Non-modal linear stability analysis of thin film spreading by Marangoni stresses

The spontaneous spreading and stability characteristics of a thin Newtonian liquid film partially coated by an insoluble surfactant monolayer are investigated in this thesis. Thin films sheared by Marangoni stresses ire characterized by film thinning in the upstream region near the terminating edge of the initial monolayer and an advancing ridge further downstream. For sufficiently thin films, experiments have shown there develops dendritic fingering patterns upstream of the ridge.; To probe the mechanisms responsible for unstable flow, a non-modal linear stability analysis is required because the base-states describing these flows are space and time-dependent. A new measure of disturbance amplification is introduced, based on the relative kinetic energy of the perturbations to the base-states, to analyze surfactant monolayers spreading either from a finite or infinite source. These studies reveal that disturbance amplification is most significant in highly curved regions of the film characterized by a large: change in the shear stress, which can develop at the advancing ridge and at the edge of the initial monolayer. For spreading from both a finite and infinite source, disturbances that convect through the ridge undergo transient amplification but eventually decay to restore film stability. By contrast, disturbances that localize to the thinned region undergo sustained amplification when surfactant is continuously supplied to the liquid film thereby promoting film instability.; By focusing on these susceptible regions, the relevant evolution equations are simplified to extract more information about the mechanism leading to instability. The length-scale controlling these “inner” regions represents the balance of viscous, capillary and Marangoni stresses. Simplification of these equations allows identification of steady travelling wave solutions whose linearized stability behavior shows that a flat film subject to a jump increase in shear stress is asymptotically unstable.; This thesis concludes by comparing recent experiments in our laboratory of a droplet of low surface tension liquid (oleic acid) spreading on a thin Newtonian film (glycerol) before the onset of instability with numerical simulations. Similar power law behavior for the ridge advance and qualitatively similar film profiles shapes occur when the simulations utilize a non-linear equation of state for the surfactant monolayer.