Stability of gravity-driven viscous films flowing down a soft cylinder

We study the stability of gravity-driven viscous liquid films flowing down a vertical cylinder that is uniformly coated with a thin layer of elastic solids. Combining the gravity-driven viscous flows with the elastic deformation of the coated soft layer, we formulate a long-wave model to describe the evolution of a film flow-soft structure coupled system. Based on the model, we systematically examine the impact of the coating properties, including the elasticity and thickness on the temporal and spatiotemporal stability. Temporal stability analysis shows that the soft layer plays a dual role, namely, the elasticity acts as a destabilizing factor, leading to large deformations of both film interface and soft surface. However, due to the geometrical effect, increasing the layer thickness stabilizes the Rayleigh-Plateau instability. By contrast, the linear phase speed is always enhanced with increasing the elasticity or the thickness of the coated layer. We then analyze the spatiotemporal nature of free-surface instabilities and find that the elasticity can trigger the film flows from being absolutely unstable to convectively unstable. Transient numerical solutions of the full asymptotic model further verify the predictions from linear stability analysis, and more importantly, reveal the nonlinear effect of the softness. Compared to liquid films falling down the cylinder with rigid walls, the soft surface can enhance the coalescence of faster, larger sliding droplets with preceding slower, smaller sliding ones, thus resulting in a more unstable system. Our study highlights the potential of coating a thin layer of soft materials onto the walls of substrate to regulate the dynamics of liquid film systems, and may have implications for the emerging bioinspired applications; for instance, the large-scale collection and transport of water on flexible microfiber arrays.