Modelling and analysis of ophthalmic fluid dynamics

Mathematical models and numerical methods are developed for analysing and simulating the spatio-temporal evolution of the tear film coating the anterior surface of the human eye during an interblink period. The novelty of the work is on two distinct fronts.
• First, a systematic approach is taken to ensure that the (coupled) model evolution equations — one each for film thickness and lipid-surfactant concentration — arising from asymptotic thin-film approximations of the Navier-Stokes equations, are uniformly valid when realistic ophthalmic data are used in the parameterisation. In this way, the present model does not — as occurs in related literature —
yield results that are in conflict with a priori approximation hypotheses. More specifically, novel results are obtained on the effects of substrate curvature by proposing a specific coordinate system in which: the influence of curvilinearity on
the evolution of the tear film can be parameterised, and; the limiting case recovers the Cartesian models of related literature. Additionally, the evolution equations are developed using sophisticated bespoke computer-algebra (MAPLE) techniques that permit the correct a priori scalings — of the competing effects of gravity, inertia, evaporation and surface tension — that guarantee the above-mentioned uniform validity. A novel consideration of the physical viability of boundary conditions at three-phase contact line on the eyelid in the existing mathematical literature leads to the proposal, implementation and investigation of novel Neumann boundary conditions that are supported by the results of recent in vitro experimental work.
• Second, bespoke spectral numerical methods are developed for solving the thinfilm approximations, yielding hitherto-unseen explicit formulæfor high-order Chebyshev differentiation matrices. Inherent errors are quantified, thereby yielding
an explicit understanding of both the modelling limitations and the plausibility of results. A suite of post-processing tools is developed to negotiate the complexities of implementing the novel boundary conditions in a spectral environment. All
numerical techniques are validated on test problems; a high degree of both accuracy and efficiency is demonstrated. An analysis is presented of the errors incurred in the numerical approximation of the (steep) film-profile gradients near the eyelids; the results of this error analysis prompt questions on the accuracy of many of the results of previously published models. Through the combination of new, uniformly valid, thin-film approximations and bespoke, fully validated numerical methods, the coupled evolution equations for the thin-film
thickness and lipid surfactant concentration are solved with confidence that the results obtained are credible. The novel boundary conditions lead to results that predict behaviours of the tear film that, whilst unseen in all prior related mathematical literature, encouragingly align with in vivo experimental observations in the ophthalmic literature.
As a result, a novel hypothesis is presented for the behaviour of the tear-film contact line, through which predictions are made regarding the development and treatment of dry-eye
pathologies. Suggestions for future work conclude the thesis.