[Paper Review] Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

Helmut Abels, Harald Garcke+1 more|arXiv (Cornell University)|Jan 1, 2011

Lattice Boltzmann Simulation StudiesEngineering17 references20 citations

TL;DR

This paper presents a thermodynamically consistent, frame-indifferent diffuse interface model for incompressible two-phase flows with different densities, derived via rational continuum mechanics. Using matched asymptotic expansions, it rigorously derives sharp interface limits, including a novel term from surface diffusion, and proves all resulting models satisfy natural energy inequalities.

ABSTRACT

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

Motivation & Objective

• To develop a diffuse interface model for incompressible two-phase flows with distinct fluid densities that satisfies thermodynamic consistency and frame indifference.
• To extend existing models like Model H and Lowengrub-Truskinovsky by incorporating variable density effects while preserving physical consistency.
• To rigorously derive sharp interface limits through matched asymptotic expansions, accounting for different scalings of mobility.
• To identify and include a new surface diffusion term in the sharp interface limit when mobility scales appropriately.
• To prove that all derived sharp interface models satisfy natural energy inequalities, ensuring thermodynamic stability.

Proposed method

• Formulates a diffuse interface model using rational continuum mechanics, ensuring local and global dissipation inequalities are satisfied.
• Introduces a free energy density proportional to $\rho\hat{\sigma}(\varepsilon^{-1}\psi(c) + \varepsilon|\nabla c|^2/2)$, where $\rho$ is the variable density.
• Derives the momentum equation with a capillary force term $-\hat{\sigma}\varepsilon\operatorname{div}(\rho\nabla c \otimes \nabla c)$, ensuring frame indifference.
• Uses the Cahn-Hilliard equation for phase evolution with a mobility $m(c)$, and defines chemical potential $\mu$ to include pressure and curvature contributions.
• Applies matched asymptotic expansions in the limit $\varepsilon \to 0$ to derive sharp interface models from the diffuse interface system.
• Analyzes scaling of the mobility $m$ to distinguish between classical models and those allowing bulk or surface diffusion in the sharp limit.

Experimental results

Research questions

• RQ1 How can a diffuse interface model for two-phase incompressible flows with different densities be constructed to satisfy thermodynamic consistency and frame indifference?
• RQ2 What are the sharp interface limits of the diffuse interface model under different scalings of the mobility coefficient?
• RQ3 Does the inclusion of surface diffusion in the sharp interface limit lead to a new physical term in the momentum balance at the interface?
• RQ4 Can all derived sharp interface models be shown to satisfy natural energy inequalities?
• RQ5 How does the variable density model differ mathematically and physically from classical Model H and Lowengrub-Truskinovsky models?

Key findings

• The proposed diffuse interface model satisfies local and global dissipation inequalities and is frame indifferent, ensuring thermodynamic and physical consistency.
• Through matched asymptotic expansions, the model yields sharp interface limits that depend on the scaling of the mobility $m$ in the Cahn-Hilliard equation.
• When mobility scales as $\varepsilon^2$, a new term from surface diffusion emerges in the momentum balance at the interface, modifying the classical sharp interface model.
• In the classical scaling, the model reduces to a standard sharp interface model with surface tension and bulk forces.
• All derived sharp interface models fulfill natural energy inequalities, confirming their thermodynamic stability.
• The analysis confirms that the variable density model correctly captures the physics of two-phase flows with different densities, including the correct pressure and chemical potential coupling.