Thin film dynamics on a prolate spheroid with application to the cornea
The tear film on the front of the eye is critical to proper eyesight; in many mathematical models of the tear film, the tear film is assumed to be on a flat substrate. We re-examine this assumption by studying the effect of a substrate which is representative of the human cornea. We study the flow of a thin fluid film on a prolate spheroid which 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 partial differential equation (PDE) valid for relatively large substrate curvature; the other retains the next higher-order terms resulting in a fourth-order parabolic PDE for the film dynamics. The former is studied for both Newtonian and Ellis (shear thinning) fluids; for typical tear film parameter values, the shear thinning is too small to be significant in this model. For larger shear thinning, we find a significant effect on finite-time singularities. The second model is studied for a Newtonian fluid and allows for a meniscus at one end of the domain. We do not find a strong effect on the thinning rate at the center of the cornea. We conclude that the corneal shape does not have a significant effect on the thinning rate of the tear film for typical conditions.
KeywordsCornea Curved substrate Lubrication theory Tear film Thin film flow
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- 3.Ehlers N (1965) The precorneal film: biomicroscopical, histological and chemical investigations. Acta Ophthalmol Suppl 81: 3–135Google Scholar
- 4.Norn MS (1979) Semiquantitative interference study of fatty layer of precorneal film. Acta Ophthalmol 57: 766–774Google Scholar
- 7.Chen H-B, Yamabayashi S, Ou B, Tanaka Y, Ohno S (1997) Structure and composition of rat precorneal tear film: a study by in vivo cryofixation. Investig Ophthalmol Vis Sci 38: 381–387Google Scholar
- 19.McCulley JP, Shine W (1997) A compositional based model for the tear film lipid layer. Trans Am Ophthalmol Soc XCV: 79–93Google Scholar
- 29.McDonald JE, Brubaker S (1971) Meniscus-induced thinning of tear films. Am J Ophthalmol 72: 139–146Google Scholar
- 44.Berger RE (1973) Pre-corneal tear film mechanics and the contact lens. Dissertation, Johns Hopkins UniversityGoogle Scholar
- 46.Carney LG, Mainstone JC, Henderson BA (1997) Corneal topography and myopia: a cross-sectional study. Investig Ophthalmol Vis Sci 38: 311–320Google Scholar
- 49.Cross MM (1965) Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems. J Colloid Interface Sci 20: 417–437Google Scholar
- 57.Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric fluids. Vol I–fluid mechanics. Wiley, New YorkGoogle Scholar
- 58.Macosko CW (1994) Rheology: principles, measurements and applications. Wiley, New YorkGoogle Scholar