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

TL;DR

This paper derives the Boussinesq approximation for the Cahn-Hilliard-Navier-Stokes equations, enabling simplified modeling of complex multiphase systems with evolving interfaces, relevant for processes like dissolution, nucleation, and phase changes.

Contribution

It provides a rigorous derivation of the Boussinesq approximation for the full Cahn-Hilliard-Navier-Stokes equations using multiple-scale analysis, applicable to various thermodynamic and hydrodynamic phenomena.

Findings

Derived the Boussinesq approximation for complex multiphase flow equations.

Established a universal model for systems with evolving interfaces.

Facilitated analysis of dissolution, nucleation, and phase change processes.

Abstract

We study the interactions between the thermodynamic transition and hydrodynamic flows which would characterise a thermo- and hydro-dynamic evolution of a binary mixture in a dissolution/nucleation process. The primary attention is given to the slow dissolution dynamics. The Cahn-Hilliard approach is used to model the behaviour of evolving and diffusing interfaces. An important peculiarity of the full Cahn-Hilliard-Navier-Stokes equations is the use of the full continuity equation required even for a binary mixture of incompressible liquids, firstly, due to dependence of mixture density on concentration and, secondly, due to strong concentration gradients at liquids' interfaces. Using the multiple-scale method we separate the physical processes occurring on different time scales and, ultimately, provide a strict derivation of the Boussinesq approximation for the Cahn-Hilliard-Navier-Stokes equations. This approximation forms a universal theoretical model that can be further employed for a thermo/hydro-dynamic analysis of the multiphase systems with strongly evolving and diffusing interfacial boundaries, i.e. for the processes involving dissolution/nucleation, evaporation/condensation, solidification/melting, polymerisation, etc.