Influence of van der Waals forces on the instability of a liquid film in a tube

This paper demonstrates through theoretical analysis and numerical simulations that van der Waals forces significantly enhance the instability of liquid films in nanotubes by accelerating perturbation growth, reducing dominant wavelengths, and suppressing satellite lobes, while both rupture and collapse processes follow a universal self-similar scaling law with a 1/3 exponent.

The Gist

Imagine a tiny, invisible tube, like a microscopic straw, coated on the inside with a very thin layer of liquid. You might think this liquid layer would just sit there, but in the world of the very small (the nanoscale), it's actually quite restless. It wants to break apart, form blobs, or collapse into a solid plug.

This paper is a deep dive into why this happens and how a hidden, invisible force called van der Waals (vdW) forces changes the rules of the game.

Here is the story of the liquid film, explained simply:

1. The Setting: The Restless Coat

Think of the liquid film as a coat of paint on the inside of a pipe. Usually, this coat is unstable because of surface tension (the same force that makes water droplets bead up). The coat naturally wants to shrink its surface area, so it starts to wiggle and form waves, eventually breaking into separate droplets or plugs. This is a well-known phenomenon called the Rayleigh-Plateau instability.

2. The New Player: The Invisible Magnet

In the past, scientists mostly looked at big tubes. But in nanotubes (tiny tubes found in nature and high-tech devices), the walls are so close that a new force kicks in: van der Waals forces.

• The Analogy: Imagine the liquid molecules and the tube wall are like people at a crowded party. If they are strangers, they might ignore each other. But if they are "attracted" (like magnets), they pull on each other strongly when they get close.
• The Effect: These forces act like an invisible magnet.
  • If the liquid is attracted to the wall, it tries to stick to it, causing the film to rupture (break a hole) and touch the wall.
  • If the liquid molecules are attracted to each other (across the gap), they pull the film inward, causing it to collapse into a solid plug in the center of the tube.

3. The Old Map vs. The New GPS

For a long time, scientists used a simplified "map" (called the lubrication model) to predict how these films behave. They thought this map worked for everything.

• The Discovery: The authors of this paper built a much more detailed "GPS" (based on Stokes equations) and found the old map was wrong for thicker films.
• The Result: The old map said the waves would be big and grow slowly. The new GPS says: "No! Because of those invisible magnets, the waves are actually tiny and they grow super fast."

4. Two Ways to Fail: The Rupture and The Collapse

The paper shows that the liquid film can fail in two distinct ways, depending on how thick it is and which "magnet" is stronger:

• Scenario A: The Rupture (The Hole Pops)

  • When: The film is very thin.
  • What happens: The attraction between the liquid and the wall pulls the thinnest part of the film until it snaps and touches the wall.
  • The Surprise: Without these forces, the film would form little "satellite" droplets (like a main drop with tiny baby drops attached). But with strong van der Waals forces, the film snaps so fast that these baby droplets never get a chance to form. It's like a balloon popping so quickly that the rubber doesn't have time to stretch into weird shapes.

• Scenario B: The Collapse (The Plug Forms)

  • When: The film is thicker.
  • What happens: The liquid molecules pull on each other across the gap, squeezing the film inward until it seals off the tube, forming a solid plug.
  • The Surprise: These forces make the plug form much faster and allow even thinner films to collapse than we previously thought possible.

5. The Universal Rule: The "1/3" Law

One of the coolest findings is about speed.

• When a film is about to break or collapse, it speeds up dramatically.
• The authors found that whether it's a rupture or a collapse, the speed follows a universal mathematical rule (a "power law" of 1/3).
• The Analogy: It's like a car accelerating toward a crash. No matter if it's a small car or a big truck, in the final split second before impact, they both follow the exact same acceleration curve. This suggests that at the very end, the invisible magnets take total control, overriding everything else.

Why Does This Matter?

This isn't just about math; it's about real life:

• Medicine: It helps us understand how airways in the lungs might collapse (a problem in asthma or premature babies).
• Technology: It's crucial for designing better fuel cells, oil recovery systems, and micro-chips where tiny amounts of liquid need to be controlled.
• Nature: It explains how water moves through the tiny tubes in trees and plants.

In Summary:
This paper tells us that in the microscopic world, invisible forces (van der Waals) are the puppet masters. They make liquid films break faster, form smaller waves, and collapse more easily than we ever imagined. By understanding these "magnetic" pulls, we can better predict and control fluids in the tiniest of tubes.

Technical Summary

1. Problem Statement

The study investigates the stability and evolution of liquid films coating the interior of cylindrical tubes (nanotubes), specifically focusing on the role of van der Waals (vdW) forces. While classical theories (Rayleigh–Plateau instability) describe film breakup driven by surface tension, they often fail to accurately predict behavior at the nanoscale where intermolecular forces become dominant.
Key challenges addressed include:

• The limitations of traditional lubrication models for films that are not extremely thin or where radial flow is significant.
• The distinction between two evolutionary modes: rupture (film thinning against the tube wall) and collapse (film thickening toward the central axis to form a liquid plug).
• The lack of a unified theoretical framework that accurately captures both linear instability and nonlinear singularity scaling in the presence of vdW forces.

2. Methodology

The authors employed a combined approach of theoretical analysis and direct numerical simulation (DNS).

A. Mathematical Modeling

• Governing Equations: The study utilizes the incompressible Navier-Stokes equations under the assumption of axisymmetry. For the linear regime, the equations are simplified to the Stokes equations (neglecting inertia, valid for high Ohnesorge numbers typical of nanoscale flows).
• Disjoining Pressure: vdW forces are modeled via the Derjaguin approximation, introducing a disjoining pressure term $\Pi = A/d^3$. The model distinguishes between two types of interactions:
  • $A_1$: Solid-liquid interactions (tube wall to film).
  • $A_2$: Liquid-liquid interactions (opposing sides of the film interface).
• Linear Stability Analysis: A normal mode method was applied to derive an explicit dispersion relation ($\omega$ vs. $k$) for the growth rate of perturbations. This was derived for both the full Stokes model and a lubrication approximation to compare their validity.

B. Numerical Simulation

• Solver: Direct Numerical Simulations (DNS) of the Navier-Stokes equations were performed using COMSOL Multiphysics with the Arbitrary Lagrangian-Eulerian (ALE) method to track the moving liquid-gas interface.
• Regimes: Simulations covered both the linear stage (small perturbations) and the nonlinear stage (large deformations leading to rupture or collapse).
• Scaling: The study utilized adaptive meshing and remeshing near singularities to resolve the thinning necks accurately.

3. Key Contributions

1. Development of a Stokes-Based Linear Stability Framework: The authors derived a rigorous dispersion relation based on the full Stokes equations, demonstrating that it is superior to lubrication models for films with moderate thickness or strong vdW forces.
2. Quantification of vdW Effects: The study systematically decoupled the effects of solid-liquid ($A_1$) and liquid-liquid ($A_2$) vdW forces on instability growth, dominant wavelength, and critical collapse thickness.
3. Universal Scaling Law: The paper establishes that both rupture and collapse singularities, when dominated by vdW forces, follow a universal self-similar scaling law with an exponent of 1/3 ($h_{min} \propto \tau^{1/3}$), differing from the classical lubrication theory predictions (1/5 or 1/2).
4. Suppression of Satellite Lobes: The research identifies a mechanism by which strong vdW forces suppress the formation of secondary satellite lobes during the nonlinear rupture process.

4. Key Results

A. Linear Regime (Small Perturbations)

• Growth Rate & Wavelength: vdW forces significantly increase the perturbation growth rate ($\omega$) and reduce the dominant wavelength ($\lambda_{max}$).
  • $A_1$ (solid-liquid) is the primary driver for wavelength reduction in ultra-thin films.
  • $A_2$ (liquid-liquid) becomes increasingly dominant in thicker films, reducing the wavelength as the opposing interfaces draw closer.
• Model Accuracy: The Stokes-based model predicts lower growth rates and longer wavelengths than the lubrication model for thicker films, aligning better with experimental data (e.g., Tomo et al., 2022). The lubrication model overestimates instability in these regimes.

B. Nonlinear Regime (Rupture vs. Collapse)

• Critical Thickness ($\epsilon_c$): The existence of a critical film thickness distinguishing stable collars from collapsing plugs is confirmed. However, the presence of $A_2$ lowers this critical threshold, allowing thinner films to collapse into plugs.
• Rupture Dynamics ($A_1$ dominant):
  • Films rupture towards the tube wall.
  • Strong $A_1$ forces accelerate thinning and suppress the formation of satellite lobes (secondary droplets), leading to a direct rupture.
  • Without vdW forces, satellite lobes form due to axial flow; vdW forces induce strong radial flow that prevents this.
• Collapse Dynamics ($A_2$ dominant):
  • Films collapse towards the central axis to form liquid plugs.
  • $A_2$ forces accelerate the coalescence process and sharpen the interface gradient.

C. Singularity Scaling and Self-Similarity

• Scaling Laws:
  • Without vdW: Rupture follows $h \sim \tau^{1/2}$ (viscous-capillary) and collapse follows $h \sim \tau^{1/5}$ (lubrication) initially, transitioning to linear scaling.
  • With vdW: Both rupture and collapse converge to a $\tau^{1/3}$ power law as the singularity is approached.
• Universality: By rescaling variables using the characteristic vdW length $a = \sqrt{\tilde{A}/6\pi\gamma}$, data from both rupture and collapse cases collapse onto a single universal curve ($h_a \sim \tau_a^{1/3}$). This confirms that vdW forces dominate the final stages of singularity, overriding geometric constraints.

5. Significance

• Theoretical Advancement: The work corrects the limitations of lubrication theory in nanoscale confined geometries, providing a more accurate Stokes-based framework for predicting instability.
• Practical Applications: The findings are critical for applications involving nanofluidics, such as carbon nanotube fluid transport, pulmonary airway closure (asthma/respiratory distress), and microfluidic patterning. Understanding how vdW forces alter rupture times and satellite lobe formation allows for better control of these processes.
• Experimental Validation: The theoretical predictions regarding dominant wavelengths and scaling laws show excellent agreement with recent experimental observations in graphene nanochannels, validating the model's predictive power.
• Mechanistic Insight: The identification of the $\tau^{1/3}$ scaling law unifies the understanding of rupture and collapse under vdW dominance, offering a new perspective on interfacial singularities in confined fluids.