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A generalized, quasi-universal subgrid model for macroscopic moving contact line flows

This paper presents a generalized, quasi-universal subgrid model that couples Cox's matched asymptotic relation with empirical contact angle models to enable predictive, grid-independent simulations of macroscopic moving contact line flows without requiring case-specific calibration or phenomenological parameters.

Original authors: Vyankatesh Manoj Mundhada, Manoj Kumar Tripathi

Published 2026-09-29✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Vyankatesh Manoj Mundhada, Manoj Kumar Tripathi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a drop of water landing on a flat surface. As it spreads, the edge where the liquid, the air, and the solid meet does not simply slide like a car tire on pavement. Instead, this boundary, known as a contact line, behaves in a way that defies the standard rules of fluid mechanics used to describe most flowing liquids. For decades, scientists have struggled to simulate this motion on computers because the physics at the very edge of the drop involves scales so tiny—down to the size of individual molecules—that trying to calculate every detail would require more computing power than exists. To get around this, researchers have used shortcuts, or "subgrid models," to guess what happens at that microscopic edge. However, these shortcuts have a major flaw: the results they produce often change depending on how finely the computer divides the space into a grid, making the simulations unreliable for real-world predictions.

A team of researchers at the Indian Institute of Science Education and Research in Bhopal has developed a new way to handle this problem. They created a model that allows computers to simulate moving contact lines accurately without needing to know the impossible-to-measure details of the microscopic world. Their approach bridges the gap between the tiny, invisible physics at the edge of a drop and the larger, visible behavior of the fluid. By doing so, they have produced simulations that remain consistent and accurate regardless of the computer's grid size, a crucial step toward predicting how liquids behave in everything from inkjet printing to oil recovery, without needing to tweak the model for every single new situation.

The core of the problem lies in a conflict between how fluids move and how they touch solid surfaces. In standard fluid mechanics, a rule called the "no-slip condition" states that a fluid touching a solid surface must stop moving completely at that boundary. While this works well for most flows, it breaks down at the contact line of a moving drop, creating a mathematical singularity where the forces become infinite. To fix this, scientists have historically assumed that the fluid slips a tiny bit at the surface, but the distance of this slip is so small it is impossible to measure directly. Consequently, previous computer models had to guess these microscopic values or rely on fitting parameters to match experimental data. This meant that if a researcher wanted to simulate a different type of liquid or a different speed, they often had to start over and recalibrate the model, stripping it of its ability to predict new scenarios.

The researchers in this study realized that the inconsistency in previous simulations came from a misunderstanding of how the angle of the drop changes as you move away from the contact line. They drew upon a well-established theory from the 1980s which describes how the shape of the liquid interface bends due to the competition between surface tension and the fluid's internal friction. This theory shows that the angle you see depends on how far away you are looking from the edge. The researchers combined this theoretical understanding with existing empirical formulas—rules derived from observing real drops—to create a new, unified model. Instead of forcing the computer to guess the microscopic slip length or the exact angle at the wall, their model calculates the correct angle to use based on the size of the computer's grid cells.

To test their idea, the team ran a series of computer simulations of a droplet spreading on a flat plate. They compared their new method against older techniques using different grid sizes, ranging from coarse to very fine. In the older methods, changing the grid size significantly altered the results, causing the simulated drop to spread at different rates or take on different shapes. With the new model, however, the results remained virtually identical across all grid sizes. The simulated drop behaved consistently, whether the computer was using a coarse grid or a fine one. This demonstrated that the model had successfully removed the artificial dependence on the grid, a phenomenon known as grid-dependence.

The study also explored how this new model performs under various conditions, such as different speeds of flow and different types of liquids. They tested scenarios where the liquid was spreading out (advancing) and where it was pulling back (receding). In every case, the new model produced stable, grid-independent results. A key finding was that the model did not require any "fitting parameters" derived from specific experiments to work. Once the researchers established the scale at which the original empirical formulas were derived, the model could predict the behavior of the flow without further adjustment. This is a significant departure from previous methods, which often required a unique calibration for each new set of conditions.

The researchers also investigated why some older simulations seemed to work well in the past, even without this new correction. They found that in certain situations, particularly with gas-liquid systems, the error introduced by the grid size was naturally small enough to go unnoticed. Additionally, if the computer grid happened to match the specific scale at which the original experimental data was collected, the results would coincidentally align with reality. However, this was a matter of luck rather than a robust solution. The new model removes this element of chance, providing a reliable framework that works across a wide range of physical conditions, including different densities and viscosities.

One of the most promising aspects of this work is its potential for predictive power. Because the model does not rely on knowing the microscopic slip length or the exact wall contact angle—quantities that are notoriously difficult to measure—it can be applied to new problems where these values are unknown. The researchers showed that by simply knowing the scale at which the original experimental observations were made, the model could be applied universally. This means that engineers and scientists could use these simulations to design better systems for coating surfaces, managing fluid flow in porous rocks, or understanding biological processes in the lungs, confident that the results are not artifacts of the computer's grid.

The study concludes that while the first-order approximation of their model works well for most angles, more complex scenarios involving very large contact angles might require a slightly more detailed mathematical correction. However, the fundamental approach of linking the microscopic theory with macroscopic observations has proven successful. By treating the contact line not as a fixed point with mysterious properties, but as a dynamic feature that responds to the scale of observation, the researchers have provided a tool that is both accurate and broadly applicable. This work represents a shift from merely fitting data to understanding the underlying physics, offering a path toward simulations that can truly predict how liquids will behave in the real world.

Technical Summary: A Generalized, Quasi-Universal Subgrid Model for Macroscopic Moving Contact Line Flows

Problem Statement
Numerical simulation of moving contact line (MCL) flows is fundamentally challenged by the stress singularity arising from the no-slip boundary condition on solid surfaces and the inherently multiscale nature of the phenomenon. Resolving the microscopic length scales (nanometer-scale slip lengths) required to eliminate this singularity renders macroscopic simulations computationally intractable. Consequently, subgrid models are necessary to bridge the gap between microscopic physics and macroscopic resolution. However, existing approaches often suffer from two critical limitations:

Grid-Dependence: Numerical results, particularly the contact line position and interface shape, vary significantly with mesh resolution when empirical contact angle models are applied directly without accounting for the scale at which they were derived.
Lack of Predictivity: Many scale-invariant models require the prescription of phenomenological parameters, such as the microscopic slip length ( $\lambda$ or $s_m$ ) and the microscopic wall contact angle ( $\theta_w$ ), which are difficult to determine experimentally or a priori. This reliance on case-specific calibration limits the predictive capability of simulations.

Methodology
The authors propose a quasi-universal subgrid model that couples Cox's matched asymptotic theory with empirical dynamic contact angle models. The core methodology involves the following steps:

Theoretical Framework: The study leverages Cox's hydrodynamic theory, which relates the apparent contact angle $\theta(s)$ at a distance $s$ from the contact line to the microscopic wall angle $\theta_w$ and the contact line capillary number ( $Ca_{CL}$ ). The theory introduces a logarithmic dependence on the length scale ratio ( $s/s_m$ ).
Coupling Strategy: Instead of prescribing unknown microscopic parameters ( $\theta_w$ and $s_m$ ), the authors derive a relationship that eliminates these variables. By recognizing that empirical models (e.g., Kistler, Jiang, Tanner) provide the contact angle $\theta_0$ at a specific observation scale $s_0$ (typically the resolution of the experimental data used to formulate the model), they establish a bridge between the empirical model and the grid scale.
Derivation of the Modified Model: The authors introduce a scale-dependent correction factor $\chi(Ca_{CL}, s)$ to the capillary number in the empirical relation. Using a first-order Taylor expansion of Cox's function $G(\theta)$ , they derive a modified capillary number:
$\chi(s) = Ca_{CL} \left[ 1 + \zeta \ln\left(\frac{s}{s_0}\right) \right]$
where $\zeta$ is a parameter derived analytically from the specific empirical model and the viscosity ratio. This allows the imposition of a grid-size-dependent contact angle $\theta(s)$ that satisfies Cox's theory without requiring knowledge of $s_m$ or $\theta_w$ .
Numerical Implementation: The model is implemented within a Volume-of-Fluid (VoF) framework using the Basilisk code. The contact line velocity is calculated simultaneously with the contact angle and position at each time step to ensure consistency, avoiding errors associated with temporally staggered quantities. The approach is tested on axisymmetric droplet spreading and withdrawing plate scenarios across a wide range of Reynolds ( $Re$ ) and Weber ( $We$ ) numbers.

Key Results

Grid Independence: The proposed "modified" contact angle models successfully eliminate grid dependence. Simulations conducted on grids ranging from $\Delta_5$ to $\Delta_9$ (where $\Delta_i = D_0/2^i$ ) show that the contact line position and droplet interface shape converge to a single solution. In contrast, using original (unmodified) empirical models results in significant errors (up to ~1.5% in interface shape and higher in contact line position) that do not converge with mesh refinement.
Error Reduction: For liquid-liquid systems (where grid dependence is most severe), the mean relative errors in contact line position ( $E_r$ ) and interface shape ( $E_s$ ) were reduced by approximately one order of magnitude compared to simulations using original models. For instance, at $Re=100$ , $We=1$ , and $\theta_e=60^\circ$ , the contact line position error dropped from ~0.69% (original Kistler) to ~0.05% (modified Kistler).
Robustness Across Regimes: The model demonstrated effectiveness across varying $Re$ (1–200), $We$ (0.5–100), and equilibrium contact angles ( $30^\circ$ – $135^\circ$ ).
Higher-Order Corrections: The study identified that for large obtuse contact angles ( $\ge 120^\circ$ ), the first-order linear approximation of the correction factor $\zeta$ introduces higher errors. A second-order correction was derived and shown to further reduce errors in these specific regimes, though the first-order approximation remains sufficient for acute and moderate angles.
Gas-Liquid Systems: The authors note that for gas-liquid systems, the original numerical method often exhibits negligible grid dependence without modification, likely because the variation of contact angle with distance is small and the grid size incidentally matches the experimental observation scale.

Significance and Claims
The paper claims to present the first framework capable of predictive simulations of moving contact line flows without the need for case-specific calibration or the a priori knowledge of microscopic parameters ( $\theta_w$ and $s_m$ ).

Quasi-Universality: The model is termed "quasi-universal" because it relies only on the validity of Cox's theory and the chosen empirical contact angle model, independent of specific flow parameters like $Re$ , $We$ , or density/viscosity ratios.
Observation Scale Dependency: The method requires only a one-time knowledge of the observation length scale ( $s_0$ ) associated with the empirical model's derivation. Once $s_0$ is fixed for a specific model, the subgrid model can be applied to any macroscopic flow problem.
Elimination of Fitting Parameters: Unlike previous scale-invariant approaches that required fitting parameters to match experimental data, this method derives the necessary corrections analytically, allowing for true predictive capability.
Generalizability: While demonstrated in axisymmetric 2D, the formulation is derived from local contact-line descriptions and is not specific to axisymmetric geometry, suggesting potential for extension to three-dimensional contact lines.

In summary, the work addresses the fundamental bottleneck of grid-dependence in MCL simulations by rigorously coupling hydrodynamic theory with empirical data, offering a path toward reliable, parameter-free numerical predictions of complex multiphase flows.

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