Linear stability analysis of a vertical liquid film over a moving substrate

TL;DR

This study analyzes the linear stability of liquid films over moving substrates, revealing how Kapitza number and surface tension influence stability, instability mechanisms, and the absolute/convective thresholds in industrial coating processes.

Contribution

It provides a detailed linear stability analysis considering different liquids and film thicknesses, highlighting the effects of Kapitza number and surface tension on stability and instability mechanisms.

Findings

Low Kapitza number liquids have smaller stability regions.

Surface tension can both stabilize and destabilize depending on wave length.

A window of absolute instability exists in the Re-h space, with specific thresholds for different liquids.

Abstract

The stability of liquid film flows are important in many industrial applications. In the dip-coating process, a liquid film is formed over a substrate extracted at a constant speed from a liquid bath. We studied the linear stability of this film considering different thicknesses $\hat{h}$ for four liquids, spanning a large range of Kapitza numbers ($\rm Ka$). By solving the Orr-Sommerfeld eigenvalue problem with the Chebyshev-Tau spectral method, we calculated the neutral curves, investigated the instability mechanism and computed the absolute/convective threshold. The instability mechanism was studied through the analysis of vorticity distribution and the kinetic energy balance of the perturbations. It was found that liquids with low $\rm Ka$ (e.g. corn oil, $\text{Ka}$ = 4) have a smaller area of stability than a liquid at high $\rm Ka$ (e.g. Liquid Zinc, $\rm Ka$ = 11525). Surface tension has both a stabilizing and a destabilizing effect, especially for large $\rm Ka$. For long waves, it curves the vorticity lines near the substrate, reducing the flow under the crests. For short waves, it fosters vorticity production at the interface and creates a region of intense vorticity near the substrate. In addition, we discovered that the surface tension contributes to both the production and dissipation of perturbation's energy depending on the $\rm Ka$ number. In terms of absolute/convective threshold, we found a window of absolute instability in the $\text{Re}-\hat{h}$ space, showing that the Landau-Levich-Derjaguin solution ($\hat{h}=0.945 \text{Re}^{1/9}\text{Ka}^{-1/6}$) is always convectively unstable. Moreover, we show that for $\text{Ka}<17$, the Derjaguin's solution ($\hat{h}=1$) is always convectively unstable.