Modelling the evaporation of a tear film over a contact lens

Abstract

A contact lens (CL) separates the tear film into a pre-lens tear film (PrLTF), the fluid layer between the CL and the outside environment, and a post-lens tear film (PoLTF), the fluid layer between the CL and the cornea. We examine a model for evaporation of a PrLTF on a modern permeable CL allowing fluid transfer between the PrLTF and the PoLTF. Evaporation depletes the PrLTF, and continued evaporation causes depletion of the PoLTF via fluid loss through the CL. Governing equations include Navier-Stokes, heat and Darcy's equations for the fluid flow and heat transfer in the PrLTF and porous layer. The PoLTF is modelled by a fixed pressure condition on the posterior surface of the CL. The original model is simplified using lubrication theory for the PrLTF and CL applied to a sagittal plane through the eye. We obtain a partial differential equation (PDE) for the PrLTF thickness that is first-order in time and fourth-order in space. This model incorporates evaporation, conjoining pressure effects in the PrLTF, capillarity and heat transfer. For a planar film, we find that this PDE can be reduced to an ordinary differential equation (ODE) that can be solved analytically or numerically. This reduced model allows for interpretation of the various system parameters and captures most of the basic physics contained in the model. Comparisons of ODE and PDE models, including estimates for the loss of fluid through the lens due to evaporation, are given.

Droplet spreading and absorption on rough, permeable substrates

“…Although we apply a no-slip condition here, we note that a slip boundary condition for permeable substrates may be needed in some cases (Beavers & Joseph 1967;Spannuth et al 2009;Talbott et al 2015).…”

Unsupervised neural networks for Maxwell fluid flow and heat transfer over a curved surface with nonlinear convection and temperature‐dependent properties

“…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.…”

Mathematical modelling of contact lens moulding

“…There have been extensive experimental and theoretical studies on squeeze layers. Mathematical modelling has been applied to a tear film over a contact lens (Talbott et al, 2014), but not to the moulding of contact lenses. A general review of modelling squeeze layers is given by Engmann et al (2005), viscoelastic fluids are examined by Leider and Bird (1974) and Venerus et al (2000) and fluids with wall slip are analysed by Lawal and Kalyon (2000).…”