Dynamics of a pre-lens tear film after a blink: Model, evolution, and rupture

Abstract

A mathematical model is developed to investigate the dynamics and rupture of a pre-lens tear film on a contact lens. The contact lens is modeled as a saturated porous medium of constant, finite thickness and is described by the Darcy-Brinkman equations with stress-jump condition at the interface. The model incorporates the influence of capillarity, gravitational drainage, contact lens properties such as the permeability, the porosity, and the thickness of the contact lens on the evolution and rupture of a pre-lens tear film, when the eyelid has opened after a blink. Two models are derived for the evolution of a pre-lens tear film thickness using lubrication theory and are solved numerically; the first uses shear-free surface condition and the second, the tangentially immobile free surface condition. The results reveal that life span of a pre-lens tear film is longer on a thinner contact lens for all values of permeability and porosity parameter considered. An increase in permeability of contact lens, porosity or stress-jump parameter increases the rate of thinning of the film and advances the rupture time. The viscous-viscous interaction between the porous contact lens and the pre-lens tear film increases the resistance offered by the frictional forces to the rate of thinning of pre-lens tear film. This slows down the thinning process and hence delays the rupture of the film as compared to that predicted by the models of Nong and Anderson [SIAM. J. Appl. Math. 70, 2771–2795 (2010)] derived in the framework of Darcy model.

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“…The governing equations obtained from the mathematical modeling that describe the physics are Navier-Stokes equations over a porous surface. 1,2 Also, when we deal with fluids like tears on the surface of the eye, we deal with non-Newtonian fluids.…”

“…To model and understand the behavior of ocular fluids, researchers in this field apply concepts from fluid mechanics. The governing equations obtained from the mathematical modeling that describe the physics are Navier–Stokes equations over a porous surface 1,2 . Also, when we deal with fluids like tears on the surface of the eye, we deal with non‐Newtonian fluids.…”