Bubbles and drops between circular frames: shape, force and stability analysis.

Interactions between bubbles and drops play an important role in many physical phenomena. Whether we consider the interaction between two bubbles (drops) or between many (foams or emulsions), these interactions are complex and still poorly understood. An interesting case arises when two equal- and constant-volume bubbles (drops) interact with each other while being held by two axisymmetrically positioned frames of circular opening - a configuration which is frequently used in characterisation devices. The minimisation of the surface energy of this "double bubble" (or "double drop") configuration, constrained by the fixed volume and the frame boundaries, creates a complex landscape of shape spaces where physically stable shapes are separated by different types of instabilities. Combining experiments, finite element simulations and theory, we provide here for the first time a complete analysis of these shape spaces, considering the adhesive energy between the bubbles and the drops (expressed by the contact angle) as an additional control parameter. We provide the full shape diagrams for different contact angles (0°, 60° and 90°), including a detailed discussion of four types of instabilities. Two of these instabilities break the axisymmetry, while the two others break the connectivity of the ensemble. As far as we are aware, two of these instabilities have never been reported before. We accompany the shape and stability analysis with detailed mechanical characterisation using force and pressure measurements. Experiments, simulations and theory showing excellent agreement. This work will not only be useful in guiding the exploitation of double bubble (double drop) experiments on frames, but it also opens the possibility to exploit these configurations for the characterisation of increasingly complex bubble or drop interactions. Since the contact angle of 90° corresponds to an "imaginary" film separating the two bubbles (drops), our analysis naturally includes the shape and stability of a capillary bridge between two circular frames.