Structure preserving finite element schemes for the Navier-Stokes-Cahn-Hilliard system with degenerate mobility

TL;DR

This work develops two structure-preserving finite element schemes for the Navier-Stokes-Cahn-Hilliard system with degenerate mobility, exploiting a truncated non-degenerate mobility $M_\varepsilon(\phi)$ and singular functionals $G_\varepsilon(\phi)$ and $J_\varepsilon(\phi)$ to enforce approximate maximum principles. Both the $G_\varepsilon$-scheme and the $J_\varepsilon$-scheme conserve mass and are energy-stable at the discrete level, with discrete bounds on $\phi$ and iterative solvers built around decoupled, Newton-based updates. Numerical experiments in 2D demonstrate accurate temporal convergence (order one), energy dissipation without external forcing, and bounds preservation for degenerate mobility, along with qualitative differences from the constant-mobility model in droplets merging, coarsening, and rotating-fluid scenarios. The results establish the first finite-element NSCH schemes with degenerate mobility that maintain conservation, energy stability, and approximate maximum principles, providing reliable and physically consistent simulations for two-phase flows with variable mobility.

Abstract

In this work we present two new numerical schemes to approximate the Navier-Stokes-Cahn-Hilliard system with degenerate mobility using finite differences in time and finite elements in space. The proposed schemes are conservative, energy-stable and preserve the maximum principle approximately (the amount of the phase variable being outside of the interval [0,1] goes to zero in terms of a truncation parameter). Additionally, we present several numerical results to illustrate the accuracy and the well behavior of the proposed schemes, as well as a comparison with the behavior of the Navier-Stokes-Cahn-Hilliard model with constant mobility.

Figures (16)

• Figure 1: Example I. Experimental Order of Convergence. Reference solution for $\phi$ (left), $|{\textbf{u}}|$ (center) and $p$ (right).
• Figure 2: Example II. Merging droplets. Evolution in time of $\phi$ and ${\textbf{u}}$ for $G_\varepsilon$-scheme at times $t=0, 0.2, 0.5, 1.5$ and $5$.
• Figure 3: Example II. Merging droplets. Evolution in time of $\phi$ and ${\textbf{u}}$ for $J_\varepsilon$-scheme at times $t=0, 0.2, 0.5, 1.5$ and $5$.
• Figure 4: Example II. Merging droplets. Evolution in time of $\phi$ and ${\textbf{u}}$ for CM-scheme at times $t=0, 0.2, 0.5, 1.5$ and $5$.
• Figure 5: Example II. Merging droplets. Evolution of the energies and the volume of the system

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Definition 3.1
• Remark 3.2