A cohesive framework is developed and presented for the mathematical modelling and computational simulation of the evolution of the human tear film: the thin layer of viscous fluid that coats the corneal surface of the eye between the eyelids. The evolution of the free surface of the tear film is governed by a nonlinear spatio-temporal evolution equation wherein gravitational, evaporative, capillary and Navier-slip effects are incorporated.
A thorough review of the boundary conditions enforced in related ophthalmic literature reveals that the ubiquitously used ``pinning'' (Dirichlet) condition at the eyelids contradicts not only physical intuition but also in vivo observations. Accordingly, the analysis and formulation herein departs from all prior ophthalmic modelling via the introduction of the novel-to-the-area Cox-Voinov condition, which allows for evolution of the tear film at the boundary in response to the evolving contact angle of the tear film. Since the contact-angle evolution can be independently constructed from in vivo data, a novel boundary-condition calibration is conducted herein.
Additionally, a novel approach to non-dimensionalisation and scaling is conducted that leads to a tear-film evolution equation in which all dominant balances are proven to be consistent when quantified by real fluid properties and ophthalmic parameters.
Since no numerical framework for solving the mathematically intractable ophthalmic problem is provided in related literature, a full numerical modus operandi is derived, implemented and validated herein. Specifically, a Chebyshev-differentiation-matrix method is used to approximate, to spectral accuracy, the spatial component of the evolution equations. In particular, a bespoke extension of a relatively recently introduced rectangular-collocation method is developed to facilitate enforcement of the nonlinear spatio-temporally-dependent Cox-Voinov condition. Both novel and existing accuracy-enhancement techniques are analysed and implemented on all spatial-discretisation tools to ensure that numerically approximated derivatives are computed with an error of the order of machine precision. Notably, the Chebyshev matrices constructed herein are evidenced to perform numerical differentiation with greater accuracy than Matlab's intrinsic routines.
Application of the bespoke numerical methods to the tear-film-evolution equation reveals novel tear-film dynamics for a range of physically meaningful initial conditions. Numerical simulations predict behaviour that agrees with not only related literature but also in vivo observations. Moreover, a comparison between the Cox-Voinov and pinning condition reveals that the latter, despite its ubiquitous enforcement in related literature, predicts dynamics that contradict in vivo observations. A novel analysis quantifying the effects of gravitational influence, corneal slip and contact-angle specification on tear-film rupture is given, and future extensions to the present work are discussed.
