Multi-Scale Modeling of the Gas-Liquid Interface Based on Mathematical and Thermodynamic Approaches~!2009-11-08~!2010-03-22~!2010-05-17~!

Abstract

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 gasliquid 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 various 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 coalescence/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 electrostatic potential due to the contamination at the interface, is derived based on a lattice gas model. Free energy is incorporated into the conventional Navier-Stokes equation as new terms using Chapman-Enskog expansion based on the multiscale 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.

Macroscopic Gas-Liquid Interfacial Equation Based on Thermodynamic and Mathematical Approaches

“…In a previous study (Yonemoto & Kunugi, 2010a), the multi-scale multiphase flow equation was derived based on the phase field theory (Cahn & Hilliard, 1958). In the derivation, we assumed that the interface has a finite thickness similar to that of a fluid membrane www.intechopen.com (mesoscopic interface).…”

“…(Yonemoto & Kunugi, 2010a). The symbols of z i [-], e [C], V e [V], and ψ [-] are the charge number, elementary charge, electrostatic potential, and order parameter, respectively.…”

“…(1)) is associated with thermodynamic pressure using the Maxwell relation. This equation is the multi-scale multiphase flow equation (Yonemoto & Kunugi, 2010a …”

“…But the multi-scale concept was not expressed in concrete terms. In a previous study (Yonemoto & Kunugi, 2010a), a thermodynamic and mathematical interfacial model that takes into account the multi-scale concept has been developed on the basis of the phase field theory (Cahn & Hilliard, 1958). In this model, we assumed that the interface has a finite thickness and that free energy is defined at the interface.…”

“…In particular, an interfacial jump condition treated by thermodynamics was derived from the multi-scale multiphase flow equation, and the thermodynamic interfacial jump condition was then compared with the conventional jump condition. In addition, we derived the Kelvin equation based on both the multi-scale multiphase flow equation (Yonemoto & Kunugi, 2010a) and the thermodynamic jump condition. The present results indicate that our interfacial model can theoretically support various interfacial phenomena characterized by thermodynamics from a multi-scale viewpoint (micro to macro).…”