Thermal and solutal capillary effects in tear film dynamics

Abstract

An analytical and numerical study is carried out for the stability of a thin tear film considering a single-layered model. The mass, momentum, and energy equations are simplified under the lubrication approximation to obtain nonlinear partial differential spatiotemporal evolution equations for the film height and surfactant concentration. These evolution equations involve various physical mechanisms such as thermo- and solutocapillary stresses, van der Waals forces, surface tension forces, and slip at the corneal surface. Linear stability analysis reveals that solutocapillary stresses enhance the stability of the tear film by driving fluid from thicker to thinner regions. The thermocapillary stresses are found to enhance the instability, where the fluid is driven from a thinner (low surface tension) region to a thicker (high surface tension) region. Convective cooling due to cold wind flow also affects the growth rate of perturbations, with higher convection leading to a higher growth rate. The solutocapillary stresses dominate over the thermocapillary stresses beyond a certain critical value of the solutal Marangoni number. This critical threshold decreases with increasing Péclet number, indicating that the influence of solutocapillary effects becomes more pronounced under stronger advective transport. Numerical computations are carried out and show that the nonlinear stability results are in good agreement with those obtained from linear stability analysis. Furthermore, the computations reveal that the rupture time decreases with increasing thermal Marangoni number and slip coefficient, whereas it increases with the solutal Marangoni number.

Schematic representation of the tear film with lipid (surfactant) molecules, along with the temporal evolution of film thickness and surfactant distribution