Inertial and dimensional effects on the instability of a thin film

TL;DR

This study investigates how inertia influences the instability of thin liquid films, revealing that inertia mainly reduces growth rates and can alter characteristic wavelengths when considering bidimensional effects, with implications for film stability analysis.

Contribution

It combines linear stability analysis, bidimensional modeling, and nonlinear simulations to clarify inertia's role in thin film instability beyond traditional lubrication theory.

Findings

Inertia decreases growth rates of unstable modes.

Maximum growth wavelength remains unchanged in long wave approximation.

Bidimensional effects cause wavelength variation with inertia when aspect ratio is not small.

Abstract

We consider here the effects of inertia on the instability of a flat liquid film under the effects of capillary and intermolecular forces (van der Waals interaction). Firstly, we perform the linear stability analysis within the long wave approximation, which shows that the inclusion of inertia does not produce new regions of instability other than the one previously known from the usual lubrication case. The wavelength, $\lambda_m$, corresponding to he maximum growth, $\omega_m$, and the critical (marginal) wavelength do not change at all. The most affected feature of the instability under an increase of the Laplace number is the noticeable decrease of the growth rates of the unstable modes. In order to put in evidence the effects of the bidimensional aspects of the flow (neglected in the long wave approximation), we also calculate the dispersion relation of the instability from the linearized version of the complete Navier-Stokes (N-S) equation. Unlike the long wave approximation, the bidimensional model shows that $\lambda_m$ can vary significantly with inertia when the aspect ratio of the film is not sufficiently small. We also perform numerical simulations of the nonlinear N-S equations and analyze to which extent the linear predictions can be applied depending on both the amount of inertia involved and the aspect ratio of the film.