An overset grid method for the study of reflex tearing

Mathematical Medicine and Biology

Volumn 25, Issue 3, 2008, Pages 187-214

  Maki, K L ^a   Braun, R J ^a   Driscoll, T A ^a   King Smith, P E ^b  

^a UNIVERSITY OF DELAWARE   (United States)

^b OHIO STATE UNIVERSITY   (United States)

Author keywords

Black line; Lubrication theory; Overset grid; Reflex tearing; Tear film

Indexed keywords

FLOW OF FLUIDS; LUBRICATION; THICKNESS MEASUREMENT;

BLACK LINE; FILM-THICKNESS; FOURTH-ORDER; FREE SURFACES; LUBRICATION THEORY; OVERSET GRID; OVERSET GRID METHODS; REFLEX TEARING; SURFACE EVOLUTION; TEAR FILMS;

FILM THICKNESS;

SURFACTANT;

ACCURACY; ARTICLE; BINOCULAR CONVERGENCE; CONTROLLED STUDY; CORNEA; EYELID; IN VIVO STUDY; LACRIMATION; LUBRICATION; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; MEASUREMENT; QUALITATIVE RESEARCH; REFLEX; SURFACE TENSION; TEAR FILM; TEAR FLOW; THEORY; THICKNESS;

EID: 51349112345     PISSN: 14778599     EISSN: 14778602     Source Type: Journal    
DOI: 10.1093/imammb/dqn013     Document Type: Article

References (41)

1. 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.
2. BERKE, A. & MUELLER, S. (1998) The kinetics of lid motion and its effects on the tear film. Lacrimal Gland, Tear Film, and Dry Eye Syndromes 2 (D. A. Sullivan, D. A. Dartt & M. A. Meneray eds). New York: Plenum, pp. 417-424.
3. BERRUT, J. & TREFETHEN, L. (2004) Barycentric Lagrange interpolation. SIAM Rev., 46, 501-517.
4. BERTOZZI, A. L., BRENNER, M. P., DUPONT, T. F. & KADANOFF, L. P. (1994) Singularities and similarities in interface flows. Trends and Perspectives in Applied Mathematics (L. Sirovich ed.). New York: Springer, pp. 155-208.
5. BITTON, E. & LOVASIK, J. V. (1998) Longitudinal analysis of precorneal tear film rupture patterns. Lacrimal Gland, Tear Film and Dry Eye Syndromes 2 (D. A. Sullivan, D. A. Dartt & M. A. Meneray eds). New York: Plenum Press, pp. 381-389.
6. BRAUN, R. J. (2006) Models for human tear film dynamics. Wave Dynamics and Thin Film Flow Systems (R. Usha, A. Sharma & B. S. Dandapat eds). Chennai, India: Narosa, pp. 404-434.
7. BRAUN, R. J. & FITT, A. D. (2003) Modelling drainage of the precorneal tear film after a blink. Math. Med. Biol., 20, 1-28.
8. 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.
9. CHESSHIRE, G. & HENSHAW, W. (1990) Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys. 90, 1-64.
10. 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.
11. DIEZ, J. A. & KONDIC, L. (2002) Computing three-dimensional thin film flows including contact lines. J. Comput. Phys., 183, 274-306.
12. DOANE, M. G. (1980) Interaction of eyelids and tears in corneal wetting and the dynamics of the normal human eyeblink. Am. J. Ophthalmol., 89, 507-516.
13. FARRIS, R. L., STUCHELL, R. N. & MANDEL, I. D. (1981) Basal and reflex human tear analysis. I. Physical measurements: Osmolarity, basal volumes, and reflex flow rate. Ophthalmology, 88, 852-857.
14. FAST, P. & SHELLEY, M. J. (2004) A moving overset grid method for interface dynamics applied to non-Newtonian Hele-Shaw flow. J. Comput. Phys., 195, 117-142.
15. HALEY, P. & MIKSIS, M. J. (1991) Effect of the contact line on droplet spreading. J. Fluid Mech., 223, 57-81.
16. HENSHAW, W. D. (2002) OGEN: The OVERTURE overlapping grid generator. Technical Report UCRL-MA132237. Livermore, CA: Lawrence Livermore National Laboratory.
17. HERYUDONO, A., BRAUN, R. J., DRISCOLL, T. A., MAKI, K. L., COOK, L. P. & KING-SMITH, P. E. (2008) Single-equation models for the tear film in a blink cycle: Realistic lid motion. Math. Med. Biol., 24, 347-377.
18. JENSEN, O. E. & GROTBERG, J. B. (1993) The spreading of heat or soluble surfactant along a thin liquid film. Phys. Fluids A, 5 58-68.
19. JOHNSON, M. E. & MURPHY, P. J. (2006) Temporal changes in the tear menisci following a blink. Exp. Eye Res., 83, 517-525.
20. 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.
21. 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.
22. 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.
23. LIU, H., BEGLEY, C. G., CHALMERS, R., WILSON, G., SRINIVAS, S. P. & WILKINSON, J. A. (2006) Temporal progression and spatial repeatability of tear breakup. Optom. Vis. Sci., 83,723-730.
24. MADSEN, N. K. & SINCOVEC, R. K. (1979) Algorithm 540: PDECOL, general collocation software for partial differential equations. ACM Trans. Math. Softw., 5, 326-351.
25. MAITCHOUK, D. Y., BEUERMAN, R. W., OHTA, T., STERN, M. & VARNELL, R. J. (2000)Tear production after unilateral removal of the main lacrimal gland in squirrel monkeys. Arch. Ophthalmol., 118, 246-252.
26. MATHERS, W. D. (1993) Ocular evaporation in meibomian gland dysfunction and dry eye. Ophthalmology, 100, 347-351.
27. MAURICE, D. M. (1973) The dynamics and drainage of tears. Int. Ophthalmol. Clin., 13, 103-116.
28. MCDONALD, J. E. & BRUBAKER, S. (1971)Meniscus-induced thinning of tear films. Am. J. Ophthalmol., 72, 139-146.
29. MILLER, K. L., POLSE, K. A. & RADKE, C. J. (2002) Black-line formation and the perched human tear film. Curr. Eye Res., 25 155-162.
30. MISHIMA, S., GASSET, A., KLYCE JR, S. D., & BAUM, J. L. (1966) Determination of tear volume and tear flow. Invest. Ophthalmol., 5, 264-276.
31. SHAMPINE, L., REICHELT, M., & KIERZENKA, J. (1999) Solving index-1 DAEs in MATLAB and Simulink. SIAM Rev., 41, 538-522.
32. SHARMA, A., TIWARI, S., KHANNA, R., & TIFFANY, J. M. (1998) Hydrodynamics of meniscus-induced thinning of the tear film. Lacrimal Gland, Tear Film, and Dry Eye Syndromes 2 (D. A. Sullivan, D. A. Dartt & M. A. Meneray eds). New York: Plenum Press, pp. 425-431.
33. STEGER, J. L. & BENEK, J. A. (1987) On the use of composite grid schemes in computational aerodynamics. Comput. Methods Appl. Mech. Eng., 64, 301-320.
34. TREFETHEN, L. N. (2000)Spectral Methods in MATLAB. Philadelphia, PA: SIAM.
35. VOLKOV, P. A. (1966) A finite difference method for finite and infinite regions with piecewise smooth boundaries. Dokl. Akad. Nauk USSR, 168, 744-747.
36. VOLKOV, P. A. (1968) The method of composite meshes for finite and infinite regions with piecewise smooth boundaries. Proc. Steklov Inst. Math., 96, 145-185.
37. WITELSKI, T. P. & BOWEN, M. (2003) ADI schemes for higher-order nonlinear diffusion equations. Appl. Numer. Math., 45, 331-351.
38. WONG, H., FATT, I. & RADKE, C. J. (1996)Deposition and thinning of the human tear film. J. Colloid Interface Sci., 184, 44-51.
39. YE, Y. & CHANG, H. (1999) A spectral theory for fingering on a prewetted plane. Phys. Fluids, 11, 2494-2515.
40. YEO, Y. L., CRASTER, R. V. & MATAR, O. K. (2003) Marangoni instability of a thin liquid film resting on a locally heated horizontal wall. Phys. Rev. E, 67, 056315.
41. ZHORNITSKAYA, L. & BERTOZZI, A. L. (2000) Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal., 37, 523-555.