A hybrid Volume of Fluid Phase-Field method for Direct Numerical Simulations of soluble surfactant-laden interfacial flows

This paper presents a hybrid Volume-of-Fluid Phase-Field method with adaptive mesh refinement for direct numerical simulations of soluble surfactant-laden flows, which accurately captures the coupling between bulk and interfacial transport to demonstrate how Marangoni stresses significantly alter bubble rise dynamics in three-dimensional geometries.

The Gist

The Big Picture: A New Way to Simulate "Soapy" Water

Imagine you are watching a bubble rise through a glass of water. If the water is pure, the bubble shoots straight up, wobbling a bit but moving fast. But if you add soap (surfactants) to the water, the bubble behaves differently. It might slow down, wobble more, or even change its path.

This happens because soap molecules love to stick to the surface of the bubble. As the bubble moves, these molecules get pushed around, creating uneven "tension" on the skin of the bubble. This uneven tension acts like invisible hands pushing the bubble in different directions, changing how it moves.

The Problem:
Simulating this on a computer is incredibly hard. It's like trying to film a soap bubble with a camera that has two conflicting settings:

1. The Sharp Camera: Needs to see the bubble's edge as a razor-thin line (to calculate pressure and shape).
2. The Blurry Camera: Needs to see the soap molecules spreading out smoothly over that edge (to calculate how the soap moves).

Most computer methods force you to choose one camera setting, making the simulation either physically inaccurate or computationally impossible to run for complex 3D shapes.

The Solution:
The authors of this paper built a hybrid method. Think of it as a "split-screen" simulation that uses the best of both worlds simultaneously:

• The Sharp Edge (Volume-of-Fluid): They use a method that keeps the bubble's edge sharp and perfectly conserves the amount of liquid (like a high-definition outline).
• The Smooth Soap (Phase-Field): They use a second, "fuzzy" layer that acts as a smooth highway for the soap molecules to travel on. This allows the soap to move naturally between the water and the bubble surface without getting stuck or lost.

How It Works: The "Traffic Controller" Analogy

To make this work, the authors created a digital traffic system for the soap molecules:

1. The Highway (The Interface): The bubble's surface is a busy highway. The soap molecules are cars.
2. The On-Ramps and Off-Ramps (Adsorption/Desorption):
   • Adsorption: Soap molecules from the water (the bulk) want to jump onto the highway (the bubble surface).
   • Desorption: Soap molecules get tired and jump off the highway back into the water.
   • The new method calculates exactly how many cars jump on or off at every single moment, ensuring no cars disappear or appear out of thin air.
3. The Traffic Jam (Marangoni Stress): When too many soap cars pile up in one spot on the bubble, that spot gets "sticky" (high tension). The bubble skin tries to pull away from the sticky spot, creating a force that slows the bubble down or makes it wobble. The simulation captures this tug-of-war perfectly.

What They Tested (The "Driving Tests")

Before letting their new car drive on the highway, they ran it through a driving school with three specific tests:

1. The Stretch Test (Expanding Sphere): They blew up a bubble covered in soap. They checked if the soap spread out evenly as the bubble got bigger. The simulation matched the math perfectly.
2. The Spin Test (Rotating Bubble): They spun a bubble with soap on it. They checked if the soap moved correctly around the circle without leaking. Again, the simulation was spot-on.
3. The Exchange Test (Flat Wall): They watched soap move from the water to a flat wall and back. They tested three scenarios:
   • Only jumping on: Does the soap stick? Yes.
   • Only jumping off: Does the soap leave? Yes.
   • Both: Does it find a balance? Yes.

The Main Event: The Rising Bubble

Finally, they let their new method simulate a bubble rising in a 3D tank of water.

• The Clean Bubble: It rose relatively fast and straight.
• The "Insoluble" Soap Bubble: The soap was stuck to the surface and couldn't leave. It created a strong "traffic jam" at the back of the bubble, slowing it down significantly.
• The "Soluble" Soap Bubble (The Real Deal): This is where the new method shines. The soap could jump on and off the bubble.
  • If the soap jumped off easily (high "desorption"), the bubble behaved almost like a clean one.
  • If the soap jumped on easily (high "adsorption"), the bubble behaved like the stuck-soap version.
  • In the middle, the bubble showed a complex dance: it slowed down, changed its path, and left a "trail" of soap in the water behind it as it rose.

Why This Matters (According to the Paper)

The authors claim this method is robust, scalable, and accurate.

• Robust: It doesn't crash when the bubble gets weird shapes or breaks apart.
• Scalable: It can run on supercomputers to handle huge, complex 3D simulations efficiently.
• Accurate: It correctly predicts how fast bubbles rise and how they wiggle, matching real-world physics.

In short: They built a new digital engine that can finally simulate how soap bubbles behave in 3D space, handling the tricky dance between the bubble's shape and the soap molecules moving on and off its skin, all without losing accuracy or crashing the computer.

Technical Summary

Technical Summary: A Hybrid Volume-of-Fluid and Phase-Field Method for Soluble Surfactant-Laden Interfacial Flows

Problem Statement
The accurate numerical simulation of interfacial flows involving soluble surfactants presents a formidable challenge due to the multiscale and multiphysics nature of the problem. Unlike insoluble surfactants, soluble species are governed by a coupled system involving bulk convection–diffusion, interfacial transport, and kinetic exchange (adsorption/desorption) processes. These interactions generate spatially and temporally varying surface tension fields, leading to Marangoni stresses that significantly alter flow topology, stability, and motion. Existing numerical methods face distinct limitations:

• Boundary Integral/Element Methods: Restricted to Stokes flow regimes and struggle with topology changes (breakup/coalescence).
• Front-Tracking: High accuracy but complex implementation in 3D, requiring explicit topological treatment and facing load-balancing issues in parallel computing.
• Level-Set Methods: Naturally handle topology changes but often suffer from mass conservation issues requiring reinitialization and post-hoc corrections.
• Phase-Field Methods: Handle topology changes naturally but often rely on thermodynamic equilibrium assumptions that can be violated by strong convection, leading to mass non-conservation.
• Volume-of-Fluid (VoF) Methods: Excellent for mass conservation and sharp interfaces but traditionally struggle with defining surface differential operators required for surfactant transport on a zero-thickness interface.

There is a specific need for a geometric VoF-based method that supports Adaptive Mesh Refinement (AMR), handles soluble surfactants with dynamic bulk-interface exchange, and operates within a scalable, open-source framework.

Methodology
The authors present a hybrid numerical framework implemented in the open-source code Basilisk, which combines a geometric Volume-of-Fluid (VoF) method with a diffuse Phase-Field approach.

1. Interface Representation:

   • VoF ($c$): A scalar field representing the volume fraction of one fluid, used for conservative interface tracking and momentum solution. It ensures exact mass conservation and supports geometric reconstruction.
   • Phase-Field ($\phi$): An auxiliary variable evolving via an Allen-Cahn-based Accurate Conservative Diffuse-Interface (ACDI) formulation. It serves as a smooth carrier for surfactant transport, providing regularized normal vectors, curvature, and surface gradients without the need for explicit surface derivative computations on a sharp interface.
   • Coupling: The Phase-Field is periodically reinitialized from the VoF field using a signed-distance function to maintain consistency.

2. Governing Equations:

   • Fluid Motion: Incompressible Navier-Stokes equations with spatially varying density and viscosity. Surface tension forces are incorporated as volumetric body forces: an isotropic capillary force ($\sigma \kappa \mathbf{n} \delta_s$) and a tangential Marangoni force ($\nabla_s \sigma \delta_m$).
   • Surfactant Transport: Two scalar fields are solved: bulk concentration ($F$) and interfacial concentration ($f$, a volumetric representation of $\Gamma$).
     • The interfacial transport equation is regularized to avoid surface derivatives, converting the surface concentration $\Gamma$ into a volumetric field $f = \Gamma \delta_\phi$ localized within the diffuse interface.
     • Kinetics: Adsorption and desorption are modeled via regularized source/sink terms ($J$) localized at the interface, linking bulk and interfacial concentrations. The model supports general equations of state (e.g., Langmuir isotherm) to relate surface tension to concentration.

3. Numerical Discretization:

   • Spatial: Finite-volume discretization on dynamically refined quadtree/octree grids (AMR). Resolution is concentrated near the interface to resolve steep gradients in concentration and flow.
   • Temporal: Operator splitting is used. Advection is treated explicitly (geometric unsplit Bell–Collela–Glaz scheme), while diffusion, anti-diffusion, and source terms are treated implicitly using a modified multigrid Poisson solver.
   • Stability: A Courant–Friedrichs–Lewy (CFL) condition is enforced based on an effective velocity that includes both convective and anti-diffusive contributions to ensure stability in the presence of strong interface gradients.

Key Contributions

• Hybrid Framework: Development of a method that retains the mass-conserving and sharp-interface advantages of VoF while utilizing a Phase-Field variable solely as a smooth carrier for surfactant transport, avoiding the geometric complexities of surface derivatives on piecewise-linear interfaces.
• Soluble Surfactant Capability: Extension of previous VoF-Phase-Field work (limited to insoluble surfactants) to the fully soluble case, incorporating coupled bulk advection-diffusion and kinetic exchange terms.
• Scalability and Adaptivity: The framework is fully adaptive (AMR) and supports 2D, axisymmetric, and 3D configurations with high parallel scalability, enabling the resolution of multiscale gradients in both bulk and interfacial fields.
• Open-Source Implementation: The method is implemented in Basilisk, providing a reproducible and accessible tool for the community.

Results and Validation
The paper validates the method through a series of benchmark tests and application cases:

1. Validation without Flow:

   • Expanding Sphere (3D): Verified the recovery of analytical solutions for insoluble surfactant dilution during isotropic expansion.
   • Rotating Interface (2D): Confirmed second-order convergence for insoluble surfactant advection-diffusion on a moving interface.
   • Adsorption/Desorption on Flat Interface: Validated the coupled bulk-interface solver against analytical solutions for constant flux adsorption, pure desorption, and full Langmuir kinetics. The method correctly captures the transition to equilibrium and the formation of depletion/enrichment layers in the bulk.

2. Axisymmetric Rising Bubble:

   • Insoluble Limit: Demonstrated the reduction of terminal rise velocity due to Marangoni stresses, with results converging to the clean-interface limit as surfactant concentration decreases.
   • Soluble Dynamics: Systematically varied the Biot number (desorption) and Damköhler number (adsorption).
     • High Biot (Fast Desorption): The system approaches the clean-interface limit as surfactants are rapidly removed from the interface.
     • High Damköhler (Fast Adsorption): The system approaches the surface-incompressible (insoluble) limit as the interface rapidly saturates.
   • Flow Structure: Visualizations showed the formation of "stagnation caps" of surfactant at the bubble rear, which suppress internal recirculation and modify the wake structure.

3. Three-Dimensional Rising Bubble:

   • Simulations of a bubble rising in 3D with soluble surfactants captured complex phenomena including shape deformation, lateral trajectory deviations (helicoidal motion), and the formation of trailing plumes due to desorption.
   • The method successfully recovered both clean and insoluble asymptotic limits in 3D, demonstrating robustness in handling unsteady flow patterns and deforming geometries.

Significance and Claims
The paper claims to provide a robust and versatile numerical framework that bridges the gap between classical insoluble surfactant models and realistic soluble formulations. Its significance lies in:

• Unified Treatment: It handles nonlinear coupling, evolving interfaces, and fully three-dimensional configurations within a single open-source platform.
• Physical Fidelity: It captures the complex interplay between hydrodynamics, bulk/interfacial transport, and Marangoni stresses without relying on turbulence closure models or empirical approximations.
• Practical Applicability: The method is suitable for free-surface flows involving interface reconnection (e.g., jet breakup, breaking waves) in the presence of industrially and biologically relevant soluble surfactants.

The authors modestly note that the current framework is limited to moderate Péclet numbers due to inherent numerical diffusion in the discretization. They identify the accurate resolution of sharp interfacial gradients in the high Péclet (convection-dominated) regime as a challenge for future work, aiming to develop formulations that minimize numerical diffusion.