---
title: "Thin-Film Instability Interactive Lab | Sandbox Physics"
description: "Why does a nearly flat liquid film grow a pattern instead of smoothing out?"
canonical_url: "https://sandboxphysics.com/labs/thin-film-instability/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/thin-film-instability/"
last_modified: "2026-10-28"
---

FR18 · A liquid film chooses a spacing

# Thin-Film Instability

**Why does a nearly flat liquid film grow a pattern instead of smoothing out?** Surface tension smooths short ripples, but attraction to the substrate can amplify longer ones. Their competition selects the fastest-growing spacing. Liquid moves from thinning regions into thicker ridges, with the total volume conserved.

Release a slightly uneven film and watch liquid collect into ridges. Compare the growth of its Fourier modes with the small-amplitude prediction, or drag the film to change its starting profile.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This periodic, one-dimensional lubrication model describes a nonvolatile Newtonian liquid with no slip and negligible inertia. A chosen wetting potential leaves a precursor layer instead of a dry contact line. Heights, time and horizontal distance are dimensionless; the vertical scale is enlarged. Growth rates are fitted from saved simulated height profiles during the first six time units. Finite amplitudes and the numerical grid can shift them from the continuum reference. Evolution stops if the thickness or small-slope checks fail. Three-dimensional holes, molecular rupture and thermal fluctuations are outside this model.

## Which ripple is growing fastest?

Cyan bars: cosine peak amplitudes from the latest stored height profile. Amber points: growth fitted to stored profiles in the first six time units. White line: the continuum small-amplitude prediction.

Amplitude by mode number $a_n/h_0$

> Fourier amplitudes calculated from saved simulated heights.

Early growth, first seven modes $\sigma_n$

> Early fitted rates compared with the analytic reference.

Reading the comparison

Positive rates amplify a mode; negative rates damp it. The finite grid approximates spatial derivatives, and finite initial ripples couple modes. Choose Small-amplitude test to reduce the second effect. Later bars show the evolving pattern; rate points keep the early fitting window.

All displayed profiles are simulated. No experimental height measurements or molecular rupture positions are shown.

Physics tutorial

## Two tendencies choose a pattern

**Background** A thin coating can look smooth and still be unstable. A small thickness difference changes both the curvature of its surface and its interaction with the solid underneath.

**Why it matters** The first ridges need not follow the biggest initial bump. A smaller ripple can win if it grows faster.

Start with the essentials

- **Focus question**: Why does a nearly flat liquid film grow a pattern instead of smoothing out?
- **One-sentence intuition**: Short ripples cost curvature energy; longer ones can lower the wetting energy by separating thick and thin regions.

### Core mathematical model

#### Conserved liquid flow

$$
\partial_t h=\partial_x\!\left(h^3\partial_x\mu\right),\qquad \mu=-\gamma\partial_{xx}h+f'(h)
$$

Height is measured in the mean initial thickness. Horizontal distance and time use the stated model scales. A gradient of the chemical potential drives liquid along the substrate; the periodic flux conserves volume.

#### A wetting potential with a thin remaining layer

$$
f(h)=-\frac{A}{2h^2}+\frac{A b^3}{5h^5}
$$

Attraction favors thinning; the short-range repulsive term grows rapidly at small thickness and has its minimum at the precursor height. The chosen potential is a teaching model, with no fitted material parameters.

#### Growth of a small sinusoidal ripple

$$
\sigma(k)=-k^2\!\left[\gamma k^2+f''(1)\right],\qquad k_{\max}=\sqrt{\frac{-f''(1)}{2\gamma}}
$$

Wavenumber is the number of radians per model length. Positive growth amplifies a ripple. The fastest mode exists when the wetting-energy curvature is negative. The white curve below is this continuum prediction; colored points come from saved simulation profiles.

### Common difficulties

#### Surface tension and instability

**Typical misconception** Surface tension makes the film unstable because it pulls the liquid into drops.

**Better mental model** In this flat-film model capillarity damps every Fourier mode. The wetting potential supplies the instability. Turn the substrate attraction to zero and watch all ripples shrink.

### Run the experiment

1. 01

   #### Predict a spacing

   Choose Growing ridges. Compare the tallest spectrum bars with the early growth-rate points.

   **What to observe:** Several modes grow at different speeds. Later, nonlinear flow changes their amplitudes and couples the modes.

2. 02

   #### Remove the attraction

   Choose Ripples flatten and watch the same seeded starting noise relax.

   **What to observe:** The remaining capillary force smooths the film. The spectrum bars fall, and the free energy decreases.

3. 03

   #### Make a new starting profile

   Drag across the liquid surface, then release it to resume the flow. Use Small-amplitude test to compare growth rates more closely.

   **What to observe:** Dragging adds a localized volume-neutral perturbation and starts a new record. Smaller starting amplitudes approach the linear prediction more closely.

Physical conclusion

A pattern emerges when the wetting-energy gain exceeds the capillary cost for some wavelengths.

## Related experiments and articles

Dynamic interfaces · Reading path

### Curvature, films, thermal flow and liquid threads

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Reduce a flowing layer to a height equation**](https://sandboxphysics.com/stories/dynamic-interfaces/lubrication-flow/)
2. [Article **Which wavelength grows out of noise first?**](https://sandboxphysics.com/stories/dynamic-interfaces/mode-selection/)
3. [Article **How a drop edge moves across a solid**](https://sandboxphysics.com/stories/dynamic-interfaces/contact-lines/)
4. [Article **How to compare growth rates with nanofilm spectra**](https://sandboxphysics.com/stories/dynamic-interfaces/paper-reading/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)