Dynamics and stability of flow down a flexible incline

The flow of a thin liquid film down a flexible inclined wall is examined. Two configurations are studied: constant flux (CF) and constant volume (CV). The former configuration involves constant feeding of the film from an infinite reservoir of liquid. The latter involves the spreading of a drop of constant volume down the wall. Lubrication theory is used to derive a pair of coupled two-dimensional nonlinear evolution equations for the film thickness and wall deflection. The contact-line singularity is relieved by assuming that the underlying wall is pre-wetted with a precursor layer of uniform thickness. Solution of the one-dimensional evolution equations demonstrates the existence of travelling-wave solutions in the CF case and self-similar solutions in the CV case. The effect of varying the wall tension and damping coefficient on the structure of these solutions is elucidated. The linear stability of the flow to transverse perturbations is also examined in the CF case only. The results indicate that the flow, which is already unstable in the rigid-wall limit, is further destabilized as a result of the coupling between the fluid and underlying flexible wall.