Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage

Inspired by the antiwetting property of pitcher plants, specialists have designed different functional material with slippery surfaces, and a directional slippery surface has been fabricated. This paper considers a gravity-driven liquid film coating the interior surface of a vertical tube, and different slippery lengths in the azimuthal direction and the axial direction are taken into account. The evolution equation of coating flow is derived using the thin film model, and time responses for two dimensional flow are calculated. Linear stability analysis (LSA) is given based on the traveling wave solutions, demonstrating that the axial slippery effect suppresses the flow instability and causes a larger traveling wave speed. Simultaneously, the azimuthal slippery effect plays a destabilizing role for perturbations with small wavenumbers and it is stabilizing for large wavenumbers. Direct simulations of the fingering flow patterns agree well with the linear stability analysis. Our results offer insight into the influence of wall slippage on the flow stability of liquid in petroleum engineering.