Effect of a Spring-Bocked Membrane on a Falling Film over a Slippery Tilted Plane

This research work investigates the influence of a spring-blocked membrane on a perturbed gravity-driven viscous fluid flow down a slippery inclined plane. The spring-bocked membrane on a falling film flow over a slippery plane can affect the flow dynamics and stability of the film. The instability of such a flow can be controlled either by modifying the behavior of the lower wall, altering the surface waves at the free surface using structures, or both, as done here by incorporating a spring-backed membrane at the top of the liquid layer and a slippery substrate. A linear stability analysis is performed utilizing the normal mode approach, with the free surface modified by a spring-blocked membrane and the lower boundary modeled as a slippery substrate. The associated Orr-Sommerfeld system is solved numerically using the spectral collocation method. The results reveal that velocity slip at the lower wall has a non-trivial impact on flow stability: it destabilizes at the onset of instability, then stabilizes at higher Reynolds numbers. Membrane tension is modeled as a stress jump at the free surface, and the mass of the membrane is also taken into account. The findings demonstrate that increasing the dimensionless spring-blocked membrane tension  reduces the growth rate of the most unstable mode, thereby enhancing flow stability. Thus, combination of a spring-blocked membrane at the free surface and a slippery base exerts a significant passive control on flow stability. The study provides an insight into how such configurations can be utilized to either suppress or amplify interfacial instabilities in gravity-driven flows.