Foams are dense suspensions of gas bubbles in a liquid. They have many industrial applications, as end products (shaving cream, shampoo, fire suppression) and as useful tools (used for separating components in froth flotation or in transporting sand into oil wells). They also can be impediments to manufacturing, forming when fluids are mixed energetically. Despite their simple ingredients (air, water, and surfactant) foams behave in quite complicated ways. At low stresses they resist shear like solids. At higher stress, they flow like liquids. They also can be compressed like gases. Deciphering the mechanisms underlying this complicated rheology is of importance to the understanding and application of foams.
Most models of foams have been in the ``dry'' limit, where the bubble sides are polygonal films and the volume fraction of liquid is small. The relevant dynamical objects in a two dimensional dry foam model are the vertices where bubbles sides meet. Geometrical difficulties have restricted these models to two dimensions, and modifications that relax the constraint are awkward. In contrast, a new model by Durian makes the bubbles themselves the dynamical objects of interest, handles wet foams (), and works equally well in two and three dimensions. Concentrating on bubbles is motivated by the experimental observation that foams flow by a series of bubble rearrangement events. The important features of the flow should therefore be reproduced by a model that ignores fine scale details (such as the hydrodynamics of fluid in the films) but gets the forces between bubbles roughly correct. In Durian's model, each bubble has a center and a radius and no other properties. In lieu of deforming, overlapping bubbles repel one another, and in lieu of film hydrodynamics, overlapping bubbles exert drag forces on one another.
Currently, a numerical simulation of Durian's model in two dimensions is being used to measure viscoelastic properties and characterize the rearrangement events in slowly driven two dimensional foams. Although the model is simply stated, there are subtleties in writing an efficient simulation program. Keeping track of neighboring bubbles requires careful bookkeeping, and sparse matrix equations must be constructed and solved at every time step. The program solves for the bubble positions as a function of time under a variety of boundary conditions: constant shear rate, fixed displacement, constant shear stress, or constant spring pulling speed. The latter mimics conditions in friction experiments, where one end of a spring is attached to an object and the other end is pulled at constant velocity.
Analogies between models for earthquakes, for flow of granular materials, and for friction between sliding surfaces are being pursued. The extension to three dimensions should happen when it is felt that the two dimensional simulation is under control.