A high-order spectral element based phase field method for two-phase flows containing soluble surfactants with large solubility ratio

This study presents a high-order spectral element based phase field method for soluble surfactant transport in two-phase flows under significant solubility difference. We reformulate the phase-field model for surfactant-laden two-phase flows by introducing an asymmetric term to characterize the solubility difference. Equilibrium properties are derived from the extremum conditions of the energy functional, including the profiles of the interface capturing phase variable and surfactant concentration, relations between the solubility ratio and energy parameter. To address the complex transport behavior of the surfactant and its nonlinear coupling with the interfacial dynamics, we employ the Legendre polynomial based spectral element method to achieve high-order spatial discretization of the governing equations. Additionally, the time-independent coefficient matrix technique and the bound-preserving interfacial energy are used, significantly enhancing computational efficiency while maintaining robustness. Benchmarking examples are organized to validate the present method, demonstrating its accuracy in predicting the surfactant distribution, capturing the interface evolution and maintaining the conservation of phase volume and surfactant transfer. We explore the buoyancy-driven bubble rising process considering the surfactant effects, demonstrating that the present method maintains its accuracy for interface capturing and produces realistic surfactant dynamics for scenarios with density ratios up to 1000 and a solubility ratio of 0.001.