Numerical methods for simulating tear-film rupture in ophthalmic flows

Although there have been 2-D numerical simulations of this 3-D problem, the vast majority of existing literature considers the formulation and solution of idealised 1-D models, which are used to identify fundamental features associated with the breakage of thin films. Typically this breakage is a direct consequence of the imposition of a universally employed "pinning" boundary condition, in which the tear film is attached to the upper and lower eyelids at fixed points. Interestingly, there does not seem to be a convincing scientific explanation of the origin of this assumption, the nature of which is critical to the evolution of the tear-film flow.

The proposed project will be on three fundamental fronts. First, the governing coupled partial differential equations and boundary conditions for the evolution of both the tear film and the surfactant will be reviewed. Second, an investigation of the efficacy of existing numerical methods, e.g. finite differences and/or Chebyshev methods, will be conducted, and the limitations of these --- given the extreme gradients and curvatures associated with rupture --- will be sought. Third, brand new adaptive-mesh, high-resolution finite element methods will be developed for both 1-D and 2-D versions of the problem, with a view to finding novel solutions arising from non-idealised problem specifications, even in the presence of the expected extreme free-surface derivatives.