Thermodynamically consistent modeling for dissolution/growth of bubbles in an incompressible solvent

TL;DR

This paper develops thermodynamically consistent mathematical models for bubble dissolution and growth in liquids, starting from a compressible two-phase system and simplifying to incompressible and dilute cases.

Contribution

It introduces a comprehensive PDE model for bubble dynamics that ensures thermodynamic consistency and derives simplified models for practical scenarios.

Findings

Derivation of a full compressible two-phase PDE system with mass transfer.

A limiting process to obtain an incompressible solvent model.

Simplified PDE system for dilute solutions.

Abstract

We derive mathematical models of the elementary process of dissolution/growth of bubbles in a liquid under pressure control. The modeling starts with a fully compressible version, both for the liquid and the gas phase so that the entropy principle can be easily evaluated. This yields a full PDE system for a compressible two-phase fluid with mass transfer of the gaseous species. Then the passage to an incompressible solvent in the liquid phase is discussed, where a carefully chosen equation of state for the liquid mixture pressure allows for a limit in which the solvent density is constant. We finally provide a simplification of the PDE system in case of a dilute solution.