A Thermodynamically Consistent Free Boundary Model for Two-Phase Flows in an Evolving Domain with Bulk-Surface Interaction

This paper derives a thermodynamically consistent free boundary model for two-phase flows in an evolving domain that incorporates bulk-surface interactions via convective Cahn--Hilliard equations and generalized Navier slip conditions, unifying previous models through rigorous derivation via Lagrange Multiplier and Energetic Variational approaches.

The Gist

Imagine you are watching a drop of oil floating in water, or a tiny bacterium swimming through a pond. In the real world, these things aren't static; they wiggle, stretch, and change shape. The fluids inside them mix and separate, and the surface they sit on might be soft and squishy, moving along with the fluid.

This paper is about building a mathematical rulebook to describe exactly how these complex, moving, two-part systems behave. The authors, Patrik Knopf and Yadong Liu, have created a new, highly accurate model that connects the physics of the "inside" (the bulk) with the physics of the "skin" (the boundary).

Here is the breakdown of their work using simple analogies:

1. The Big Picture: A Moving, Breathing World

Most old models treated the container (like a glass of water) as a rigid, unchanging box. But in nature, containers often move.

• The Analogy: Think of a bacterium. It's not just a blob of water in a jar; the bacterium is the jar. Its skin (cell wall) moves because the water inside pushes against it.
• The Innovation: This paper creates a model where the "jar" (the domain) is alive. It expands, contracts, and flows exactly as the fluid inside pushes it. The boundary isn't a wall; it's a surfboard riding on the waves of the fluid inside.

2. The Two Layers: The Ocean and the Shore

The model looks at two distinct places that talk to each other:

• The Bulk (The Ocean): The fluid inside the domain.
• The Surface (The Shore): The boundary layer where the fluid meets the outside world.

The Problem with Old Models:
Imagine a shoreline where the sand (the boundary) and the water (the bulk) never mix. In old math models, the water hitting the shore was forced to stop dead (no-slip), and the "mixing" of materials between the water and the sand was impossible. It was like saying a sponge can't absorb water from a cup.

The New Solution:
This model treats the boundary as a living, breathing skin.

• Material Transfer: Just like a sponge soaking up water, this model allows material to move from the "ocean" (bulk) to the "shore" (surface) and vice versa. This is crucial for things like cells absorbing nutrients or a droplet soaking into a soft surface.
• Variable Contact Angles: In old models, a water droplet always hit a wall at a perfect 90-degree angle (like a square block). In reality, a droplet can be flat or round. This model allows the droplet to "lean" at any angle, changing dynamically as it moves.

3. The "Traffic Rules" (The Equations)

To make this work, the authors combined two famous sets of physics rules:

1. Navier-Stokes (The Traffic Flow): This describes how the fluid moves (like cars on a highway).
2. Cahn-Hilliard (The Mixing Paint): This describes how two fluids (like oil and water) separate or mix, creating a fuzzy boundary rather than a sharp line.

The "Secret Sauce":
The authors added Dynamic Boundary Conditions.

• Think of it like a dance floor: In old models, the dancers on the edge of the floor (the boundary) were frozen in place. In this new model, the dancers on the edge can slide, spin, and even swap partners with the dancers in the middle.
• They also added a Generalized Navier Slip. Imagine a car driving on a road. Old models said the tires must be perfectly stuck to the road (no sliding). This model says, "Hey, tires can slide a little bit!" This is essential for describing how a contact line (where the fluid meets the solid) actually moves.

4. How They Built It: The "Thermodynamic" Guarantee

The authors didn't just guess the equations. They built them from the ground up using two different "construction methods" to ensure they were physically honest:

• Method A (The Accountant): They used "Lagrange Multipliers" to balance the books. They made sure that mass (stuff) and energy (movement/heat) are conserved perfectly. Nothing disappears into thin air.
• Method B (The Architect): They used the "Energetic Variational Approach," which is like designing a building to be the most energy-efficient structure possible. They ensured the system always loses energy in a natural way (like friction slowing down a spinning top), which is a fundamental law of the universe.

5. Why Does This Matter?

This isn't just abstract math; it's a tool for the future.

• Biology: It helps us understand how bacteria move, how cells divide, or how proteins move along a cell wall.
• Engineering: It helps design better micro-fluidic devices (tiny chips that move fluids) or understand how droplets spread on soft, deformable materials (like skin or soft robotics).
• Nature: It explains how a drop of rain behaves when it hits a leaf that is bending in the wind.

Summary

In short, this paper gives us a thermodynamically consistent, moving, 3D movie of how two fluids interact when they are inside a container that is also moving and changing shape. It fixes the "frozen boundary" problem of the past, allowing for realistic sliding, soaking, and angle-changing, making our mathematical understanding of the fluid world much closer to reality.

Technical Summary

1. Problem Statement

The paper addresses the mathematical modeling of two-phase flows (mixtures of two viscous, incompressible fluids) occurring within a time-evolving domain $\Omega(t)$. Unlike traditional models where the domain is fixed, this work considers scenarios where the domain boundary $\Gamma(t) = \partial\Omega(t)$ moves and deforms in response to the fluid velocity field.

Key challenges addressed include:

• Bulk-Surface Interaction: Modeling the exchange of material between the bulk fluid and the boundary surface (e.g., absorption, adsorption, or chemical reactions on the surface).
• Variable Contact Angles: Moving beyond the restrictive assumption of a fixed 90-degree contact angle, allowing the interface between the two phases to meet the boundary at a dynamic angle.
• Thermodynamic Consistency: Ensuring the model strictly adheres to the laws of thermodynamics, specifically mass conservation and energy dissipation, in a moving domain setting.
• Moving Contact Lines: Accurately describing the motion of the contact line where the phase interface meets the moving boundary, which requires relaxing the standard no-slip boundary condition.

2. Methodology

The authors derive the governing equations using two distinct mathematical approaches, both starting from local mass balance laws:

A. The Lagrange Multiplier Approach (General Case)

This approach is used to derive the full model without restrictive assumptions on density or mass flux.

• Local Energy Dissipation: The authors formulate local energy dissipation inequalities for both the bulk and the surface.
• Constitutive Assumptions: They introduce Lagrange multipliers (chemical potentials $\mu, \theta$ and surface pressure $q$) to enforce constraints (incompressibility, mass conservation).
• Flux Identification: By ensuring the dissipation inequality holds, they identify the constitutive relations for stress tensors, mass fluxes, and boundary conditions. This allows for a general derivation that includes unmatched densities and mass transfer between bulk and surface.

B. The Energetic Variational Approach (EnVarA) (Special Case)

This approach is applied under specific simplifying assumptions: matched densities ($\tilde{\rho}_1 = \tilde{\rho}_2$) and no mass flux between bulk and surface.

• Least Action Principle: Used to derive inertial and conservative forces from the kinetic and free energy functionals.
• Maximum Dissipation Principle (Onsager's Principle): Used to derive dissipative forces (viscosity, friction) from the Rayleigh dissipation functional.
• Force Balance: The final equations are obtained by balancing the sum of inertial, conservative, and dissipative forces to zero.

3. Key Contributions and Model Formulation

The paper presents a novel Navier–Stokes–Cahn–Hilliard (NS-CH) system coupled with dynamic boundary conditions.

The Governing System

The model consists of:

1. Bulk Dynamics:
   • Navier–Stokes Equation: Describes the momentum balance of the mixture with phase-dependent density $\rho(\phi)$ and viscosity $\nu(\phi)$. It includes an additional stress term due to interfacial tension (Korteweg stress).
   • Convective Cahn–Hilliard Equation: Describes the evolution of the bulk phase-field $\phi$, coupled with the fluid velocity.
2. Surface Dynamics:
   • Surface Cahn–Hilliard Equation: A dynamic boundary condition describing the evolution of the surface phase-field $\psi$.
   • Coupling: The bulk and surface equations are coupled via boundary conditions that allow for material transfer (controlled by parameters $K$ and $L$).
3. Boundary Conditions:
   • Generalized Navier Slip: Replaces the no-slip condition to allow for moving contact lines. The tangential velocity is proportional to the tangential stress plus a forcing term related to surface tension gradients.
   • Dynamic Boundary Conditions: The normal velocity of the boundary is determined by the normal component of the fluid velocity ($u \cdot n = V$), making it a free boundary problem.
   • Contact Angle Dynamics: The boundary conditions for $\phi$ and $\mu$ (controlled by parameters $K, L$) allow the contact angle to evolve dynamically rather than being fixed at 90 degrees.

Key Physical Properties

The derived model satisfies:

• Mass Conservation: Total mass is conserved; if $L=\infty$, bulk and surface masses are conserved separately.
• Energy Dissipation: The system satisfies a global energy dissipation law, proving thermodynamic consistency. The total energy (kinetic + bulk free energy + surface free energy) decreases over time due to viscosity, friction, and diffusion.
• Volume and Area Preservation: Due to the incompressibility constraints ($\text{div } u = 0$ and $\text{div}_\Gamma u = 0$), the volume of the domain and the area of the boundary are preserved.

4. Results

• Derivation Verification: The authors successfully derived the same system using two different methods, validating the robustness of the model.
• Generalization: The model is shown to generalize existing literature:
  • Setting $K=L=\infty$ recovers models with fixed 90-degree contact angles and no surface mass transfer.
  • Fixing the domain boundary ($V=0$) recovers the standard NS-CH models with dynamic boundary conditions found in previous works (e.g., [50]).
  • The AGG model (Abels-Garcke-Grün) is recovered as a special case in a fixed domain.
• Parameter Analysis: The paper details how the parameters $K$ and $L$ control the physical behavior:
  • $K=0$: Imposes a linear relation between bulk and surface phase fields (rapid equilibration).
  • $L \in (0, \infty)$: Allows for a finite rate of mass transfer (absorption/adsorption).
  • $L=\infty$: Enforces zero mass flux across the boundary.

5. Significance

• Biological and Physical Applications: The model is particularly relevant for simulating biological systems (e.g., moving bacteria, cell membranes) and soft matter physics (e.g., droplets on deformable substrates) where the domain geometry changes dynamically due to fluid motion.
• Theoretical Advancement: It provides a rigorous thermodynamic framework for two-phase flows on evolving manifolds, a topic previously lacking a consistent formulation that handles both bulk-surface mass transfer and variable contact angles simultaneously.
• Flexibility: The inclusion of parameters $K$ and $L$ allows the model to be tuned for various physical scenarios, from impermeable boundaries to highly absorptive surfaces.
• Future Directions: The paper sets the stage for future analytical work (proving well-posedness) and numerical simulations to approximate these complex coupled dynamics on evolving geometries.

In summary, this paper establishes a comprehensive, thermodynamically consistent mathematical framework for two-phase flows in evolving domains, bridging the gap between bulk fluid dynamics and surface phenomena through a sophisticated coupling of Navier–Stokes and Cahn–Hilliard equations.