Simulation of Parabolic Flow on an Eye-Shaped Domain with Moving Boundary

TL;DR

This paper develops a spectral collocation method to simulate thin tear film flow on a dynamically changing eye-shaped domain, effectively combining geometric mapping and numerical techniques for accurate modeling.

Contribution

It introduces a novel approach using conformal maps and spectral collocation to efficiently simulate nonlinear PDEs on a moving eye-shaped boundary.

Findings

High accuracy in solving linear and nonlinear diffusion equations

Effective handling of complex eye-shaped geometry

Potential for future application to tear film dynamics

Abstract

During the upstroke of a normal eye blink, the upper lid moves and paints a thin tear film over the exposed corneal and conjunctival surfaces. This thin tear film may be modeled by a nonlinear fourth-order PDE derived from lubrication theory. A challenge in the numerical simulation of this model is to include both the geometry of the eye and the movement of the eyelid. A pair of orthogonal and conformal maps transform a square into an approximate representation of the exposed ocular surface of a human eye. A spectral collocation method on the square produces relatively efficient solutions on the eye-shaped domain via these maps. The method is demonstrated on linear and nonlinear second-order diffusion equations and shown to have excellent accuracy as measured pointwise or by conservation checks. Future work will use the method for thin-film equations on the same type of domain.