Partial differential equations of thin liquid films analysis and numerical simulation /

by Levy, Rachel.

LEVY, RACHEL. Partial Differential Equations of Thin Liquid Films: Analysis and Numerical Simulation. (Under the direction of Michael Shearer.) We consider four problems related to Marangoni-driven thin liquid films. The first compares two models for the motion of a contact line, the triple point at which solid, liquid and air meet, which is a major outstanding problem in the fluid mechanics of thin films. The precursor model replaces the contact line by a sharp transition between the bulk fluid and a thin layer of fluid, effectively pre-wetting the solid; the Navier slip model replaces the usual no-slip boundary condition by a singular slip condition that is effective only very near the contact line. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. This involves analyzing third order ODE that depend on several parameters. We find that the range of effective contact slopes in both models is confined to an interval bounded away from zero. Moreover, the precursor model breaks down in an unexpected way as the precursor height is decreased for a given upstream height: below a critical precur- sor height, there are no traveling waves from the upstream height to the precursor, even when both heights are equilibria of the ODE. This is explained with the aid of Poincaré sections of the phase diagram of the ODE. In the second problem, we apply a theory of scalar conservation laws that includes a kinetic relation and nucleation condition, to our model for the flow of thin liquid films. As in the previous problem, the context is the flow of a thin liquid film up an inclined planar solid substrate, the flow being driven by a surface tension gradient against the action of gravity. Our goal is to use theory from hyperbolic conservation laws to map the rich set of wave structures that arise as solutions to the Cauchy problem, including classical shocks, nonclassical shock waves (known as undercompressive shocks) and rarefactions. To create such a ’Riemann map’, we employ a kinetic relation that describes admissible nonclassical shock waves, and a nucleation condition that determines when a nonclassical solution is selected. The hyperbolic theory is able to capture features observed in thin film flow, such as multiple long-time solutions for the same initial upstream and downstream states. The third problem incorporates localized heating by an infrared (IR) laser to the model of a Marangoni-driven thin film from the previous problems. We analyze two types of steady state solutions, using a dynamical systems approach to explain homoclinic solutions and PDE simulations to explain heteroclinic solutions. The Riemann map developed in the previous problem predicts the effect of initial conditions on the development of wave structures. We discuss several methods for controlling the downstream height and the strength of forcing required to create homoclinic solutions from uniform or monotonic initial data. The fourth problem explores a model for a different physical scenario, in which a thin film is driven down a solid substrate by gravity and surfactant, surface-active-agent that lower surface tension. The problem is motivated by application of surfactants in medicine to improve lung function in premature infants by lowering surface tension in the alveoli. The model couples the thin film PDE for the height of the film with an equation for the transport of surfactant. Solutions of the parabolic-hyperbolic system include a complicated double wave solution, with discontinuities in the height and surfactant concentration gradient. We explain the solutions analytically using two types of discontinuous solutions and traveling waves. Agreement of the analytical solutions with data from numerical simulations indicates that we have successfully modeled longtime wave structures, including a linear increase in the maximum of the surfactant concentration.