A unified relationship for the bubble departure from a wall*

Bubble departure is a ubiquitous phenomenon in nature and industry. In addition to the conventional mechanism, i.e., isolated bubble departure driven by buoyancy, coalescence induced bubble departure can be a predominant process while the underlying mechanics remaining unknown. Here, we report a new regime of bubble departure owing to the coalescence of two bubbles, where the departure diameter can be much smaller than the conventional buoyancy limit. We show that the significant reduction of bubble base area promotes the bubble departure, which is governed by two chracterisitic timescales of bubble coalescence. More importantly, using the distribution of bubble nucleation sites, we combine the buoyancy driven and the coalescence induced bubble departure modes and develop a unified relationship between the departure diameter and nucleation site density. This work elucidates how coalescing bubbles depart from a wall and provides design guidelines for achieving efficient bubble departure in applications such as boiling and gas-evolving reactions.