Model problems for the tear film in a blink cycle: single-equation models

We consider model problems for the tear film over multiple blink cycles in limits F that yield a single equation for the tear film; the single nonlinear partial differential equation that governs the film thickness arises from lubrication theory. The two models arise from considering the absence of naturally occurring surfactant and the case when the surfactant strongly affects the surface tension. The film is considered on a sinusoidally varying domain length with specified film thickness and volume flux at each end; only one end of the domain is moving, which is analogous to the upper eyelid moving with each blink. A main contribution of this article is computation of solutions for multiple complete blink cycles; the results of these non-trivial computations show a distinct similarity to quantitative in vivo observations of the tear film under partial blink conditions. A transition between periodic and non-periodic solutions has been estimated and this may be a criterion for what is effectively a full blink according to fluid dynamic considerations.