# Oscillatory thermocapillary instability of a film heated by a thick substrate

## TL;DR

This paper investigates oscillatory thermocapillary instabilities in heated liquid films, revealing conditions under which these instabilities occur, especially emphasizing the role of substrate thickness and thermal properties in their emergence.

## Contribution

It introduces a nonlinear PDE model coupling film dynamics with substrate heat transfer, analyzing oscillatory instability conditions via a non-self adjoint eigenvalue problem.

## Key findings

- Oscillatory instabilities are more prominent with insulating substrates.
- Large substrate thicknesses can lead to complex eigenvalues indicating oscillatory behavior.
- The model provides a comprehensive stability map for different parameters.

## Abstract

In this work we consider a new class of oscillatory instabilities that pertain to thermocapillary destabilization of a liquid film heated by a solid substrate. We assume the substrate thickness and substrate-film thermal conductivity ratio are large so that the effect of substrate thermal diffusion is retained at leading order in the long-wave approximation. As a result, system dynamics are described by a nonlinear partial differential equation for the film thickness that is nonlocally coupled to the full substrate heat equation. Perturbing about a steady quiescent state, we find that its stability is described by a non-self adjoint eigenvalue problem. We show that, under appropriate model parameters, the linearized eigenvalue problem admits complex eigenvalues that physically correspond to oscillatory (in time) instabilities of the thin film height. As the principal results of our work, we provide a complete picture of the susceptibility to oscillatory instabilities for different model parameters. Using this description, we conclude that oscillatory instabilities are more relevant experimentally for films heated by insulating substrates. Furthermore, we show that oscillatory instability where the fastest-growing (most unstable) wavenumber is complex, arises only for systems with sufficiently large substrate thicknesses.