Effect of bubble deformation on the coalescence of two ascending bubbles in a viscous liquid

We conduct a trajectory analysis for ascending bubbles with deformation at low Reynolds number $Re≪1$ to understand the effect of deformation on the cross section for coalescence. We derive equations of motion for the separation and shape of two bubbles with three controlling parameters: Bond number (⁠$Bo$⁠), size ratio (λ), and initial separation. These equations can be reduced to equations of motion of separation and shape contrast for two bubbles. A theoretical examination of the reduced equations provides the critical initial separation for collision as a function of $Bo$ and λ. We develop a new model of the cross section for the coalescence of deformable bubbles by comparing the critical initial separation with experimental data. The cross section shows three different types of behavior depending on $Bo$⁠. For small $Bo$⁠, the cross section is independent of $Bo$ and is proportional to λ. For moderate $Bo$⁠, the cross section is proportional to $Bo(1−λ)−1$⁠. For large $Bo$⁠, the cross section is independent of $Bo$ and is proportional to $(1−λ)−4$⁠.