PDE analysis of thin liquid films: Steady states, long-time behavior, and blow-up

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This award will support the PI's studies of mathematical models for thin films of viscous liquids. A characteristic feature of these problems is that there is a competition between different physical effects, such as gravity and surface tension. Specifically, the partial differential equations that describe the thin film evolution equation have both linearly stabilizing terms (from surfacetension) that lead to smooth film profiles and terms (from gravity, for example) that destabilize long length scales. The models thus exhibit a finite band of linearly unstable modes, leading to interesting steady states or long-time dynamics and to new finite-time singularities. These models willbe studied with analytical, numerical, and asymptotic techniques.Pattern formation in fluid films is very familiar. For example, when a liquidis spilled on a table, what determines the size, distribution, and shape of the droplets that form? How does the composition of a paint affectthe irregularities that form when it dries? What properties of an inhalant will allow it to spread most effectively on lung tissue? In all these situations there is a competition between "smoothing" and "roughening" physical effects. These effects are often unavoidable andtypically lead to the formation of persistent or transient patterns. The award will support mathematical work on the formation and developmentof such patterns in thin liquid films. Understanding such phenomena is of mathematical and scientific interest, as well as of engineering, industrial, and medical use.