Thermodynamically consistent phase-field modeling and numerical simulation for two-phase fluid-solid dynamics

TL;DR

The paper develops a thermodynamically consistent diffuse-interface model coupling Cahn–Hilliard phase-field dynamics with Navier–Stokes flow to capture two-phase fluid–solid interactions, including a solid phase that is effectively immobile. It introduces a fully discrete, energy-stable finite element scheme built on a semi-implicit time stepping and a convex–concave split of the double-well potential, proving a discrete free-energy dissipation inequality. A preprocessing strategy enables generating initial phase fields from sharp-interface descriptions, facilitating simulations in complex geometries, while monolithic and partitioned solution strategies are analyzed and compared. The method is demonstrated on lid-driven cavity and channel-flow setups, including extensions to precipitation/dissolution and chemically reacting ion transport, illustrating robust energy decay, accurate interface dynamics, and potential for reactive-flow applications.

Abstract

We introduce a coupled Cahn-Hilliard Navier-Stokes model that governs the two-phase dynamics of a system that consists of a fluid and a solid phase and prove its thermodynamic consistency. Moreover, we present an associated fully-discrete numerical method that relies on a continuous finite element approach and a semi-implicit time-stepping method. As the main theoretical result we show that the fully-discrete method satisfies a discrete analog of the free energy dissipation inequality. Numerical experiments confirm the theoretical findings and show the applicability of the method for realistic settings including an extension to chemically reacting flow. In this context, we provide a preprocessing strategy that enables computing fluid flow in complex geometries given a sharp-interface formulation of the initial phase distribution. Moreover, we briefly introduce different solution strategies for the novel discretization based on the monolithic and partitioned solution paradigms and assess these in a comparative study.

Figures (9)

• Figure 1: Lid-driven cavity inspired setup with one solid inclusion. On the left, we depict the result of the first time step after the inflow is stopped. On the right, we show the result of the final time step. The fluid phase is colored red and the solid phase blue. The phase transition is illustrated by the transition in color. The arrows are scaled according to the magnitude of the conserved quantity $\mathbf{w} := \tilde{\phi}_f \mathbf{v}$. The color scaling is due to the data ranges presented on the right of the respective subfigure. Note that the scaling is two orders of magnitude smaller in the right subfigure.
• Figure 2: Energy plot for the lid-driven cavity inspired setup with one solid inclusion and stopped inflow (dashed line).
• Figure 3: Lid-driven cavity inspired setup with two solid inclusions. On the left, we depict the result of the first time step after the inflow is stopped. On the right, we show the result of the time step where the smaller inclusion is completely vanished. The arrows are scaled according to the magnitude of the conserved quantity $\mathbf{w} := \tilde{\phi}_f \mathbf{v}$. The color scaling is due to the data ranges presented on the right of the respective subfigure, which is an order of magnitude smaller in the right subfigure.
• Figure 4: Energy plot for the lid-driven cavity inspired setup with two solid inclusions and stopped inflow (dashed line).
• Figure 5: Sharp-interface configuration of the channel-flow inspired setup.

Theorems & Definitions (6)

• Remark 2.1: Modeling aspects
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Lemma A.1