Liquid-vapor transformations with surfactants. Phase-field model and Isogeometric Analysis

Surfactants are compounds that find energetically favorable to be located at the boundaries between fluids. They are able to modify the properties of those interfaces, for example, reducing surface tension. Here, we propose a new model for liquid-vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid. The aforementioned model is derived from a free energy functional by using a Coleman-Noll approach. The proposed theory emanates from the isothermal Navier-Stokes-Korteweg equations, which describe single-component two-phase flow and naturally allow for phase transformations. We believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes. From a numerical point of view, the proposed model poses significant challenges to existing discretization methods, including stiffness in space and time, internal and boundary layers as well as higher-order partial differential operators. To overcome these challenges we propose algorithms based on Isogeometric Analysis, which permit an accurate and efficient discretization. Finally, we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions.