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.

[1]  H. Callen.  Thermodynamics and an Introduction to Thermostatistics , 1988 .

[2]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[3]  David Jacqmin,et al.  Contact-line dynamics of a diffuse fluid interface , 2000, Journal of Fluid Mechanics.

[4]  M. Fiałkowski,et al.  Scaling of the Euler characteristic, surface area, and curvatures in the phase separating or ordering systems. , 2001, Physical review letters.

[5]  Vincent S. J. Craig,et al.  Bubble coalescence and specific-ion effects , 2004 .

[6]  J. Brackbill,et al.  A continuum method for modeling surface tension , 1992 .

[7]  Ingo Steinbach,et al.  Phase Field Simulation of Equiaxed Solidification in Technical Alloys , 2006 .

[8]  Dieter Mewes,et al.  The influence of electrolytes on gas hold-up and regime transition in bubble columns , 2007 .

[9]  D. Beaglehole,et al.  An investigation of the critical thickness of film rupture and drainage phenomena using dual wavelength ellipsometry , 1998 .

[10]  M. Plapp,et al.  Three-dimensional phase-field simulations of directional solidification , 2007 .

[11]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .

[12]  L. Onsager.  Reciprocal Relations in Irreversible Processes. II. , 1931 .

[13]  Rudolf Podgornik,et al.  Statistical thermodynamics of surfaces, interfaces, and membranes , 1995 .

[14]  T. Yabe,et al.  The constrained interpolation profile method for multiphase analysis , 2001 .

[15]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[16]  W. Wulff.  COMPUTATIONAL METHODS FOR MULTIPHASE FLOW , 1990 .

[17]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[18]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[19]  D. Durran.  Numerical Methods for Fluid Dynamics , 2010 .

[20]  O. Reynolds.  I. On the theory of lubrication and its application to Mr. Beauchamp tower’s experiments, including an experimental determination of the viscosity of olive oil , 1886, Proceedings of the Royal Society of London.

[21]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[22]  D. Juric,et al.  A front-tracking method for the computations of multiphase flow , 2001 .

[23]  D. Koch,et al.  Bubble-size dependence of the critical electrolyte concentration for inhibition of coalescence. , 2004, Journal of colloid and interface science.

[24]  Tomoaki Kunugi,et al.  MARS for multiphase calculation , 2000 .

[25]  Andrew J. Bernoff,et al.  Dynamics of three-dimensional thin film rupture , 2000 .

[26]  John S. Rowlinson,et al.  Molecular Theory of Capillarity , 1983 .

[27]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[28]  Phase Field Simulation of the Effect of Anisotropy in Grain Boundary Energy on Growth kinetics and Morphology of Grain Structure , 2005 .

[29]  Richard R. Lessard,et al.  Bubble Coalescence and Gas Transfer in Aqueous Electrolytic Solutions , 1971 .

[30]  S. Marčelja.  Short-range forces in surface and bubble interaction , 2004 .

[31]  R. Callen,et al.  Thermodynamics and an Introduction to Thermostatistics, 2nd Edition , 1985 .

[32]  R. Thomson,et al.  A phase-field model for the solidification of multicomponent and multiphase alloys , 2005 .