Origin of filaments in finite-time in Newtonian and non-Newtonian thin-films*

The sticky fluids found in pitcher plant leaf vessels can leave fractal-like filaments behind when dewetting from a substrate. To understand the origin of these filaments, we investigate the dynamics of a retreating thin-film of aqueous polyethylene oxide (PEO) solutions which partially wet polydimethyl siloxane (PDMS) substrates. Under certain conditions the retreating film generates regularly-spaced liquid filaments. The early-stage thin-film dynamics of dewetting are investigated to identify a theoretical criterion for liquid filament formation. Starting with a linear stability analysis of a Newtonian or simple non-Newtonian (power-law) thin-film, a critical film thickness is identified which depends on the Hamaker constant for the fluid-substrate pair and the surface tension of the fluid. When the measured film thickness is smaller than this value, the film is unstable and forms filaments as a result of van der Waals forces dominating its behaviour. This critical film-height is compared with experimental measurements of film thickness obtained for receding films of Newtonian (glycerol-water mixtures) and non-Newtonian (PEO) solutions generated on substrates inclined at angles $0^{circ}$, $30^{circ}$, and $60^{circ}$ to the vertical. The observations of filament and its absence show good agreement with the theory. Further analysis of the former case, involving a stability analysis of the contact line, yields a prediction of the spacing (wavelength) $hat{lambda_{f}}$ between filaments as $hat{lambda} extsubscript{f}hat{eta}/hat{gamma} propto Ca$, where $hat{Ca}$ is the capillary number for contact line motion: our experiments yield $hat{lambda} extsubscript{f}hat{eta}/hat{gamma} propto Ca^{1.08}$ and earlier studies in the literature reported $hat{lambda} extsubscript{f}hat{eta}/hat{gamma} propto Ca^{0.945}$. The evolution of the thin-film shape is modelled numerically to show that the formation of filaments arises because the thin-film equation features a singular solution after a finite-time, hence termed a ``finite-time singularity''.