Journal of Engineering Mathematics

, Volume 73, Issue 1, pp 121–138 | Cite as

Thin film dynamics on a prolate spheroid with application to the cornea

  • R. J. BraunEmail author
  • R. Usha
  • G. B. McFadden
  • T. A. Driscoll
  • L. P. Cook
  • P. E. King-Smith


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.


Cornea Curved substrate Lubrication theory Tear film Thin film flow 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • R. J. Braun
    • 1
    Email author
  • R. Usha
    • 2
  • G. B. McFadden
    • 3
  • T. A. Driscoll
    • 1
  • L. P. Cook
    • 1
  • P. E. King-Smith
    • 4
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  3. 3.Mathematical and Computational Sciences DivisionNational Institute of Standards and TechnologyGaithersburgUSA
  4. 4.College of OptometryThe Ohio State UniversityColumbusUSA

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