Thin Film Evolution over a Thin Porous Layer: Modeling a Tear Film on a Contact Lens

Abstract

Abstract.We examine a mathematical model describing the behavior of the precontact lens tear film of a human eye. Our work examines the effect of contact lens thickness and lens permeability on the film dynamics. Also investigated are gravitational effects and the effects of different slip models at the fluid-lens interface. A mathematical model for the evolution of the tear film is derived using a lubrication approximation applied to the hydrodynamic equations of motion in the fluid film and the porous layer. The model is a nonlinear fourth-order partial differential equation subject to boundary conditions and an initial condition for post-blink film evolution. The evolution equation is solved numerically, and the effects of various parameters on the rupture of the thin film are studied. We find that increasing the lens thickness, permeability, and slip all contribute to an increase in the film thinning rate, although for parameter values typical for contact lens wear, these modifications are minor. Gravity plays a role similar to that for tear films in the absence of a contact lens. The presence of the contact lens does, however, fundamentally change the nature of the rupture dynamics as the inclusion of the porous lens leads to rupture in finite time rather than infinite time. [5], and Cher [6]) suggest a somewhat more complex system. In particular, the current view replaces the mucus and aqueous layers with a mucoaqueous layer in which mucins secreted from goblet cells are distributed throughout the bulk of the tear film and epithelial mucins form a complex barrier at the corneal surface. Measurements of the overall thickness of a human tear film range from a few microns to as many as 40 (see the review by Bron et al. [5]). The corneal surface itself has 100-nm-scale microbumps and rod-like mucins of length 200-500 nm that extend into the tear film [4]. These mucins have multiple functions ranging from cleanup and removal of debris from the tear film, stabilization of the tear film, and, with particular attention to the membrane-associated mucins, maintaining wettability at the corneal surface. The lipid layer, whose thickness has been estimated to range between 13-100 nm, serves to further stabilize the film and to slow evaporative mass loss (Bron et al. [5]).Dry eye syndrome is a common disorder of the human tear film that results from decreased tear production, excessive tear evaporation, and/or an abnormality in the

Gravity-driven flow over heated, porous, wavy surfaces

“…Trevelyan et al 53 consider both constant temperature and specified heat flux boundary conditions, and note that specifying the heat flux is more appropriate when modeling flows along a wall that loses heat to both the fluid and the ambient air. However, there are other applications where the constant temperature boundary condition may be more appropriate, such as in the problem considered by Nong et al 68 involving a tear layer flowing over a contact lens. In this case, the eye would be maintained at a constant temperature as a result of the body heating the eye.…”

The effect of surface tension on the gravity-driven thin film flow of Newtonian and power-law fluids

“…Among these are paints [1], contact lens manufacture [2] and microchip fabrication [3]. Gravity-driven thin film flow also occurs throughout nature, including a variety of gravity currents, such as lava flow and glacier flow [4–5].…”

Porous squeeze-film flow

“…where α (> 0) is the dimensionless Beavers-Joseph constant (see the original work by Beavers & Joseph (1967), the historical and critical note by Nield (2009), and the many papers using this condition, including, for example, Sherwood (1990), Wu (1972), Prakash & Vij (1974), Lin et al (2001), and Nong & Anderson (2010)). This condition allows for velocity slip on the interface between the fluid layer and the porous bed at z = 0 with slip length l s = k 1/2 /α.…”