Diffuse-interface modeling and simulation of the freezing of binary fluids with the Marangoni effect

TL;DR

The paper develops a reduction-consistent diffuse-interface model that couples phase-field, enthalpy-based temperature, and constrained solute-transport formulations to simulate gas–liquid–solid freezing with Marangoni effects. A comprehensive lattice Boltzmann framework then solves the coupled dynamics for the gas–liquid interface, solid–liquid front, flow, temperature, and solute fields, preserving mass and capturing volume changes due to density differences. The approach is validated against classical Marangoni benchmarks, three-phase Stefan problems, and binary solidification scenarios, and applied to binary droplet freezing and impurity-laden flows, showing good agreement with theory and experiments. This work advances predictive modeling of complex freezing phenomena where Marangoni convection, phase change, and solute redistribution interact in multiphase environments, with implications for manufacturing, energy storage, and environmental processes.

Abstract

This paper proposes a diffuse-interface model for simulating gas-liquid-solid multiphase flows involving solid-liquid phase change, solute transport, and the Marangoni effect. In this model, a phase-field method is employed to capture the evolution of fluid-fluid interfaces, while an enthalpy-based approach is used to describe the temperature field and implicitly track the solid-liquid interface. Solute transport is modeled using a constrained scalar-transport model combined with a pseudo-potential concentration approach. The proposed diffuse-interface model satisfies the reduction consistency, and can degenerate to the conservative phase-field method for incompressible two-phase flow and the classical enthalpy method for binary material solidification in an appropriate way. Furthermore, the model not only can preserve the mass conservation, but also can capture the volume change induced by phase change. To solve the diffuse-interface model, a lattice Boltzmann (LB) method is then developed, and the numerical tests demonstrate that the method has a good performance in the study of the freezing process coupled with Marangoni flow, phase-change-induced volume change, and solute transport. Finally, the model is applied to investigate the freezing dynamics of a system containing an insoluble impurity, revealing the complex interaction between the advancing freezing front and the impurity. It is found that the numerical results are in good agreement with experimental data.

Figures (10)

• Figure 1: Schematic diagram of the problem. (a) The freezing of a binary droplet on a cold substrate. The streamlines schematically illustrate the direction of the Marangoni flow inside the droplet. $\Omega_g$ ($\phi=-1 \cap f_s=0$), $\Omega_l$ ($\phi=1 \cap f_s=0$), and $\Omega_s$ ($\phi=1 \cap f_s=1$) are filled with gas, liquid and solid phases. $\Gamma_{sl}$ indicates the freezing front between the solid and liquid phases, and $\Gamma_{mg}$ represents the interface between the solid-liquid mixture and the gas phase. (b) A zoom-in map illustrates the interface between the solid-liquid mixture and the gas phase $\Gamma_{mg}$. From left to right, it displays the one-dimensional diffuse interface, the microstructure of this interface, and a close-up view of the microstructure, schematically indicating that the fingers are assumed to be thin structures.
• Figure 2: (a) A schematic diagram of the droplet migration under the Marangoni effect. (b) The velocity around the rising droplet at $t^* = 10$. (c) A comparison of the normalized migration velocity dimensionless time $t^*$, the black dashed line indicates the theoretical prediction in the limit of vanishing Reynolds and Marangoni numbers. (d) The distributions of velocity and isotherm around the droplet at $Ma = 1$, $Ma = 10$ and $Ma = 100$, from left to right.
• Figure 3: (a) A schematic diagram of the Marangoni convection of layered immiscible fluids. Velocity and temperature profiles along the center line (b). The distributions of temperature (c) and velocity (d) of thermocapillary flows with $\lambda_l / \lambda_h = 1.0$. The analytical and numerical results are indicated by red and blue, respectively.
• Figure 4: (a) Schematic diagram of the three-phase Stefan problem, the domains occupied by gas, liquid and solid phases are denoted by $\Omega_g$ (white region), $\Omega_l$ (blue region), and $\Omega_s$ (yellow region), $\Gamma_{sl}$ indicates the freezing front, and $\Gamma_{mg}$ represents the interface between the phase-change material and gas phase. (b) Comparisons of the freezing front evolution between numerical and analytical solutions under different values of solid–liquid density ratio $\rho_s/\rho_l$ at $Ste = 0.1$ and $\alpha_r = 1.0$.
• Figure 5: (a) A schematic diagram of binary solidification. (b) the evolution of the freezing front. (c) the distribution of solute concentration. (d) the distribution of temperature. (e) A macroscopic view of concentration distribution.