A diffuse-interface model for N-phase flows with liquid-solid phase change

This paper proposes and validates a reduction-consistent diffuse-interface model coupled with a lattice Boltzmann method to accurately and efficiently simulate N-phase flows involving liquid-solid phase change, including complex interactions with insoluble impurities.

The Gist

Imagine you are watching a complex kitchen experiment. You have a pot containing a mixture of oil, water, and air. Now, imagine you turn on the freezer. What happens? The water turns to ice, but the oil stays liquid, and the air gets trapped. As the water freezes, it expands (like a soda can bursting), pushing the oil and air around. If you have a drop of oil inside a drop of water, the freezing process gets even messier.

This paper is about building a super-smart computer simulation that can predict exactly how this chaotic freezing process happens, even when you have many different liquids mixed together (not just water and oil, but potentially dozens of different fluids).

Here is a breakdown of how the authors did it, using some everyday analogies:

1. The Problem: Freezing is Messy

In the real world, when things freeze, two things happen at once:

• The Boundary Moves: The line between liquid and solid (the "freezing front") moves forward.
• The Shape Changes: Because ice takes up more space than water (usually), the whole shape of the droplet changes. It might bulge out, crack, or form a sharp point on top.

Previous computer models were good at handling one type of liquid freezing, or they were good at handling many liquids mixing, but they struggled to do both at the same time, especially when the liquids change volume as they freeze.

2. The Solution: A "Smart Blur" Approach

The authors created a new mathematical model called a Diffuse-Interface Model.

• The Analogy: Imagine trying to draw a line between a red drop of paint and a blue drop of paint. A "Sharp Interface" model tries to draw a perfect, razor-thin line. But in reality, the colors blend a little bit at the edge.
• The Innovation: This model accepts that the edge is a "blur" or a "fuzzy zone." Instead of a hard line, it treats the boundary as a smooth gradient. This makes the math much more stable and allows the computer to handle complex shapes (like a bubble inside a droplet) without the simulation crashing.

They combined two powerful tools:

1. The Phase-Field (The Map): This acts like a GPS for the different liquids, telling the computer where the oil is, where the water is, and where the air is.
2. The Enthalpy Method (The Thermometer): This tracks the heat. It knows when a specific spot has cooled down enough to turn from liquid to solid.

3. The "Volume Change" Trick

One of the biggest headaches in freezing simulations is volume change.

• The Analogy: Think of a balloon filled with water. If you freeze it, the balloon expands. If your computer model doesn't account for this expansion, it's like trying to freeze a balloon in a rigid box—the math breaks because there's no room to grow.
• The Fix: The authors added a special "source term" (a mathematical adjustment) to their equations. It's like telling the computer, "Hey, as this part turns to ice, it's going to get bigger, so push the surrounding fluids out of the way." This ensures the simulation respects the laws of physics (conservation of mass) even when things swell up or shrink down.

4. The Engine: The Lattice Boltzmann Method (LBM)

To solve these complex equations, they used a method called Lattice Boltzmann.

• The Analogy: Instead of trying to solve the whole fluid as one giant, continuous blob (which is hard), imagine the fluid is made of millions of tiny, invisible billiard balls bouncing around on a grid. The computer tracks how these balls bounce and collide.
• Why it's cool: This method is incredibly fast and great at handling complex boundaries, like the jagged edge of a freezing front or a bubble trapped inside a droplet.

5. What Did They Test?

They ran several "experiments" in their computer to prove it works:

• The Film Test: Freezing a flat sheet of water. They checked if the computer correctly calculated how much the water expanded as it turned to ice. (It did!)
• The Single Drop: Freezing a water droplet. They saw the famous "conical tip" (a sharp point) form at the top, just like in real life.
• The Compound Drop: This was the tricky one. Imagine a drop of oil inside a drop of water. They froze it and watched how the ice pushed the oil around. The simulation showed that depending on whether the oil shrank or expanded, the final shape changed dramatically.
• The Impurity Test: They put a "foreign object" (like a bubble or a different droplet) inside the liquid. They watched the freezing front approach it.
  • The Result: If the impurity conducts heat well, the freezing front bends around it. If it's a bubble (insulator), the front gets stuck or changes shape. The simulation captured these subtle interactions perfectly.

The Big Picture

This paper is like giving scientists a high-definition, physics-accurate crystal ball.

Before this, if you wanted to study how sea ice forms with trapped air bubbles, or how to freeze-dry complex pharmaceuticals, or how to 3D print with multiple materials, you had to rely on rough approximations. Now, with this new model, researchers can simulate these complex, multi-liquid freezing scenarios with high precision.

It's a tool that helps us understand nature's freezing tricks, which could lead to better weather prediction, improved manufacturing, and more efficient energy storage.

Technical Summary

1. Problem Statement

The paper addresses the complex challenge of simulating N-phase fluid flows (involving two or more immiscible liquid phases) undergoing solid-liquid phase change (freezing).

• Context: This phenomenon is critical in natural environments (e.g., sea-ice formation) and industrial applications (e.g., additive manufacturing, electronic device fabrication).
• Challenges:
  • Multiphase Complexity: Unlike single-component freezing, multiphase systems involve complex interfacial dynamics between different liquid components and the surrounding gas/ambient fluid.
  • Volume Change: Solidification often involves density differences between liquid and solid phases, leading to volume expansion or contraction. This induces fluid motion and interface deformation that must be accurately captured.
  • Coupling: The problem requires simultaneous tracking of two dynamic interfaces: the fluid-fluid interface (gas-liquid or liquid-liquid) and the solid-liquid phase change front.
  • Existing Limitations: Previous numerical methods largely focused on pure single-component droplets or neglected the coupling between volume changes and multiphase hydrodynamics.

2. Methodology

The authors propose a comprehensive Diffuse-Interface Model coupled with a Lattice Boltzmann (LB) method to solve the governing equations.

A. Mathematical Model

The model integrates three key physical components:

1. Phase-Field Approach (Fluid-Fluid Interfaces):

   • Uses a conservative Allen-Cahn (AC) equation to track the interfaces between $N$ immiscible phases.
   • Introduces order parameters ($\phi_p$) to represent the volume fraction of each phase.
   • Ensures reduction-consistency, meaning the model rigorously degenerates to standard $N$-phase models when phase change is absent.
   • Calculates surface tension forces via a reduction-consistent total free energy functional.

2. Enthalpy-Based Approach (Solid-Liquid Interfaces):

   • Uses an enthalpy formulation to describe the phase change front.
   • Introduces a solid fraction ($f_s$) to distinguish between solid, liquid, and mushy zones.
   • Couples heat transfer with phase change, accounting for latent heat.

3. Mass Conservation and Volume Change:

   • Modifies the continuity equation to include a source term ($\dot{m}$) derived from the density difference between solid and liquid phases ($\rho_s \neq \rho_l$).
   • This source term drives the velocity field to account for volume expansion (when $\rho_s < \rho_l$) or contraction (when $\rho_s > \rho_l$), ensuring strict mass conservation.
   • Physical properties (density, viscosity, thermal conductivity) are defined as linear functions of the order parameters and solid fraction.

B. Numerical Solver

• Lattice Boltzmann Method (LBM): A coupled LB scheme is developed to solve the Navier-Stokes equations, the enthalpy equation, and the Allen-Cahn equation simultaneously.
• Forcing Terms:
  • A penalty force term is added to the momentum equation to enforce no-slip conditions in solid regions.
  • Source terms in the continuity and energy equations handle volume changes and latent heat effects.
• Boundary Conditions: The method supports complex wetting boundary conditions for N-phase flows on solid substrates.

3. Key Contributions

1. Unified N-Phase Freezing Model: The first proposed model capable of simulating N-phase flows with liquid-solid phase change while simultaneously handling volume changes due to density differences.
2. Reduction-Consistency: The model is mathematically rigorous, ensuring it reduces correctly to standard N-phase flow models (without phase change) and classical enthalpy methods (for single-phase freezing).
3. Accurate Volume Change Handling: By introducing a source term in the continuity equation, the method accurately captures the fluid motion induced by density-driven volume expansion/contraction, a feature often neglected in previous multiphase freezing simulations.
4. Versatility: The method is demonstrated to handle various complex configurations, including compound droplets (Janus, collar, lens, encapsulated modes) and systems with insoluble impurities.

4. Numerical Results and Validation

The authors validated the model through several benchmark tests and complex scenarios:

• Liquid Film Freezing:

  • Validated against analytical solutions for a single-component film.
  • Results showed excellent agreement in predicting freezing front position and gas-liquid interface movement caused by volume expansion.
  • Maximum error in solid-phase height was less than 5%.

• Multiphase Pool Freezing:

  • Simulated freezing of immiscible fluid layers and side-by-side fluids.
  • Demonstrated that the model correctly predicts interface deformation based on specific density ratios (e.g., expansion in one fluid causing upward movement of interfaces in others).

• Single and Compound Droplet Freezing:

  • Pure Droplet: Reproduced the formation of conical tips (or rounded caps depending on density ratios) and matched experimental data regarding droplet height and volume evolution.
  • Compound Droplets: Successfully simulated four distinct configurations (Janus, collar, lens, encapsulated). The model captured secondary solidification phenomena and the influence of thermal diffusivity differences on freezing rates.

• Freezing with Impurities:

  • Simulated a liquid pool containing suspended droplets and bubbles.
  • Captured the interaction between the advancing freezing front and impurities.
  • Key Finding: The model showed that impurities with higher thermal diffusivity cause the freezing front to bend (concave to convex) and can lead to the engulfment of droplets or the formation of pore defects (bubbles trapped in the solid).

5. Significance and Conclusion

• Accuracy and Efficiency: The proposed diffuse-interface LB method is proven to be accurate, efficient, and robust for studying complex N-phase systems with phase change.
• Physical Insight: The study provides new insights into how thermal diffusivity differences and density ratios influence freezing front morphology, impurity engulfment, and final solid structures in multiphase systems.
• Applications: This methodology offers a powerful tool for optimizing industrial processes like additive manufacturing and understanding natural phenomena like sea-ice formation, where multiphase interactions and phase change occur simultaneously.
• Limitations & Future Work: The current model assumes a uniform melting temperature for all species. Future work will aim to address systems with varying melting points and potentially more complex solid-phase deformations (e.g., shell rupture).

In summary, this paper presents a significant advancement in computational fluid dynamics by unifying multiphase flow physics with phase-change thermodynamics, providing a rigorous framework for simulating complex freezing scenarios involving multiple immiscible components and volume changes.