Journal of Fluid Mechanics

Papers

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

K. L. MAKIa1, R. J. BRAUNa1 c1, P. UCCIFERROa1, W. D. HENSHAWa2 and P. E. KING-SMITHa3

a1 Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA

a2 Lawrence Livermore National Laboratory, Box 808, L-550, Livermore, CA 94551-0808, USA

a3 College of Optometry, The Ohio State University, Columbus, OH 43218, USA

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.

(Received August 06 2009)

(Revised December 01 2009)

(Accepted December 03 2009)

Correspondence:

c1 Email address for correspondence: braun@math.udel.edu

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