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Tear film dynamics on an eye-shaped domain. Part 2. Flux boundary conditions

  • K. L. MAKI (a1), R. J. BRAUN (a1), P. UCCIFERRO (a1), W. D. HENSHAW (a2) and P. E. KING-SMITH (a3)...

Abstract

We model the dynamics of the human tear film during relaxation (after a blink) using lubrication theory and explore the effects of viscosity, surface tension, gravity and boundary conditions that specify the flux of tear fluid into or out of the domain. The governing nonlinear partial differential equation is solved on an overset grid by a method of lines using finite differences in space and an adaptive second-order backward difference formula solver in time. Our simulations in a two-dimensional domain are computed in the Overture computational framework. The flow around the boundary is sensitive to both our choice of flux boundary condition and the presence of gravity. The simulations recover features seen in one-dimensional simulations and capture some experimental observations of tear film dynamics around the lid margins. In some instances, the influx from the lacrimal gland splits with some fluid going along the upper lid towards the nasal canthus and some travelling around the temporal canthus and then along the lower lid. Tear supply can also push through some parts of the black line near the eyelid margins.

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Corresponding author

Email address for correspondence: braun@math.udel.edu

References

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Becker, J. & Grün, G. 2005 The thin-film equation: recent advances and some new perspectives. J. Phys. Condens. Matter 17 (9), S291S307.
Becker, J., Grün, G., Seemann, R., Mantz, H., Jacobs, K., Mecke, K. R. & Blossey, R. 2002 Complex dewetting scenarios captured by thin film models. Nature Materials 2, 5963.
Berger, R. E. & Corrsin, S. 1974 A surface tension gradient mechanism for driving the precorneal tear film after a blink. J. Biomech. 7, 225238.
Bertozzi, A. L., Brenner, M. P., Dupont, T. F. & Kadanoff, L. P. 1994 Singularities and similarities in interface flows. In Trends and Perspectives in Applied Mathematics (ed. Sirovich, L.), pp. 155208. Springer.
Braun, R. J. & Fitt, A. D. 2003 Modelling drainage of the precorneal tear film after a blink. Math. Med. Biol. 20, 128.
Braun, R. J. & King-Smith, P. E. 2007 Model problems for the tear film in a blink cycle: single equation models. J. Fluid Mech. 586, 465490.
Braun, R. J., Usha, R., McFadden, G. B., Driscoll, T. A., Cook, L. P. & King-Smith, P. E. 2009 Thin film dynamics on a prolate spheroid with application to the cornea. (submitted).
Brenan, K. E., Campbell, S. L. & Petzold, L. R. 1989 Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Elsevier.
Bron, A. J., Tiffany, J. M., Gouveia, S. M., Yokoi, N. & Voon, L. W. 2004 Functional aspects of the tear film lipid layer. Exp. Eye Res. 78, 347388.
Chesshire, G. & Henshaw, W. 1990 Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys. 90, 164.
Creech, J. L., Do, L. T., Fatt, I. & Radke, C. J. 1998 In vivo tear-film thickness determination and implications for tear-film stability. Curr. Eye Res. 17, 10581066.
Doane, M. G. 1981 Blinking and the mechanics of the lacrimal drainage system. Ophthalmology 88, 844851.
Fatt, I. & Weissman, B. A. 1992 Physiology of the Eye: An Introduction to the Vegetative Functions. Butterworth-Heinemann.
Gipson, I. K. 2004 Distribution of mucins at the ocular surface. Exp. Eye Res. 78, 379388.
Golding, T. R., Bruce, A. S. & Mainstone, J. C. 1997 Relationship between tear-meniscus parameters and tear-film breakup. Cornea 16, 649661.
Gorla, M. S. R. & Gorla, R. S. R. 2004 Rheological effects of tear film rupture. Intl J. Fluid Mech. Res. 31, 552562.
Greer, J. B., Bertozzi, A. L. & Sapiro, G. 2006 Fourth-order partial differential equations on general geometries. J. Comput. Phys. 216 (1), 216246.
Grün, G. & Rumpf, M. 2000 Nonnegativity preserving convergent schemes or the thin film equations. Numer. Math. 87, 113152.
Harrison, W. W., Begley, C. G., Lui, H., Chen, M., Garcia, M. & Smith, J. A. 2008 Menisci and fullness of the blink in dry eye. Optom. Vis. Sci. 85, 706714.
Henshaw, W. D. 2002 Ogen: the overture overlapping grid generator. Tech. Rep. UCRL-MA-132237. Lawrence Livermore National Laboratory.
Heryudono, A., Braun, R. J., Driscoll, T. A., Maki, K. L., Cook, L. P. & King-Smith, P. E. 2007 Single-equation models for the tear film in a blink cycle: realistic lid motion. Math. Med. Biol. 24 (4), 347377.
Johnson, M. E. & Murphy, P. J. 2004 Changes in the tear film and ocular surface from dry eye syndrome. Progr. Ret. Eye Res. 23, 449474.
Johnson, M. E. & Murphy, P. J. 2006 Temporal changes in the tear menisci following a blink. Exp. Eye Res. 83, 517525.
Jones, M. B., McElwain, D. L. S., Fulford, G. R., Collins, M. J. & Roberts, A. P. 2006 The effect of the lipid layer on tear film behaviour. Bull. Math. Biol. 68, 13551381.
Jones, M. B., Please, C. P., McElwain, D. S., Fulford, G. R., Roberts, A. P. & Collins, M. J. 2005 Dynamics of tear film deposition and drainage. Math. Med. Biol. 22, 265288.
Jossic, L., Lefevre, P., de Loubens, C., Magnin, A. & Corre, C. 2009 The fluid mechanics of shear-thinning tear substitutes. J. Non-Newtonian Fluid Mech. 61, 19.
King-Smith, P. E., Fink, B. A., Hill, R. M., Koelling, K. W. & Tiffany, J. M. 2004 The thickness of the tear film. Curr. Eye Res. 29, 357368.
King-Smith, P. E., Fink, B. A., Nichols, K. K., Hill, R. M. & Wilson, G. S. 2000 The thickness of the human precorneal tear film: evidence from reflection spectra. Invest. Ophthalmol. Vis. Sci. 40, 33483359.
Kondic, L. & Diez, J. 2001 Pattern formation in the flow of thin films down an incline: constant flux configuration. Phys. Fluids 13 (11), 31683184.
Lee, Y. C., Thompson, H. M. & Gaskell, P. H. 2007 An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localized topographic features. Comp. Fluids 36, 838855.
Lemp, M. A. 2007 The definition and classification of dry eye disease: report of the definition and classification subcommittee of the international dry eye workshop. Ocul. Surf. 5, 7592.
Maki, K. L. 2009 Computational solution of linear systems and models for the human tear film. PhD thesis, University of Delaware.
Maki, K. L., Braun, R. J., Driscoll, T. A. & King-Smith, P. E. 2008 An overset grid method for the study of reflex tearing. Math. Med. Biol. 25, 187214.
Maki, K. L., Braun, R. J., Henshaw, W. D. & King-Smith, P. E. 2010 Tear film dynamics on an eye-shaped domain. Part I. Pressure boundary conditions. Math. Med. Biol. (in press).
Mathers, W. D. & Daley, T. E. 1996 Tear flow and evaporation in patients with and without dry eye. Ophthalmology 103, 664669.
Maurice, D. M. 1973 The dynamics and drainage of tears. Intl Ophthalmol. Clin. 13, 103116.
Miljanović, B., Dana, R., Sullivan, D. A. & Schaumberg, D. A. 2007 Impact of dry eye syndrome on vision-related quality of life. Am. J. Ophthalmol. 143, 409415.
Miller, D. 1969 Measurement of the surface tension of tears. Arch. Ophthalmol. 82, 368371.
Mishima, S. 1965 Some physiological aspects of the precorneal tear film. Arch. Ophthalmol. 73, 233241.
Mishima, S., Gasset, A., Klyce, S. D. & Baum, J. L. 1966 Determination of tear volume and tear flow. Ophthalmol. Vis. Sci. 5, 264276.
Nagyová, B. & Tiffany, J. M. 1999 Components responsible for the surface tension of human tears. Curr. Eye Res. 19, 411.
Naire, S., Braun, R. J. & Snow, S. A. 2000 Limiting cases of gravitational drainage of a vertical free film for evaluating surfactants. SIAM J. Appl. Math. 61, 889913.
Oron, A. & Bankoff, S. G. 2001 Dynamics of a condensing liquid film under conjoining/disjoining pressures. Phys. Fluids 13 (5), 11071117.
Owens, H. & Phillips, J. 2001 Spread of the tears after a blink: velocity and stabilization time in healthy eyes. Cornea 20, 484487.
Piegl, L. A. & Tiller, W. 1997 The NURBS Book. Springer.
Read, S. A., Collins, M. J., Carney, L. G. & Franklin, R. J. 2006 The topography of the central and peripheral cornea. Invest. Ophthalmol. Vis. Sci. 47, 14041415.
Schein, O. D., Munoz, B., Tielsvh, J. M., Bandeen-Roche, K. & West, S. 1997 Prevalence of dry eye among the elderly. Am. J. Ophthalmol. 124, 723728.
Schwartz, L. W., Roy, R. V., Eley, R. E. & Petrash, S. 2001 Dewetting patterns in a drying liquid film. J. Coll. Interface Sci. 234, 363374.
Sharma, A., Tiwari, S., Khanna, R. & Tiffany, J. M. 1998 Hydrodynamics of meniscus-induced thinning of the tear film. In Lacrimal Gland, Tear Film, and Dry Eye Syndromes 2 (ed. Sullivan, D. A., Dartt, D. A. & Meneray, M. A.), pp. 425431. Plenum.
Tiffany, J. M. 1991 The viscosity of human tears. Intl Ophthalmol. 15, 371376.
Tiffany, J. M., Todd, B. S. & Baker, M. R. 1998 Computer-assisted calculation of exposed area of the human eye. In Lacrimal Gland, Tear Film, and Dry Eye Syndromes 2 (ed. Sullivan, D. A., Dartt, D. A. & Meneray, M. A.), pp. 433439. Plenum.
Wang, J., Fonn, D., Simpson, T. L. & Jones, L. 2003 Precorneal and pre- and postlens tear film thickness measured indirectly with optical coherence tomography. Invest. Ophthalmol. Vis. Sci. 44, 25242528.
Warner, M. R. E., Craster, R. V. & Matar, O. K. 2002 Dewetting of ultrathin surfactant-covered films. Phys. Fluids 14 (11), 40404054.
Witelski, T. P. & Bowen, M. 2003 ADI schemes for higher-order nonlinear diffusion equations. Appl. Numer. Math. 45 (2–3), 331351.
Wong, H., Fatt, I. & Radke, C. J. 1996 Deposition and thinning of the human tear film. J. Coll. Interface Sci. 184, 4451.
Zhornitskaya, L. & Bertozzi, A. L. 2000 Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2), 523555.
Zhu, H. & Chauhan, A. 2005 A mathematical model for tear drainage through the canaliculi. Curr. Eye Res. 30, 621630.
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Tear film dynamics on an eye-shaped domain. Part 2. Flux boundary conditions

  • K. L. MAKI (a1), R. J. BRAUN (a1), P. UCCIFERRO (a1), W. D. HENSHAW (a2) and P. E. KING-SMITH (a3)...

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