Spatiotemporal stability of a thin film in the presence of thermal and solutal Marangoni stresses

Abstract

The stability analysis of a gravity-driven thin liquid film over a uniformly heated substrate is carried out, and the effect of an insoluble surfactant is studied. The heating induces a temperature variation at the liquid–gas interface that generates a gradient in surface tension and creates a thermocapillary flow at the liquid–gas interface. When the film is perturbed the thermocapillary stress leads to the growth of the perturbation. The governing equations for the evolution of the film thickness and surfactant concentration are simplified within the lubrication approximation. Four non-dimensional groups appear in the model that affect the film dynamics and stability, namely, Marangoni numbers, M and Σ, the surface Peclet number, Pes, and Biot number, Bi. The critical conditions are explored in terms of these non-dimensional parameters for the film to become temporally unstable. Further, spatiotemporal stability analysis is performed to characterise the instability as absolute and convective instability. A critical Marangoni number is found beyond which the system is absolutely unstable independent of the inclination angle. This critical Marangoni number seems to increase with the increase in solutocapillary effect.