Contact line instability of gravity driven thin films flowing down an inclined plane with wall slippage

Abstract Coating liquid films exist widely in nature and industrial processes, such as lava flows, falling-film evaporators, and microfabrication processes. In this paper, we aim to give a comprehensive study on contact line instability of thin film flows on a slippery substrate, which has different slippery lengths in the streamwise direction and the spanwise direction. It is reported here for thin film flows with contact lines, the streamwise slippery effect plays an opposite role compared with the spanwise slippery effect, different from the conclusion by Ding and Wong that the wall slippage always promoted the instability of the thin film flow [Ding and Wong, Int. J. Heat Mass Tran. 90, 2015]. The evolution equation of the thin film flow is derived by the lubrication theory, and traveling wave solutions are obtained, showing that with a larger streamwise slippery length, the traveling wave speed is promoted and a higher capillary ridge is induced. Linear stability analysis (LSA) and energy analysis are carried out to elucidate the mechanism of the flow instability. Results indicate that for thin film flows with dynamic contact lines, the slippery effect in the streamwise and spanwise direction plays different roles in the flow instability. The wall slippage in the streamwise direction suppresses the wave height of traveling waves and impedes the instability of the flow, while the wall slippage in the spanwise direction promotes the instability with a smaller cut off wavenumber. Besides, numerical simulations for thin film flows are performed, and the flow phenomenon agrees well with the prediction by the LSA. Our findings offer insight into the influence of the wall slippage on the dynamics of thin films coating on biomaterials and compliant substrates.