Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations

Abstract

We use the Cahn-Hilliard approach to model the slow dissolution dynamics of binary mixtures. An important peculiarity of the Cahn-Hilliard-Navier-Stokes equations is the necessity to use the full continuity equation even for a binary mixture of two incompressible liquids due to dependence of mixture density on concentration. The quasicompressibility of the governing equations brings a short time-scale ͑quasiacoustic͒ process that may not affect the slow dynamics but may significantly complicate the numerical treatment. Using the multiple-scale method we separate the physical processes occurring on different time scales and, ultimately, derive the equations with the filtered-out quasiacoustics. The derived equations represent the Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations. This approximation can be further employed as a universal theoretical model for an analysis of slow thermodynamic and hydrodynamic evolution of the multiphase systems with strongly evolving and diffusing interfacial boundaries, i.e., for the processes involving dissolution/nucleation, evaporation/condensation, solidification/melting, polymerization, etc.

On the phase-field modelling of a miscible liquid/liquid boundary

“…In the current work, however, this approach is used as a comprehensive physical model capable of describing the thermo-and hydrodynamic evolution of multiphase binary mixtures with undergoing phase transformations. Within the phase-field model, the diffusion process is defined through the gradient of the chemical potential, taking into account the effect of barodiffusion [6,7]. This model also captures the surface tension effects existent at miscible boundaries, thus 35 permitting us to model the inclined shape of the solute/solvent boundary.…”

“…Later, on the basis of the multiple-scale method, the quasi-acoustic effects were filtered out, so the Boussinesq approximation of the full equations was obtained [6].…”

“…The evolution of the multiphase liquid/liquid binary mixture is defined by the Boussinesq approximation of the the Cahn-Hilliard-Navier-Stokes equations [6],…”

“…[6] 3 This parameter should not be mixed with another classical parameter, also called as the capillary number, and defined as ηV /σ, with η, V , and σ being the typical values of viscosity, velocity, and surface tension. The classical capillary number is used in the sharp-interface model to define the ratio between the viscous and capillary forces.…”

Phase-field modelling of a miscible system in spinning droplet tensiometer

“…The full system of equations is compressible (due to dependence of mixture density on concentration) and its direct numerical solution is hardly feasible. The Boussinesq approximation of the Cahn-Hilliard-NavierStokes equations was derived in [27]. The equations in the Boussinesq approximation for a heterogeneous mixture of miscible liquids are similar to the equations previously used by e.g.…”