Thin Films with Non-conservative Effects

Abstract

Thin viscous fluids films (or “thin films”) spreading over a solid domain have appli- cations in many physical and biological systems. The dynamics of these thin films can be subject to fluid evaporation and vapor condensation with applications to pre- corneal tear film and thermal management. In this thesis, we study three models that arise from applications of thin fluid films with non-conservative effects. First, inspired by an evaporating thin film model studied by Ajaev in 2005, we investigate the pattern formation of a thin film equation with evaporation or condensation effects. Unlike the conservative thin film models where the steady states usually satisfy a second-order ODE, our model has a rich family of steady state solutions that satisfy a fourth-order ODE. Using bifurcation theory and stability analysis, we show that the coexistence of these different types of steady states yields interesting bifurcation structures and dynamics. Second, an interesting finite-time singularity phenomenon in this model motivates us to investigate various types of rupture solutions of a family of generalized thin film equations with non-conservative effects. Third, we consider a tear film model proposed by Peng in 2014 that characterizes the tear film rupture driven by locally-elevated evaporation effects. In addition to analytical results to a related generalized model, we show an interesting rupture-shock dynamics in the film thickness and osmolarity.