Global existence of solutions to a tear film model with locally elevated evaporation rates

TL;DR

This paper proves the global existence of solutions for a generalized tear film model with evaporation effects and explores its dynamics, including rupture phenomena and convergence behaviors, through analytical and numerical methods.

Contribution

It extends the mathematical analysis of tear film models by establishing global solutions under various conditions and validating findings with simulations.

Findings

Proved global existence of solutions for the model.

Identified finite-time rupture due to elevated evaporation.

Observed convergence to equilibrium and infinite thinning in simulations.

Abstract

Motivated by a model proposed by Peng et al. [Advances in Coll. and Interf. Sci. 206 (2014)] for break-up of tear films on human eyes, we study the dynamics of a generalized thin film model. The governing equations form a fourth-order coupled system of nonlinear parabolic PDE for the film thickness and salt concentration subject to non-conservative effects representing evaporation. We analytically prove the global existence of solutions to this model with mobility exponents in several different ranges and the results are then validated against PDE simulations. We also numerically capture other interesting dynamics of the model, including finite-time rupture-shock phenomenon due to the instabilities caused by locally elevated evaporation rates, convergence to equilibrium and infinite-time thinning.