Color-gradient lattice Boltzmann modeling of wetting boundary condition on curved solid boundaries

This paper introduces a wetting boundary condition for curved solid surfaces in the color-gradient lattice Boltzmann method by updating order parameters on ghost nodes, a scheme validated on GPU hardware to effectively handle large density and viscosity contrasts while minimizing spurious currents and accurately reproducing both static and dynamic contact line behaviors.

The Gist

Imagine you are trying to simulate how a drop of water behaves when it hits a curved surface, like a raindrop landing on a leaf or a bubble sliding down a curved glass. To do this on a computer, scientists use a method called the Lattice Boltzmann Method. Think of this method as a giant, invisible grid of tiny tiles covering the computer screen. Each tile holds a little bit of "fluid" information, and the computer updates these tiles step-by-step to see how the fluid moves.

The tricky part is the boundary condition—specifically, how the fluid behaves when it touches a solid wall. In the real world, water doesn't just stop dead at a wall; it forms a specific angle (called the contact angle) depending on whether the surface is wet (like clean glass) or dry (like a waxed car).

The Problem: The "Ghost" in the Machine

In the computer simulation, the solid wall isn't a smooth line; it's jagged because it's made of square grid tiles. To make the math work, the computer needs to know what the fluid is doing inside the solid wall, even though there is no fluid there. These imaginary spots inside the wall are called "ghost nodes."

Previous methods for telling these ghost nodes what to do had some flaws:

• They sometimes created fake "ghost currents" (spurious velocities) where the fluid seemed to move on its own without any force.
• They struggled with curved surfaces, often acting like they were only designed for flat walls.
• They sometimes required special, complicated math just to handle a neutral angle (where the water neither spreads out nor beads up).

The Solution: A New Rule for the Ghosts

The authors of this paper introduced a new, simpler rule for these ghost nodes.

The Analogy: Imagine the fluid has a "mood" (represented by a color, from 0 for gas to 1 for liquid). In the real world, this mood changes smoothly from gas to liquid as you cross the surface.

• Old Method: It was like trying to guess the mood of a person standing behind a wall by shouting a random guess.
• New Method: The authors realized that if you know the "mood" of the person standing just outside the wall (in the fluid), you can mathematically extend that smooth mood curve through the wall to the ghost node. They simply ask: "If the fluid wants to form a 45-degree angle here, what must the ghost node's mood be to make that happen?"

This new rule is like a seamless bridge. It extends the natural shape of the fluid drop right up to and slightly into the solid wall, ensuring the angle the drop makes with the wall is exactly what the scientist requested.

What They Tested

To prove their new rule works, they ran several simulations on a very powerful computer chip (an NVIDIA A100 GPU):

1. The Static Drop: They put a drop of water on a flat plate and a curved cylinder. They checked if the drop settled at the exact angle they asked for.
   • Result: Their new rule was more accurate than the previous best method, especially when the angle was very sharp (like a beading drop) or very flat (like a spreading drop).
2. The Floating Particle: They simulated a cylinder floating at the boundary between oil and water.
   • Result: Their method calculated the position of the water line more accurately than before.
3. The Falling Drop: They simulated a drop falling and hitting a cylinder, watching it splash and spread.
   • Result: The drop behaved realistically, and the new rule didn't cause any weird, fake movements in the fluid.

Key Takeaways

• Accuracy: The new method handles curved surfaces much better than older methods, keeping the fluid's angle correct whether the wall is flat or round.
• Stability: It creates very little "fake noise" (spurious currents) in the simulation, meaning the fluid looks more natural.
• Simplicity: It avoids the need for special, complicated math when the contact angle is exactly 90 degrees (neutral), which was a headache for previous methods.
• Speed: By using modern computer chips (GPUs) and a specific programming style, they made the simulations run very fast. They found that using a slightly less precise number format (single precision) made the computer run twice as fast without ruining the results for most tests.

In short, the authors built a better "rulebook" for how computer simulations handle the edge where liquid meets a solid wall, making the digital drops look and act more like real ones, even on curved surfaces.

Technical Summary

Technical Summary: Color-Gradient Lattice Boltzmann Modeling of Wetting Boundary Conditions on Curved Solid Boundaries

Problem Statement
Accurate modeling of wetting phenomena is critical for simulating technologically relevant processes such as porous media flow, microfluidics, and colloidal film drying. In lattice Boltzmann method (LBM) frameworks utilizing diffuse interfaces, implementing wetting boundary conditions on curved solid boundaries presents significant challenges. Existing approaches often suffer from limitations including the introduction of artificial mass sources or sinks, the generation of spurious (parasitic) velocities near interfaces, restrictions to planar geometries, and inconsistencies in enforcing no-slip and wetting conditions simultaneously. Furthermore, many methods require special handling for neutral contact angles ($\theta = 90^\circ$) or rely on linear extrapolations that introduce errors near solid boundaries.

Methodology
The authors introduce a wetting boundary condition specifically designed for curved solid boundaries within a two-dimensional color-gradient LBM framework. The core of the methodology involves updating the order parameter (color-field, $\phi$) on "ghost nodes" located inside the solid phase.

• Ghost Node Update Rule: Instead of prescribing a constant value or using linear extrapolation, the scheme extends the equilibrium color profile into the solid phase. The value of the color-field at a ghost node ($x_G$) is calculated based on the color-field at a corresponding point in the fluid domain ($x_F$) located a distance $\Delta x$ away along the solid normal.
• Mathematical Formulation: Assuming the color distribution is near equilibrium, the derivative of the order parameter in the direction of the solid normal is derived from the equilibrium profile and the prescribed contact angle $\theta$. Integrating this relationship yields a non-linear update rule:
  $$\phi_G = \frac{\phi_F}{\phi_F + (1 - \phi_F) \exp(\varepsilon)}$$
  where $\varepsilon = -4\Delta x \cos \theta / W$, and $W$ is the interface width.
• Key Features: This formulation ensures the ghost node color-field remains bounded between 0 and 1, preventing artificial mass transfer. Crucially, the rule does not require special treatment when $\theta = 90^\circ$ (where $\varepsilon = 0$), unlike previous schemes. The method is implemented using a JAX-based framework running on NVIDIA A100 GPUs, utilizing kernel fusion and automatic differentiation for high performance.

Key Contributions and Results
The proposed scheme was validated against analytical solutions and the benchmark scheme of Fakhari et al. [20] across several test cases:

1. Static Drop on a Flat Plate: The scheme consistently produced lower errors in measured contact angles compared to Fakhari et al. across a range of $30^\circ$ to $150^\circ$. It also generated smaller spurious currents ($5.74 \times 10^{-7}$ to $7 \times 10^{-7}$ lattice units) compared to the benchmark scheme.
2. Drop on a Solid Cylinder: For a drop sitting on a curved cylinder, the proposed method demonstrated improved accuracy in predicting the droplet apex height ($h_{max}$), particularly at extreme contact angles ($30^\circ$ and $150^\circ$), where errors were reduced from $\sim 0.76\%$ to $\sim 0.35\%$ compared to the benchmark.
3. Fixed Particle at Liquid-Gas Interface: In simulations of a periodic array of cylinders trapped at an interface, the proposed scheme reduced errors in vertical interface displacement to $\sim 9\%$ at extreme angles, compared to $\sim 14\%$ for the benchmark.
4. Dynamic Drop Impact: The scheme successfully simulated drop impact on a cylinder at high Reynolds and Bond numbers, capturing complex dynamic behaviors such as droplet spreading on hydrophilic surfaces and splitting on hydrophobic surfaces. The results aligned well with analytical scaling laws for film thickness.
5. Computational Performance: GPU implementation showed that single-precision (FP32) computations were at least twice as fast as double-precision (FP64) while maintaining accuracy for static and moderate dynamic cases. However, FP32 became unstable for the high Reynolds/Weber number drop impact test, necessitating FP64 for such high-energy scenarios.

Significance and Claims
The paper claims that the proposed wetting boundary condition offers a robust and accurate alternative for simulating immiscible two-phase flows on curved solid boundaries. By extending the equilibrium profile into the solid phase via a specific ghost node update rule, the method:

• Maintains the model's ability to handle large density and viscosity contrasts.
• Produces relatively small spurious currents (orders of magnitude lower than popular pseudopotential models).
• Avoids the numerical singularities associated with neutral contact angles found in other methods.
• Demonstrates superior accuracy over existing schemes (specifically Fakhari et al.) for both static and dynamic contact lines on curved geometries.

The authors note that while the current implementation is restricted to single-GPU execution and two dimensions, the underlying physics is directly applicable to phase-field and free-energy models and can be extended to three dimensions. The work provides a foundation for future simulations involving solid particles at liquid-gas interfaces, particularly when coupled with advancements in contact line capillary force calculations and hysteresis schemes.