Coupled volume of fluid and phase field method for direct numerical simulation of insoluble surfactant-laden interfacial flows and application to rising bubbles

Improved numerical methods are needed to understand the effect of surfactants in interfacial fluid mechanics, with various applications including thin films, inkjet printing, and ocean-atmosphere interactions. We provide a three-dimensional coupled volume of fluid (VoF) and phase field numerical approach to simulate the effects of insoluble surfactant-laden flows. The framework is validated against analytical cases for surfactant transport and Marangoni stresses. We then systematically investigate a single surfactant-laden rising bubble. The characteristics of a clean bubble rising in a quiescent liquid are governed by nondimensional numbers, i.e., the Galileo number $\text{Ga}$, which compares inertial and viscous effects, and the Bond number $\text{Bo}$, which compares gravitational and surface tension stresses. The effect of insoluble surfactants introduces an additional independent parameter, the Marangoni number $\text{Ma}$, comparing the change in surface tension forces due to gradients in surfactants concentration with viscous forces. We apply our numerical methods to investigate the influence of surfactants (through the Marangoni number) on rising bubbles in otherwise quiescent fluids. We observe that an increase in the Marangoni number first decreases the rise velocity before reaching a limiting value at high $\text{Ma}$. The value of $\text{Ma}$ necessary to observe a significant slowdown increases with $\text{Ga}$. We discuss the associated surfactant accumulation and the vortical dynamics when a steady state is reached. Finally, we perform three-dimensional simulations and demonstrate that Marangoni effects can induce a change in the rise trajectory from spiraling to zigzagging for set values of Bo and Ga, consistent with experimental results.