Theoretical study on Rayleigh-Plateau instability on a solid plate and satellite droplet formation

Abstract

We theoretically studied how a liquid filament on a substrate breaks down. This instability problem is a branch of Rayleigh-Plateau instability(RP instability) and this problem has been broadly explored due to several applications particularly for surface patterning, e.g., metal nanoparticle patterning, multiple-droplet coating, and liquid metal wires for flexible devices. Previous studies are mostly done by numerical simulation and experiments, and few studies predict satellite droplet formation on a solid surface. Thus, to obtain the surface profile, we developed a theoretical model for the instability of a liquid filament on a substrate where the solid-liquid interaction is crucial. Based on the lubrication theory, linearization, perturbation theory, and additional no-slip boundary condition, finally, we obtained simplified governing equations, which are the only functions of the axial direction and time. From our theoretical model, Ohnesorge number and wavenumber are the important parameters on RP instability on a solid surface because they are the dimensionless variables determining the equations. By solving the governing equations, we found that the theoretical model can predict a breakup of a filament and a satellite droplet formation, which shows that the boundary condition on a stationary surface plays an important role to make satellite droplets. Remarkably, when changing the wavenumber of a problem, the satellite droplet radius changes and we suggest that it has exponential relation with wavenumber. Also, we evaluate the effect of Ohnesorge number

satellite droplet radius decreases when Ohnesorge number decreases.