A study of the dynamics of foam growth: Analysis of the growth of closely spaced spherical bubbles

Abstract A mathematical analysis of bubble growth in an expanding foam is presented. The analysis is based on a cell model whereby the foam is divided into spherical microscopic unit cells of equal and constant mass, each consisting of a liquid envelope (or shell) and a concentric spherical gas bubble. Expansion occurs by diffusion of a dissolved gas from the supersaturated envelope into the bubble. This cell model is capable of describing important qualitative features of a real system of numerous bubbles growing in close proximity to one another, and is intended as the building block of a global analysis of macroscopic foam expansion. The coupled algebraic and differential equations governing the growth of a cell are derived and solved numerically. Five dimensionless parameters are identified for the case of constant temperature and pressure outside the cell, and their effects are demonstrated through computer simulations of the system. Of these parameters, surface tension and initial radius prove to be of relatively little importance in the practical cases considered. The other parameters are the thermodynamic driving force, the cell mass (inversely proportional to the number density of bubbles), and the ratio of characteristic times for mass and momentum transport.