The rheology of two-dimensional foams

Why study the rheology of foams? The motivations are widespread and diverse (Weaire and Hutzler 1999). Foams are common in oil extraction and industrial cleaning. Closer to home, an understanding of flowing foams helps to extinguish fires more efficiently and to generate the perfect pint of beer. At low shear-rates a foam behaves as an elastic solid, while at high shearrates it yields like a liquid, generating a rich range of behaviours. Foams are non-Newtonian, but have a specific structure which allows analysis of their nonNewtonian behaviour. Foams can thus serve as a prototype for other non-Newtonian fluids and even for granular materials. In addition to rheology, we seek to understand the drainage (usually gravitational) of liquid through the foam structure. High flow-rate drainage can cause the structure itself to flow; these convective instabilities (Weaire et al. 2003) further motivate our interest in rheology. Is the study of two-dimensional (2D) foams, or froths, useful? As in many fields, the restriction to 2D allows easier visualization of both experiment and simulation, and often simplifies the theory, allowing us to isolate and study new phenomena (though we must check for spurious 2D effects). To develop tools for 2D simulations, we first consider the foam’s liquid fraction Fl, i.e. its fractional liquid content by volume. The dry limit of low liquid fraction, Fl fi 0, is a natural idealization, in which soap films are infinitesimally thin arcs of circles. The Laplace equation then determines the curvature j of a film from the pressure difference Dp between neighbouring bubbles: