The influence of wetting effects on the stability of spanwise-confined liquid films

Abstract: We investigate the influence of side-walls wetting effects on the linear stability of falling liquid films confined in the spanwise direction. Building upon our previous stability framework, which was developed to analyze the effect of spanwise confinement on the stability, we now incorporate wetting phenomena to develop a more comprehensive theoretical model. This extended model captures the interplay of gravity, inertia, surface tension, viscous dissipation, static meniscus, and moving contact lines. The base state exhibits two key features: a curved meniscus and a velocity overshoot near the side walls. A biglobal linear stability analysis is conducted based on the linearized Navier-Stokes equations. Unlike classical stability theory, our results reveal that surface tension, when strong, suppresses the long-wave instability ($k \rightarrow 0$), significantly increasing the critical Reynolds number as it increases. Notably, this effect is more pronounced for smaller contact angles. Moreover, stabilization is present across all wavenumbers at small Reynolds numbers, however, at large Reynolds numbers, the stabilization effect weakens, even for small contact angles. Furthermore, this stabilization is governed by the ratio of the capillary length to channel width, where complete stabilization occurs when this ratio exceeds a critical value dependent on the contact angle. We attribute this behavior to a capillary attenuation mechanism that dominates at smaller contact angles. No destabilization due to velocity overshoot was observed in the linear regime. Additionally, the introduction of wetting effects results in vortical structures in the vicinity of the side walls. These vortices dissipate the perturbation energy, thereby stabilizing the flow.