Multi-scale modeling of the gas-liquid interface based on mathematical and thermodynamic approaches

A gas-liquid interface involves complex physics along with unknown phenomena related to thermodynamics, electromagnetics, hydrodynamics, and heat and mass transfer. Each phenomenon has various characteristic time and space scales, which makes detailed understanding of the interfacial phenomena very complex. Therefore, modeling the gas- liquid interface is a key issue for numerical research on multiphase flow. Currently, the continuum surface force (CSF) model is popular in modeling the gas-liquid interface in multiphase flow. However, the CSF model cannot treat the vari- ous chemical and physical phenomena at the gas-liquid interface because it is derived based only on mechanical energy balance and it assumes that the interface has no thickness. From certain experimental observations, bubble coales- cence/repulsion was found to be related to a contamination at the interface. The present study developed a new gas-liquid interfacial model based on thermodynamics via a mathematical approach, assuming that the interface has a finite thickness like a thin fluid membrane. In particular, free energy, including an elec- trostatic potential due to the contamination at the interface, is derived based on a lattice gas model. Free energy is incorpo- rated into the conventional Navier-Stokes equation as new terms using Chapman-Enskog expansion based on the multi- scale concept. Using the Navier-Stokes equation with the free energy terms, we derived a new governing equation of fluid motion that characterizes mesoscopic scale phenomena. Finally, the new governing equation was qualitatively evaluated by simulating an interaction between two microbubbles in two dimensions while also accounting for electrostatic force.