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

King, J. R.; Bowen, M.

Abstract

The thin-film equation, ht = −(h n hxxx)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. Longstanding 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.

Citation

King, J. R., & Bowen, M. (in press). Touchdown-singularity formation and criticality in the thin-film equation. Philosophical Transactions A: Mathematical, Physical and Engineering Sciences,

• Journal Article Type: Article
• Acceptance Date: Dec 19, 2024
• Deposit Date: Feb 12, 2025
• Journal: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences
• Print ISSN: 1364-503X
• Electronic ISSN: 1471-2962
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Keywords: Mathematical Modelling; Partial Differential Equations; Ordinary Differential Equations; Asymptotics; Self-similarity; Singularity formation; Film rupture; Criticality
• Public URL: https://nottingham-repository.worktribe.com/output/45310956