Stability and transient dynamics of thin liquid films flowing over locally heated surfaces

The dynamics and linear stability of a liquid film flowing over a locally heated surface are studied using a long-wave lubrication analysis. The temperature gradient at the leading edge of the heater induces a gradient in surface tension that opposes the gravitationally driven flow and leads to the formation of a pronounced capillary ridge. The resulting free-surface shapes are computed, and their stability to spanwise perturbations is analyzed for a range of Marangoni numbers, substrate inclination angles, and temperature profiles. Instability is predicted above a critical Marangoni number for a finite band of wave numbers separated from zero, which is consistent with published results from experiment and direct numerical simulation. An energy analysis is used to gain insight into the effect of inclination angle on the instability. Because the spatial nonuniformity of the base state gives rise to nonnormal linearized operators that govern the evolution of perturbations, a nonmodal, transient analysis is used to determine the maximum amplification of small perturbations to the film. The structure of optimal perturbations of different wave numbers is computed to elucidate the regions of the film that are most sensitive to perturbations, which provides insight into ways to stabilize the flow. The results of this analysis are contrasted to those for noninertial coating flows over substrates with topographical features, which exhibit similar capillary ridges but are strongly stable to perturbations.