We study the flow of a thin fluid film on a prolate ellipsoid that is a good approximation to the shape of the human cornea. Two lubrication models for the dynamics of the film are studied in prolate spheroidal coordinates which are appropriate for this situation. One is a self-consistent leading-order hyperbolic PDE for relatively large substrate curvature; the other retains another order resulting in a fourth order parabolic PDE for the film dynamics. The former is studied for both Newtonian and Ellis (shear thinning) fluids; when applied to the tear film on the eye, the shear thinning is sufficiently small that it has no significant impact at those parameters. We explore a wide parameter range of shear thinning and find a significant effect on finite-time singularities present in the model. The other model is studied for Newtonian fluids only, and in geometries that are relevant for bounding the tear film on eyes. This second model allows for presence of a meniscus at one end of the domain, but we find that it does not have a strong effect on the thinning rate at the center of the cornea. We conclude that the ellipsoidal substrate does not have a significant effect on the thinning rate of the human tear film at the center of the cornea.
Citation: Journal of Engineering Mathematics
Pub Type: Journals
capillarity, cornea, fluid flow, lubrication theory, tear films