Modeling of thin liquid film drainage governing the lifetime of a single mini-scale surface bubble

Surface bubble lifetime is determined by the thin liquid film drainage, which is driven by pressure gradients within the film. Traditional models consider gravity- and surface-tension-driven drainage for large and small bubbles, respectively. However, gravitational effects remain significant in small bubbles within highly viscous liquids because of the increased film thickness. Many models neglect the film pressure or estimate it solely based on the Young–Laplace equation, overlooking the pressure gradients along the liquid film. Moreover, despite the presence of partial slip behavior in microscale systems, which can influence film drainage rates, idealized boundary conditions such as slip or no-slip at the gas–liquid interface are often assumed. To enhance predictive accuracy, this study proposes a model that incorporates both gravity- and surface-tension-driven drainage, considers pressure gradients along the film, and applies partial-slip boundary conditions. Three key model variables—film pressure, film thinning rate, and tangential film drainage velocity—were determined by simultaneously solving three governing equations: (1) the Stokes equation for the liquid film under partial-slip conditions, (2) mass conservation equation that relates film thinning rate to film drainage velocity, and (3) force balance at the deformed free surface. The model was validated across various liquids, including water, perfluorocarbon liquid (PP11), and polydimethylsiloxane (PDMS) liquids with viscosities of 10, 100, and 1000 Pa s, yielding mean absolute percentage errors of 10.2%, 36.4%, 33.6%, 24.9%, and 32.9%, respectively. The model, derived from first principles, provides consistent and interpretable predictions across various liquids without relying on empirical fitting, demonstrating its general applicability.