Thin films flowing down inverted substrates: Three dimensional flow

TL;DR

This paper investigates the stability and pattern formation of thin fluid films flowing down inverted substrates in three dimensions, focusing on surface and transverse instabilities, and the effects of viscosity variations.

Contribution

It extends previous two-dimensional models to three dimensions, analyzing the coupling of surface and transverse instabilities with variable viscosity effects.

Findings

Flow pattern depends on inclination angle and viscosity gradient.

Transverse fingering instabilities are influenced by the additional dimension.

Viscosity variation significantly affects instability development.

Abstract

We study contact line induced instabilities for a thin film of fluid under destabilizing gravitational force in three dimensional setting. In the previous work (Phys. Fluids, {\bf 22}, 052105 (2010)), we considered two dimensional flow, finding formation of surface waves whose properties within the implemented long wave model depend on a single parameter, $D=(3Ca)^{1/3}\cot\alpha$, where $Ca$ is the capillary number and $\alpha$ is the inclination angle. In the present work we consider fully 3D setting and discuss the influence of the additional dimension on stability properties of the flow. In particular, we concentrate on the coupling between the surface instability and the transverse (fingering) instabilities of the film front. We furthermore consider these instabilities in the setting where fluid viscosity varies in the transverse direction. It is found that the flow pattern strongly depends on the inclination angle and the viscosity gradient.