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Linear stability analysis of a vertical liquid film over a moving substrate

This study employs linear stability analysis via the Orr-Sommerfeld eigenvalue problem to investigate the stability mechanisms, vorticity distribution, and energy balance of vertical liquid films on moving substrates across a wide range of Kapitza numbers, revealing that surface tension exhibits dual stabilizing and destabilizing effects and that the Landau-Levich-Derjaguin solution is inherently convectively unstable.

Original authors: Fabio Pino, Miguel Alfonso Mendez, Benoit Scheid

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Fabio Pino, Miguel Alfonso Mendez, Benoit Scheid

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 or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a giant, vertical wall being pulled out of a bathtub of liquid. As the wall rises, it drags a thin sheet of the liquid up with it. This is the "dip-coating" process, used everywhere from making shiny zinc coatings on steel to putting protective layers on food. But here's the catch: sometimes this liquid sheet gets wobbly. Instead of a smooth, even coat, it starts to ripple and wave, ruining the final product.

This paper is like a detective story about those ripples. The authors, Fabio Pino, Miguel A. Mendez, and Benoit Scheid, wanted to figure out exactly when and why these ripples grow out of control, and whether they just drift along with the flow or explode everywhere at once.

The Four Liquid Test Subjects

To solve this mystery, the team didn't just look at one liquid. They simulated four very different fluids, each with a unique personality defined by how "sticky" it is versus how much it "wants" to stay together (a property called the Kapitza number, or Ka).

Corn Oil: The "greasy" one. It has a low Ka (4). It's thick and viscous.
Water-Glycerol Mix: A middle-ground syrup. Ka is 195.
Water: The familiar stuff. Ka is 3400.
Liquid Zinc: The heavy metal. It's super hot and runny but incredibly dense. It has a massive Ka of 11,525.

They ran thousands of computer simulations (using a method called the Chebyshev-Tau spectral method) to see how these liquids behaved when the wall moved at different speeds.

The Great Ripple Battle: Surface Tension's Double Agent

One of the biggest discoveries in the paper is that surface tension (the force that makes water bead up) is a bit of a trickster. It doesn't just help or hurt; it does both, depending on the size of the ripple and the type of liquid.

For long, lazy waves: Surface tension acts like a brake. It bends the swirling motion (vorticity) of the liquid near the wall, slowing down the flow under the wave's "crest" (the top part). This tends to stabilize the film.
For short, snappy waves: Surface tension flips the switch. It actually creates more swirling motion right at the surface and even near the bottom wall, making the film more unstable.

The authors found that liquids with low Ka (like corn oil) are much more fragile. They have a smaller "safe zone" before they start rippling compared to high Ka liquids like liquid zinc. In fact, for corn oil, the instability grows fast and furious, while liquid zinc can handle a wider range of conditions before things go wrong.

The "Vortex" Dance

To understand why the ripples grow, the authors looked at the invisible "swirls" or vortices inside the liquid.

In the corn oil (low Ka): The instability is driven mostly by the sheer size of these swirls near the surface. The liquid moves in a way that pushes mass from the troughs (dips) to the crests (hills), making the waves bigger.
In the liquid zinc (high Ka): It's a different story. Here, surface tension creates a brand-new, intense zone of swirling right next to the moving wall. It's like a secret engine near the bottom that pumps energy into the ripples, making them grow even faster.

The "Absolute" vs. "Convective" Mystery

This is the most exciting part of the paper. The authors asked a crucial question: If you poke the liquid film with a finger, will the resulting wave just drift away downstream (convective instability), or will it stay right there and grow until it destroys the whole film (absolute instability)?

Think of convective instability like a leaf floating down a river. You drop it, it moves away, and the river is fine.
Think of absolute instability like a fire in a room. You drop a spark, and the fire stays right there, growing until it consumes everything.

The paper found a "window of absolute instability."

For most liquids, if the film is too thin or too thick, the ripples just drift away (convective).
But in the middle—specifically for certain film thicknesses and speeds—there is a dangerous zone where the ripples get stuck and explode (absolute).

Crucially, the paper rules out a few things:

They found that the famous "Landau-Levich-Derjaguin" solution (a standard formula for how thick the film should be) is always convectively unstable. It never sits in that dangerous "absolute" zone.
They also discovered that for liquids with a Ka less than 17 (like corn oil), the standard "Derjaguin" solution is always convectively unstable, no matter how fast the wall moves. It never becomes absolutely unstable.

The Numbers Behind the Magic

The authors didn't just guess; they calculated specific thresholds.

For the Derjaguin solution (where the film thickness is exactly 1 in their units), if the Kapitza number is below 17, the film is always safe from absolute instability.
The "window" of absolute instability appears in the Reynolds number (Re) and film thickness space. For example, for liquid zinc (Ka = 11,525), this window opens up at specific speeds and thicknesses, creating a region where the film is doomed to become wobbly and chaotic.
They even found a mathematical relationship for the lower edge of this danger zone: the film thickness scales with the Reynolds number to the power of 4/9.

What This Means for the Future

The paper doesn't claim to have "solved" the problem of coating liquids forever. Instead, it provides a precise map of where the danger lies. The authors suggest that this map is essential for designing control systems. If you know exactly where the "absolute instability" window is, you can build machines that avoid those specific speeds and thicknesses, or use sensors to stop the ripples before they take over.

In short, this paper turns the messy, wobbly world of liquid films into a clear, mathematical landscape. It shows us that while some liquids (like corn oil) are prone to chaos, others (like liquid zinc) have a more complex dance involving surface tension and wall friction. And most importantly, it tells us exactly where the "do not enter" signs should be placed to keep our industrial coatings smooth and perfect.

Technical Summary: Linear Stability Analysis of a Vertical Liquid Film over a Moving Substrate

Problem Statement
This study investigates the linear stability of a vertical liquid film formed on a substrate moving against gravity, a configuration central to the dip-coating industrial process. In this process, a liquid film is extracted from a bath, and its stability determines the uniformity of the final coating. While previous literature has addressed stability via asymptotic long-wave expansions (valid only for small Reynolds numbers and long wavelengths) or integral boundary layer models, a comprehensive analysis of the full Orr-Sommerfeld (OS) eigenvalue problem for controlled and uncontrolled film flows across a wide range of parameters has been lacking. Specifically, the paper addresses the instability mechanisms and the threshold between absolute and convective instability for liquids spanning a broad range of Kapitza numbers ($Ka$), from low-$Ka$ fluids like corn oil ( $Ka \approx 4$ ) to high-$Ka$ liquid metals like zinc ( $Ka \approx 11525$ ).

Methodology
The authors formulate the problem using the incompressible 2D Navier-Stokes equations, non-dimensionalized using reference velocity ( $U_p$ ) and a viscous-gravity balance length scale. The base state is defined as a flat film with a non-monotonic relationship between film thickness and flow rate, yielding thin ( $\hat{h} < 1$ ) and thick ( $\hat{h} > 1$ ) film branches.

To analyze stability, the authors:

Linearized the governing equations around the base state, deriving the Orr-Sommerfeld eigenvalue problem.
Solved the eigenvalue problem numerically using the Chebyshev-Tau spectral method with $N=20$ and $N=80$ polynomials to ensure accuracy and filter spurious eigenvalues.
Computed neutral curves and growth rates ( $\omega_i$ ) across the wavenumber ( $k$ ) and Reynolds number ($Re$) spaces.
Investigated instability mechanisms through long-wave asymptotic expansions, vorticity distribution analysis, and a detailed kinetic energy balance of the perturbations (accounting for kinetic energy, surface tension potential energy, shear work, Reynolds stress, and viscous dissipation).
Determined the Absolute/Convective (AC) threshold using Briggs' method. This involved mapping the complex wavenumber space to the complex frequency space to identify saddle points satisfying the causality (collision) criterion, thereby distinguishing between convectively unstable flows (disturbances swept away) and absolutely unstable flows (disturbances growing locally).

Key Results

Stability and Neutral Curves: The study reveals that liquids with low $Ka$ (e.g., corn oil) possess a smaller stability area compared to high-$Ka$ liquids (e.g., zinc). For thick films ( $\hat{h} > 1$ ), the neutral curves for different fluids converge, suggesting a stabilizing mechanism dominated by the balance of viscous and gravitational forces where surface tension effects diminish.
Instability Mechanisms:
- Vorticity and Phase Shift: For thin films, instability is driven by the amplitude of vorticity with minimal phase shift. For thick films, the vorticity amplitude decreases while the phase shift increases.
- Role of Surface Tension: Surface tension exhibits a dual stabilizing and destabilizing role, particularly for high $Ka$. For long waves, it curves vorticity lines near the substrate, reducing flow under crests (stabilizing). For short waves, it fosters vorticity production at the interface and creates intense vorticity regions near the substrate (destabilizing).
- Energy Balance: The analysis shows that surface tension contributes to both the production and dissipation of perturbation energy depending on the $Ka$. For high $Ka$ (zinc), surface tension induces significant vorticity production at both the free surface and the wall, leading to intense Reynolds stresses and dissipative effects. For low $Ka$ (corn oil), shear effects dominate the viscous terms.
Absolute/Convective Threshold:
- A window of absolute instability exists in the $\hat{h}-Re$ parameter space, bounded by lower and upper thresholds.
- Derjaguin's Solution ( $\hat{h}=1$ ): The paper finds that for $Ka < 17$, the Derjaguin solution is always convectively unstable regardless of the Reynolds number.
- Landau-Levich-Derjaguin (LLD) Solution: The LLD solution ( $\hat{h} \approx 0.945 Re^{1/9} Ka^{-1/6}$ ) is found to be always convectively unstable for zinc, water, and water-glycerol. For corn oil, it intersects the absolute instability window only at high Reynolds numbers ($100 < Re < 500$), a regime deemed unrealistic for typical experimental conditions due to the required domain length.
- Scaling: The lower and upper branches of the absolute instability window can be rescaled to collapse data for different fluids ($Ka > 195$). A specific power-law relation ( $\check{h}/\check{h}_{min} = (Re/Re_{min})^{4/9}$ ) is proposed to approximate the lower AC threshold.
- Bifurcation: For water-glycerol, a bifurcation point is identified around $Re \approx 40$ , where a second solution branch appears, independent of $Ka$ and governed by gravity-inertia balance.

Significance and Claims
The authors claim that this work provides the first thorough analysis of the full Orr-Sommerfeld eigenvalue problem for dip-coating flows across a wide range of $Ka$ numbers, addressing a gap left by previous asymptotic and integral models. The study clarifies the physical mechanisms driving instability, specifically the complex, dual role of surface tension in vorticity generation and energy balance.

Crucially, the paper establishes the existence and location of the absolute instability window in the parameter space. This is significant for the design of control laws in industrial coating processes, as the nature of the instability (absolute vs. convective) dictates whether control actions must affect the entire film or only specific regions. The findings suggest that for low-$Ka$ fluids, the standard Derjaguin solution is inherently stable against absolute instability, whereas for high-$Ka$ fluids, the stability landscape is more complex. The results are intended to serve as a benchmark for validating reduced-order models and understanding the fundamental stability properties of moving liquid films.