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

Published online by Cambridge University Press:  18 March 2010

K. L. MAKI ,
R. J. BRAUN ,
P. UCCIFERRO ,
W. D. HENSHAW  and
P. E. KING-SMITH
K. L. MAKI
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA
R. J. BRAUN*
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA
P. UCCIFERRO
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA
W. D. HENSHAW
Affiliation:
Lawrence Livermore National Laboratory, Box 808, L-550, Livermore, CA 94551-0808, USA
P. E. KING-SMITH
Affiliation:
College of Optometry, The Ohio State University, Columbus, OH 43218, USA
*
†Email address for correspondence: [email protected]
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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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 647: Dedicated to Professor S. H. Davis on his 70th birthday , 25 March 2010 , pp. 361 - 390
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Becker, J. & Grün, G. 2005 The thin-film equation: recent advances and some new perspectives. J. Phys. Condens. Matter 17 (9), S291–S307.Google Scholar
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, 59–63.Google Scholar
Berger, R. E. & Corrsin, S. 1974 A surface tension gradient mechanism for driving the precorneal tear film after a blink. J. Biomech. 7, 225–238.Google Scholar
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. 155–208. Springer.CrossRefGoogle Scholar
Braun, R. J. & Fitt, A. D. 2003 Modelling drainage of the precorneal tear film after a blink. Math. Med. Biol. 20, 1–28.Google Scholar
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, 465–490.Google Scholar
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).Google Scholar
Brenan, K. E., Campbell, S. L. & Petzold, L. R. 1989 Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Elsevier.Google Scholar
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, 347–388.CrossRefGoogle ScholarPubMed
Chesshire, G. & Henshaw, W. 1990 Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys. 90, 1–64.Google Scholar
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, 1058–1066.Google Scholar
Doane, M. G. 1981 Blinking and the mechanics of the lacrimal drainage system. Ophthalmology 88, 844–851.Google Scholar
Fatt, I. & Weissman, B. A. 1992 Physiology of the Eye: An Introduction to the Vegetative Functions. Butterworth-Heinemann.Google Scholar
Gipson, I. K. 2004 Distribution of mucins at the ocular surface. Exp. Eye Res. 78, 379–388.Google Scholar
Golding, T. R., Bruce, A. S. & Mainstone, J. C. 1997 Relationship between tear-meniscus parameters and tear-film breakup. Cornea 16, 649–661.CrossRefGoogle ScholarPubMed
Gorla, M. S. R. & Gorla, R. S. R. 2004 Rheological effects of tear film rupture. Intl J. Fluid Mech. Res. 31, 552–562.CrossRefGoogle Scholar
Greer, J. B., Bertozzi, A. L. & Sapiro, G. 2006 Fourth-order partial differential equations on general geometries. J. Comput. Phys. 216 (1), 216–246.Google Scholar
Grün, G. & Rumpf, M. 2000 Nonnegativity preserving convergent schemes or the thin film equations. Numer. Math. 87, 113–152.Google Scholar
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, 706–714.Google Scholar
Henshaw, W. D. 2002 Ogen: the overture overlapping grid generator. Tech. Rep. UCRL-MA-132237. Lawrence Livermore National Laboratory.Google Scholar
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), 347–377.Google Scholar
Johnson, M. E. & Murphy, P. J. 2004 Changes in the tear film and ocular surface from dry eye syndrome. Progr. Ret. Eye Res. 23, 449–474.Google Scholar
Johnson, M. E. & Murphy, P. J. 2006 Temporal changes in the tear menisci following a blink. Exp. Eye Res. 83, 517–525.CrossRefGoogle ScholarPubMed
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, 1355–1381.CrossRefGoogle ScholarPubMed
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, 265–288.Google Scholar
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, 1–9.Google Scholar
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, 357–368.CrossRefGoogle ScholarPubMed
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, 3348–3359.Google Scholar
Kondic, L. & Diez, J. 2001 Pattern formation in the flow of thin films down an incline: constant flux configuration. Phys. Fluids 13 (11), 3168–3184.Google Scholar
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, 838–855.Google Scholar
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, 75–92.Google Scholar
Maki, K. L. 2009 Computational solution of linear systems and models for the human tear film. PhD thesis, University of Delaware.Google Scholar
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, 187–214.Google Scholar
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).CrossRefGoogle Scholar
Mathers, W. D. & Daley, T. E. 1996 Tear flow and evaporation in patients with and without dry eye. Ophthalmology 103, 664–669.CrossRefGoogle ScholarPubMed
Maurice, D. M. 1973 The dynamics and drainage of tears. Intl Ophthalmol. Clin. 13, 103–116.Google Scholar
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, 409–415.CrossRefGoogle ScholarPubMed
Miller, D. 1969 Measurement of the surface tension of tears. Arch. Ophthalmol. 82, 368–371.Google Scholar
Mishima, S. 1965 Some physiological aspects of the precorneal tear film. Arch. Ophthalmol. 73, 233–241.Google Scholar
Mishima, S., Gasset, A., Klyce, S. D. & Baum, J. L. 1966 Determination of tear volume and tear flow. Ophthalmol. Vis. Sci. 5, 264–276.Google Scholar
Nagyová, B. & Tiffany, J. M. 1999 Components responsible for the surface tension of human tears. Curr. Eye Res. 19, 4–11.Google Scholar
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, 889–913.Google Scholar
Oron, A. & Bankoff, S. G. 2001 Dynamics of a condensing liquid film under conjoining/disjoining pressures. Phys. Fluids 13 (5), 1107–1117.Google Scholar
Owens, H. & Phillips, J. 2001 Spread of the tears after a blink: velocity and stabilization time in healthy eyes. Cornea 20, 484–487.Google Scholar
Piegl, L. A. & Tiller, W. 1997 The NURBS Book. Springer.CrossRefGoogle Scholar
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, 1404–1415.Google Scholar
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, 723–728.Google Scholar
Schwartz, L. W., Roy, R. V., Eley, R. E. & Petrash, S. 2001 Dewetting patterns in a drying liquid film. J. Coll. Interface Sci. 234, 363–374.CrossRefGoogle Scholar
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. 425–431. Plenum.Google Scholar
Tiffany, J. M. 1991 The viscosity of human tears. Intl Ophthalmol. 15, 371–376.Google Scholar
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. 433–439. Plenum.CrossRefGoogle Scholar
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, 2524–2528.Google Scholar
Warner, M. R. E., Craster, R. V. & Matar, O. K. 2002 Dewetting of ultrathin surfactant-covered films. Phys. Fluids 14 (11), 4040–4054.Google Scholar
Witelski, T. P. & Bowen, M. 2003 ADI schemes for higher-order nonlinear diffusion equations. Appl. Numer. Math. 45 (2–3), 331–351.CrossRefGoogle Scholar
Wong, H., Fatt, I. & Radke, C. J. 1996 Deposition and thinning of the human tear film. J. Coll. Interface Sci. 184, 44–51.CrossRefGoogle ScholarPubMed
Zhornitskaya, L. & Bertozzi, A. L. 2000 Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2), 523–555.Google Scholar
Zhu, H. & Chauhan, A. 2005 A mathematical model for tear drainage through the canaliculi. Curr. Eye Res. 30, 621–630.Google Scholar