Partial Differential Equations of Thin Liquid Films: Analysis and Numerical Simulation

Abstract

We consider four problems related to Marangoni-driven thin liquid films. The first compares two models for the motion of a contact line: the precursor model and the Navier slip model. 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. The range of effective contact slopes and parameter values are explained with the aid of Poincaré sections of the phase diagram of the third order ODE.

In the second problem, we use theory from hyperbolic conservation laws to map classical shocks, nonclassical shock waves (known as undercompressive shocks) and rarefactions that arise as solutions to the Cauchy problem. 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 captures 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. 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. 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 [em double wave solution], with discontinuities in the height and surfactant concentration gradient. Agreement of analytical solutions with data from numerical simulations indicates that we have successfully modeled long-time wave structures.