Dripping from a curved ceiling: a linear optimal transient growth analysis

We investigate theoretically the stability of a thin viscous film on the underside of a curved cylindrical surface. Gravity acts both as a stabilizing force originating in the progressive drainage of the film and as a destabilizing force prone to form dripping droplets. The drainage solution, derived from lubrication equations, is found asymptotically stable with respect to infinitesimal perturbations. This result first reported by Trinh et al. when studying the region near the top of a coated cylinder is here generalized to the entire structure. The governing parameters, namely the Bond number, which prescribes the relative importance of gravity and surface tension forces, and the initial film thickness to cylinder radius ratio are found not to play a role in the long time stability of the film. However, the system displays a linear transient growth potential which increases exponentially with the Bond number. Depending on its value, there is a critical initial disturbance amplitude above which non-linear effects yield the formation of droplets, suggesting that the transition to dripping is noise and roughness dependent.