# Touchdown-singularity formation and criticality in the thin-film equation

## TL;DR

This paper investigates how thin films rupture, focusing on a mathematical model to determine when film rupture is possible based on an exponent value.

## Contribution

The study reveals new touchdown behaviors and proposes conjectures about the critical exponent n=2 in thin-film rupture.

## Key findings

- New touchdown behaviors in thin-film equations were identified using analytical and numerical methods.
- The exponent n=2 is conjectured to be critical for determining film rupture possibility.
- The work advances understanding of high-order degenerate parabolic equations.

## Abstract

The thin-film equation (TFE), ht=−(hnhxxx)x, is significant physically in the description of surface-tension-driven flows of thin films of viscous liquids and has served an important role mathematically in elucidating the properties and challenges of high-order degenerate parabolic equations. Long-standing open questions nevertheless remain, of which perhaps the most important is the identification of the critical value of the exponent n above which film rupture is not possible. Here, we apply a combination of analytical and numerical methods to further the understanding of this issue, uncovering new types of touchdown behaviour that lead to concrete conjectures regarding the role of n=2 in this type of criticality.

This article is part of the theme issue ‘Science into the next millennium: 25 years on’.