The Effect of Corneal Topography and Mucins on Tear Film Rupture

Abstract

Tear film rupture on the corneal surface plays a critical role in ocular health and visual comfort. Conventional theoretical approaches often idealize the cornea as a perfectly smooth surface, ignoring the surface roughness that are characteristic of healthy as well as diseased eyes. In this study, we develop a comprehensive mathematical model to investigate tear film dynamics over the corneal surface incorporating the effects of surface roughness, slip, van der Waals forces, and lipid transport at the film-air interface. The corneal surface is represented by a small-amplitude periodic modulation. Steady-state solutions obtained using asymptotics reveal nonlinear corrections to the base profile at $O(η^2)$, which are confirmed numerically. Linear stability analysis performed using the Floquet theory demonstrates that an increase in the amplitude of roughness destabilizes the film. Specifically, both the dominant growth rate and the most unstable wavenumber increase with the roughness amplitude. Nonlinear simulations show that surface roughness significantly accelerates tear-film rupture. The slip coefficient, amplitude of roughness of the corneal surface and the initial film profile are found to significantly influence the rupture time. Moreover, the location of the rupture is sensitive to the initial disturbance. These results highlight the crucial role of surface topography and slip in determining tear film stability. The predicted rupture times are consistent with the experimental observations. The proposed model provides a realistic and accurate prediction of tear film dynamics and rupture over the corneal surface. This study offers a new perspective on tear film instability and will help address challenges such as contact lens failure which is related to tear film behavior.

Figures (12)

• Figure 1: $(a)$ Illustration of the tear film over a rough corneal epithelial surface $(b)$ schematic representation of the mathematical model describing the dynamics of the tear film.
• Figure 2: Steady-state film profiles. $(a)$ Steady states for different values of $\eta$ with $A_k=1,C=1$$(b)$ Steady states for different values of $C$ with $A_k=1,\eta=0.2$. Solid lines denote numerical solutions, while symbols indicate the corresponding asymptotic solutions.
• Figure 3: Variation of the perturbation growth rate $\sigma$ with wavenumber $q$. $(a)$ Dispersion curves for different values of the surface roughness amplitude $\eta$$(b)$ Comparison between the dispersion relation obtained using Floquet theory and the discretised eigenvalue method for $\eta =0.1$. Other parameters are $C=1, M=0.1,Pe_s=100,\beta=0.1,A_k=1$ and $\gamma_s=0.5$
• Figure 4: Variation of the maximum growth rate $\sigma_m^{Max}$ and the most unstable wavenumber with $(a)$ surface roughness amplitude $(\eta)$ and $(b)$ reduced capillary number $C$ for $\eta=0.2$. Other parameters are $C=1, M=0.1,Pe_s=100,\beta=0.1,A_k=1$ and $\gamma_s=0.5$.
• Figure 5: $(a)$ Effect of wavenumber of the roughness on the linear stability characteristics $(b)$ Validation of our in-house numerical scheme using the Fourier spectral method and the Chebyshev spectral method implemented in Mathematica. The figure shows the film profile at the time of rupture, demonstrating excellent agreement between the two methods and the result of Burelbach et alburelbach1988nonlinear