Dynamics of thin-film equations

Thin liquid films are omnipresent in nature, daily life, and industrial processes, with extensive use in various coating teachniques such as spin coating, responsive coating and biomimetic materials.

These films, ranging from nanometers to micrometers in thickness are typically exposed to external forces like temperature, surface active agents (short: surfactant), or rotation. Their ultrathin nature makes them sensitive to perturabations, potentially leading to film-rupture. Mathematically, these flows are described by quasilinear, degenerate, fourth-order partial differential equations.

When film rupture occurs, the fourth-order term is lost (degneracy), leading to an invalidity of the model equation. This adds a significant challenge to the study of thin films. Despite their importance, a rigorous mathematical analysis aligning with real-world settings is yet missing.

The proposed projects aims to fill this gab by providing a mathematical foundation and analysis of thin-films exposed to (A) temperature, (B) responsive surfactant, (C) rotation in a cylinder.

We will derive models from physical first principles and study local/global solutions, steady states, instability phenomena including film-rupture.

We use methods from functional analysis, energy estimates, dynamical systems, bifurcation as well as sprectral analysis.The project shall be carried out by the PI, a PhD student (focusing on A) and a Postdoc (supporting B,C) together with international collaborators.