Title: Simulation of Parabolic Flow on an Eye-Shaped Domain with Moving Boundary,
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.