A comparative study of a viscous froth lens in two and three dimensions

Abstract

The two-dimensional (2D) viscous froth model was initially designed to capture the dynamic behavior of dry foams within a Hele–Shaw cell, characterized by two parallel covering plates separated by a small gap. However, due to its inherent 2D nature, this model does not explicitly account for the dimension across the gap. To address this limitation, we have opted for a three-dimensional (3D) version of the viscous froth model. In this 3D model, the dynamic effect is introduced through the motion of the surface Plateau borders, while the configurations of films in bulk are determined via surface energy minimization subject to specified bubble volumes and surface Plateau border locations. We use this 3D model to simulate the motion of a viscous froth lens within a straight channel. The steady states of the viscous froth lens in 3D are primarily influenced by the driving velocity-to-gap size ratio, particularly at relatively small values of this ratio. By contrast, as the ratio becomes relatively large, the gap size begins to play a significant role in influencing the behavior of the viscous froth lens in 3D. Differences are observed in the steady-state configurations of the viscous froth lens in 3D when compared to those in 2D. However, the behavior of the viscous froth lens in 2D can be better aligned with the results in 3D by treating the drag coefficient required in the 2D viscous froth model as a fitting parameter. A further quantitative analysis indicates that the drag coefficient needed in the 2D viscous froth model may not serve as a uniform parameter for the entire foam structure. Instead, it may depend on the specific location along the evolving foam films over time.