dynamics in inviscid liquids using Hamilton's principle, focusing on reducing interface complexity and analyzing well-posedness.

Contribution

It introduces a variational framework to derive reduced bubble models with simplified interface descriptions and provides well-posedness analysis under specific conditions.

Findings

Classical sharp interface model recovered from variational principle

Reduced models with parametrized bubble surfaces derived

Well-posedness established for curl-free flow and homogeneous pressure

Abstract

In this paper, we derive reduced models for the motion of gas bubbles in an ambient inviscid liquid, using Hamilton's least action principle. We first explain how to recover from this principle the classical sharp interface model, in which the pressure is continuous across the surfaces of the bubbles. We then show how to reduce the complexity of the model, by simplifying the description of those surfaces. Namely, we impose them to evolve within a subclass of hypersurfaces described by a finite number of parameters (the simplest example being spheres, that is neglecting deviation of the bubbles from sphericity). The difficulty from a mathematical and modeling point of view is to determine the interface conditions that substitute to pressure continuity. We complete the derivation of the reduced models by some well-posedness analysis, in the case of curl-free liquid flow and homogeneous pressure in the bubbles.