Authors: S.J. COX 1, D. WEAIRE 2, G. MISHURIS 1

Abstract:

Foams are widely used in traditional applications such as fire-fighting and froth flotation [1, 2]. More recently they have begun to find new applications in the emerging field of discrete microfluidics [3, 4], in which small volumes of gas (or, equivalently, liquid) can be manipulated within narrow channels.

When monodisperse bubbles are confined in narrow channels with a low liquid fraction, they generally form ordered structures [3, 5] – see figure 1. They may be readily manipulated in networks of such channels. Such a system is suggestive of a practical system of microfluidics [6], so the motion of the bubbles, when driven by a pressure, is of practical as well as basic interest. The desire to predict the structure and dynamics of such of a foam arises from the requirement to design both chemical formulation and container geometry with maximum efficiency, to perform specific functions.

In a rectangular channel whose width is much greater than its depth, the structure in question may be regarded as essentially two-dimensional. That is, each bubble touches both of the bounding plates and the soap films that span the gap are perpendicular to those plates. Accordingly a 2D model, previously developed in the physics of foams [7] has been used in this area. This “viscous froth” model is only a provisional or skeletal one (including in particular linear relationships when power laws may be more realistic), but it has succeeded in shedding light on some observed phenomena that are seen when the flow velocity is large enough to depart from the quasistatic condition [7, 8, 9, 10].

However, it then appeared that significant qualitative details of observed structures were apparently not compatible with the model and it was not clear what needed to be added to it. The present study, which explores the high velocity limit, was stimulated in part by these anomalous structures.

We review the model in §2 and find its steady-state solutions for single lines in §3. Even this apparently straightforward problem presents quite a wealth of mathematical detail. Simplicity is restored in the high velocity limit [10], which is described in §4. Any of these steady-state solutions may be used to provide pieces of a more general solution in which the lines meet at 120◦, allowing comparison with experiment, as discussed in §5.

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