Viscous effects in the rheology of foams and concentrated emulsions

Abstract

A microscopic model for the rheology of foams and concentrated emulsions is developed by considering small deformations of an idealized material with two-dimensional spatially periodic cell structure. Most of the continuous-phase liquid is considered to be in the Plateau borders. Following Schwartz and Princen, viscous effects are modeled by the film withdrawal mechanism of Mysels, Shinoda, and Frankel. The primary flow occurs where films with inextensible interfaces are withdrawn from or recede into quasistatic Plateau borders. The quasi-steady asymptotic analysis of the flow is developed for small capillary numbers Ca based on the macroscopic deformation rate. The viscous flow induces an excess tension that varies between films and alters the foam structure. The instantaneous structure and effective stress for a foam of arbitrary orientation are determined for simple shear and planar extension. To clarify the role of viscous flow and surface tension, the effective stress is calculated by two methods, which involve the volume average of either local stress or stress power. The viscous contribution to the effective stress is O(Ca 2 3 ) and depends on the foam orientation but not its liquid content. A dilatational viscosity of O(Ca −1 3 ) is predicted for uniform expansion.