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. Braun
  • R. Usha
  • G. B. McFadden
  • T. A. Driscoll
  • L. P. Cook
  • P. E. King-Smith
Article
First Online:
Received:
Accepted:

Abstract

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.

Keywords

Cornea Curved substrate Lubrication theory Tear film Thin film flow 
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References

  1. 1.
    Holly FJ, Lemp MA (1977) Tear physiology and dry eyes. Rev Surv Ophthalmol 22: 69–87CrossRefGoogle Scholar
  2. 2.
    Mishima S (1965) Some physiological aspects of the precorneal tear film. Arch Ophthalmol 73: 233–241CrossRefGoogle Scholar
  3. 3.
    Ehlers N (1965) The precorneal film: biomicroscopical, histological and chemical investigations. Acta Ophthalmol Suppl 81: 3–135Google Scholar
  4. 4.
    Norn MS (1979) Semiquantitative interference study of fatty layer of precorneal film. Acta Ophthalmol 57: 766–774Google Scholar
  5. 5.
    Bron AJ, Tiffany JM, Gouveia SM, Yokoi N, Voon LW (2004) Functional aspects of the tear film lipid layer. Exp Eye Res 78: 347–360CrossRefGoogle Scholar
  6. 6.
    King-Smith PE, Fink BA, Nichols JJ, Nichols KK, Braun RJ, McFadden GB (2009) The contribution of lipid layer movement to tear film thinning and breakup. Investig Ophthalmol Vis Sci 50: 2747–2756CrossRefGoogle Scholar
  7. 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
  8. 8.
    Gipson IK (2004) Distribution of mucins at the ocular surface. Exp Eye Res 78: 379–388CrossRefGoogle Scholar
  9. 9.
    Govindarajan B, Gipson IK (2010) Membrane-tethered mucins have multiple functions on the ocular surface. Exp Eye Res 90: 655–663CrossRefGoogle Scholar
  10. 10.
    King-Smith PE, Fink BA, Hill RM, Koelling KW, Tiffany JM (2004) The thickness of the tear film. Curr Eye Res 29: 357–368CrossRefGoogle Scholar
  11. 11.
    King-Smith PE, Fink BA, Nichols JJ, Nichols KK, Hill RM (2006) The thickness of the human precorneal tear film: evidence from reflection spectra. J Opt Soc Am A 23: 2097–2104ADSCrossRefGoogle Scholar
  12. 12.
    Wang J, Fonn D, Simpson TL, Jones L (2003) Precorneal and pre- and postlens tear film thickness measured indirectly with optical coherence tomography. Investig Ophthalmol Vis Sci 44: 2524–2528CrossRefGoogle Scholar
  13. 13.
    Palakuru JR, Wang J, Aquavella JV (2007) Effect of blinking on tear dynamics. Investig Ophthalmol Vis Sci 48: 3032–3037CrossRefGoogle Scholar
  14. 14.
    Johnson ME, Murphy PJ (2006) Temporal changes in the tear menisci following a blink. Exp Eye Res 83: 517–525CrossRefGoogle Scholar
  15. 15.
    Harrison WW, Begley CG, Lui H, Chen M, Garcia M, Smith JA (2008) Menisci and fullness of the blink in dry eye. Optom Vis Sci 85: 706–714CrossRefGoogle Scholar
  16. 16.
    Tiffany JM (1991) The viscosity of human tears. Int Ophthalmol 15: 371–376CrossRefGoogle Scholar
  17. 17.
    Pandit JC, Nagyová B, Bron AJ, Tiffany JM (1999) Physical properties of stimulated and unstimulated tears. Exp Eye Res 68: 247–253CrossRefGoogle Scholar
  18. 18.
    Leiske DL, Raju SR, Ketelson HA, Millar TJ, Fuller GG (2010) The interfacial viscoelastic properties and structures of human and animal meibomian lipids. Exp Eye Res 90: 598–604CrossRefGoogle Scholar
  19. 19.
    McCulley JP, Shine W (1997) A compositional based model for the tear film lipid layer. Trans Am Ophthalmol Soc XCV: 79–93Google Scholar
  20. 20.
    Nagyová B, Tiffany JM (1999) Components of tears responsible for surface tension. Curr Eye Res 19: 4–11CrossRefGoogle Scholar
  21. 21.
    Berger RE, Corrsin S (1974) A surface tension gradient mechanism for driving the pre-corneal tear film after a blink. J Biomech 7: 225–238CrossRefGoogle Scholar
  22. 22.
    Owens H, Phillips J (2001) Spread of the tears after a blink: velocity and stabilization time in healthy eyes. Cornea 20: 484–487CrossRefGoogle Scholar
  23. 23.
    Mudgil P, Torres M, Millar TJ (2006) Adsorption of lysozyme to phospholipid and meibomian lipid monolayer films. Colloids Surf B 48: 128–137CrossRefGoogle Scholar
  24. 24.
    Mudgil P, Millar TJ (2008) Adsorption of apo- and holo-tear lipocalin to a bovine meibomian lipid film. Exp Eye Res 86: 622–628CrossRefGoogle Scholar
  25. 25.
    Zhang L, Matar OK, Craster RV (2003) Analysis of tear film rupture: effect of non-Newtonian rheology. J Colloids Interface Sci 262: 130–148CrossRefGoogle Scholar
  26. 26.
    Zhang L, Matar OK, Craster RV (2004) Rupture analysis of the corneal mucus layer of the tear film. Mol Simul 30: 167–172MATHCrossRefGoogle Scholar
  27. 27.
    Gorla MSR, Gorla RSR (2004) Rheological effects of tear film rupture. Int J Fluid Mech Res 31: 552–562CrossRefGoogle Scholar
  28. 28.
    Jossic L, Lefevre P, de Loubens C, Magnin A, Corre C (2009) The fluid mechanics of shear-thinning tear substitutes. J Non-Newton Fluid Mech 161: 1–9MATHCrossRefGoogle Scholar
  29. 29.
    McDonald JE, Brubaker S (1971) Meniscus-induced thinning of tear films. Am J Ophthalmol 72: 139–146Google Scholar
  30. 30.
    Wong H, Fatt I, Radke CJ (1996) Deposition and thinning of the human tear film. J Colloid Interface Sci 184: 44–51CrossRefGoogle Scholar
  31. 31.
    Sharma A, Tiwari S, Khanna R, Tiffany JM (1998) Hydrodynamics of meniscus-induced thinning of the tear film. In: Sullivan DA, Dartt DA, Meneray MA (eds) Lacrimal gland, tear film, and dry eye syndromes, vol 2. Plenum, New York, pp 425–431CrossRefGoogle Scholar
  32. 32.
    Miller KL, Polse KA, Radke CJ (2002) Black line formation and the “perched” human tear film. Curr Eye Res 25: 155–162CrossRefGoogle Scholar
  33. 33.
    Jones MB, Please CP, McElwain DLS, Fulford GR, Roberts AP, Collins MJ (2005) Dynamics of tear film deposition and drainage. Math Med Biol 22: 265–288MATHCrossRefGoogle Scholar
  34. 34.
    Jones MB, McElwain DLS, Fulford GR, Collins MJ, Roberts AP (2006) The effect of the lipid layer on tear film behavior. Bull Math Biol 68: 1355–1381MathSciNetCrossRefGoogle Scholar
  35. 35.
    Aydemir E, Breward CJW, Witelski TP (2011) The effect of polar lipids on tear film dynamics. Bull Math Biol 73: 1171–1201MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Braun RJ, Fitt AD (2003) Modelling precorneal tear film drainage after a blink. Math Med Biol 20: 1–28MATHCrossRefGoogle Scholar
  37. 37.
    Winter KN, Anderson DM, Braun RJ (2010) A model for wetting and evaporation of a post-blink precorneal tear film. Math Med Biol 27: 211–225MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Gorla MSR, Gorla RSR (2000) Nonlinear theory of tear film rupture. J Biomech Eng 122: 498–503CrossRefGoogle Scholar
  39. 39.
    Braun RJ, King-Smith PE (2007) Model problems for the tear film in a blink cycle: single equation models. J Fluid Mech 586: 465–490MathSciNetADSMATHCrossRefGoogle Scholar
  40. 40.
    Heryudono A, Braun RJ, Driscoll TA, Cook LP, Maki KL, King-Smith PE (2007) Single-equation models for the tear film in a blink cycle: realistic lid motion. Math Med Biol 24: 347–377MATHCrossRefGoogle Scholar
  41. 41.
    Maki KL, Braun RJ, Driscoll TA, King-Smith PE (2008) An overset grid method for the study of reflex tearing. Math Med Biol 25: 187–214MATHCrossRefGoogle Scholar
  42. 42.
    Maki KL, Braun RJ, Ucciferro P, Henshaw WD, King-Smith PE (2010) Tear film dynamics on an eye-shaped domain II. Flux boundary conditions. J Fluid Mech 647: 361–390MathSciNetADSMATHCrossRefGoogle Scholar
  43. 43.
    Maki KL, Braun RJ, Henshaw WD, King-Smith PE (2010) Tear film dynamics on an eye-shaped domain I. Pressure boundary conditions. Math Med Biol 27: 227–254MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Berger RE (1973) Pre-corneal tear film mechanics and the contact lens. Dissertation, Johns Hopkins UniversityGoogle Scholar
  45. 45.
    Read SA, Collins MJ, Carney LG, Franklin RJ (2006) The topography of the central and peripheral cornea. Investig Ophthalmol Vis Sci 47: 1404–1415CrossRefGoogle Scholar
  46. 46.
    Carney LG, Mainstone JC, Henderson BA (1997) Corneal topography and myopia: a cross-sectional study. Investig Ophthalmol Vis Sci 38: 311–320Google Scholar
  47. 47.
    Harris WF (2006) Curvature of ellipsoids and other surfaces. Ophthalmic Physiol Opt 26: 497–501CrossRefGoogle Scholar
  48. 48.
    King-Smith PE, Nichols JJ, Nichols KK, Fink BA, Braun RJ (2008) Contributions of evaporation and other mechanisms to tear film thinning and breakup. Optom Vis Sci 85: 623–630CrossRefGoogle Scholar
  49. 49.
    Cross MM (1965) Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems. J Colloid Interface Sci 20: 417–437Google Scholar
  50. 50.
    Myers TG (2005) Application of non-Newtonian models to thin film flow. Phys Rev E 72: 066302MathSciNetADSCrossRefGoogle Scholar
  51. 51.
    Howell PD (2003) Surface-tension-driven flow on a moving curved surface. J Eng Math 45: 283–308MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Roy RV, Roberts AJ, Simpson ME (2002) A lubrication model of coating flows over a curved substrate in space. J Fluid Mech 454: 235–261MathSciNetADSMATHCrossRefGoogle Scholar
  53. 53.
    Naire S, Braun RJ, Snow SA (2000) Limiting cases of gravitational drainage of a vertical free film for evaluating surfactants. SIAM J Appl Math 61: 889–913MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Golding TR, Bruce AS, Mainstone JC (1997) Relationship between tear-meniscus parameters and tear-film breakup. Cornea 16: 649–661CrossRefGoogle Scholar
  55. 55.
    Tomlinson A, Doane MG, McFadyen A (2009) Inputs and outputs of the lacrimal system: review of production and evaporative loss. Ocul Surf 7: 17–29CrossRefGoogle Scholar
  56. 56.
    Kimball SH, King-Smith PE, Nichols JJ (2010) Evidence for the major contribution of evaporation to tear film thinning between blinks. Investig Ophthalmol Vis Sci 51: 6294–6297CrossRefGoogle Scholar
  57. 57.
    Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric fluids. Vol I–fluid mechanics. Wiley, New YorkGoogle Scholar
  58. 58.
    Macosko CW (1994) Rheology: principles, measurements and applications. Wiley, New YorkGoogle Scholar
  59. 59.
    Perazzo CA, Gratton J (2003) Thin film of non-Newtonian fluid on an incline. Phys Rev E 67: 016307ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • R. J. Braun
    • 1
  • 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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